/
Text
LARRY GONICK
Author of The Cartoon History of the Universe
& WOOLLCOTT SMITI
Also by Larry Gonick
The Cartoon History of the Universe
The Cartoon Guide to Physics (with Art Huffman)
The Cartoon Guide to the Computer
The Cartoon Guide to Genetics (with Mark Wheelis)
The Cartoon History of the United States
The Cartoon Guide to (Non) Communication
THE CARTOON GUIDE TO
< VM...
VIELI... It’S
TwEy’gE-.
I ^EM'
"'ТСООбЛ-Л
LARRY GONICK
& WOOLLCOTT SMITH
Ым
Mi
HarperPerennial
A Division of HarperColIinsP^W/sAen
the cartoon guide to statistics. Copyright ©1993 by Larry Gonick and Woollcott Smith.
All rights reserved. Printed in the United States of America. No part of this book may be
used or reproduced in any manner whatsoever without written permission except in the
case of brief quotations embodied in critical articles and reviews. For information address
HarperCollins Publishers, Inc., 10 East 53rd Street, New York, NY 10022.
HarperCollins books may be purchased for educational, business, or sales promotional
use. For information please write: Special Markets Department, HarperCollins Publishers,
Inc., 10 East 53rd Street, New York, NY 10022.
FIRST HARPERPERENNIAL EDITION
Illustrations by Larry Gonick
Library of Congress Cataloging-in-Publication Data
Gonick, Larry.
The cartoon guide to statistics / Larry Gonick & Woollcott Smith
—1st HarperPerennial ed.
p. cm.
Includes bibliographical references and index
ISBN 0-06-273102-5 (pbk.)
1. Statistics—Caricatures and cartoons. I. Smith, Woollcott, 1941- 11. Title.
QA276.12.G67 1993
519.5—dc20 92-54683
97 LG/RRD 20 19 18 17 16
♦CONTENTS*
CHAPTER 1---------------------------------------------------1
WHAT IS STATISTICS?
CHAPTER 2---------------------------------------------------7
PATA PESCRIPTION
CHAPTER 3---------------------------------------------------27
PROBABILITY
CHAPTER 4---------------------------------------------------43
RANPOM VARIABLES
CHAPTER 5---------------------------------------------------73
A TALE OF TWO PISTRIBUTIOMS
CHAPTER 6---------------------------------------------------09
SAMPLING
CHAPTER 7---------------------------------------------------111
CONFIPENCE INTERVALS
CHAPTER 0 --------------------------------------------------137
HyPOTHESlS TESTING
CHAPTER 9 --------------------------------------------------107
COMPARING TWO POPULATIONS
CHAPTER IP--------------------------------------------------101
EXPERIMENTAL PESIGN
CHAPTER 11--------------------------------------------------107
REGRESSION
CHAPTER П------------------------------------------------211
CONCLUSION
BIBLIOGRAPHY---------------------------------------------221
INPEX ----------------------------------------------------224
Acknowledgments
WE WOULP LIKE TO THANK CAROL COMEM AT HARPERCOLLINS
FOR SUGGESTING THIS PROJECT, OUR EPITOR ERICA SPABERG
FOR PATIENTLY EN PURI NG THE LAST-MINUTE PASH TO THE
PEAPLINE, ANP VICKY BIJUR, OUR LITERARY AGENT, FOR
INITIATING THE GONICK/SMlTH COLLABORATION BY INTROPUC-
ING THE COAUTHORS.
WILLIAM FAIRLEY’S ANP LEAH SMITH'S COMMENTS IMPROVEP
EARLIER PRAFTS OF THIS BOOK.
PONNA OKINO PROVIPEP INVALUABLE ASSISTANCE AMP APVICE
IN PROPUClNG THE CARTOON PAGES. SHE SAYS THAT CREATING
A CARTOON GUIPE IS HARPER THAN RUNNING A MARATHON,
ANP SHE SHOULP KNOWi SHE’S PONE BOTH.
THE ALTSYS CORPORATION CREATEP FONTOGRAPHER, THE
WONPERFUL SOFTWARE THAT ALLOWEP US TO SIMULATE
HANP-LETTEREP TEXT ANP FORMULAS ON THE MACINTOSH.
anp, since epucation is always a two-way street, a tip
OF THE HAT TO SMITH’S LONG-SUFFERING TEMPLE UNIVERSITY
STUPENTS ANP ESPECIALLY THE FALL ’92 STUPY GROUP
ORGANIZEP BY APRIANA TORRES. THE FUTURE IS THEIRS.
♦ Chapter 1 ♦
WHAT IS STATISTICS?
WE MUPPLG THROUGH LIFE MAKING (MO\C&>
BASEP ON INCOMPLETE INFORMATION...
SHOULP I PAVE THE SOUP?
EVERYTHING ELSE \5 50
EXPENSIVE, ANP I PONT
KNOW WHO’S PAYING- ARE
STATISTICIANS 5TIN6Y? I’VE
NEVER GONE OUT WITH
ONE BEFORE... THOUGH I
ONCE KNEW А VERY
GENEROUS ACCOUNTANT...
SHOULP I HAVE THE SOUP?
27 OUT OF THE 36 TIMES
I’VE HAP IT, IT WAS PRETTY
GOOP... BUT IS MONPAY THE
REGULAR CHEF’S NIGHT
OFF? ANP WHAT IF ALL THE
MR MOLECULES IN THE
ROOM SUPPENLY FLY UP TO
THE CEILING?
0
О
О
UlV1
О
1
WHAT MAKE5 STATISTICS UNIQUE IS ITS ABILITY TO QUANTIFY UNCERTAINTY,
TO MAKE IT PRECISE. THIS ALLOWS STATISTICIANS TO MAKE CATEGORICAL
5TATEMEMT5, WITH COMPLETE ASSURANCE—ABOUT THEIR LEVEL OF
UNCERTAINTY'
6OOP CHOICE.' I’M 95% л
CONFIDENT THAT TONIGHT'S
SOUP HAS PROBABILITY
BETWEEN 73% ANP 77% OF
BE IN G REALLY PELICIOU5! y
2
THIS IS NOT JUST A MATTER OR
ORPERIN6 SOUP! STATISTICS ALSO
INVOLVES MATTERS OF LIFE ANP
PEATH...
FOR EXAMPLE, IN 1906, THE SPACE SHUTTLE CUALLEN6ER EXPLOPEP, KILLING-
SEVEN ASTRONAUTS. THE PEClSION TO LAUNCH IN 29-PEG-REE WEATHER HAP
BEEN MAPE WITHOUT POIN6 A SIMPLE ANALYSIS OF PERFORMANCE PATA AT
LOW TEMPERATURE.
oh.ThAT
WT Of THE
Cufcvs III
A MORE POSITIVE EXAMPLE IS THE ШАГ POLIO VALINE- IN 1954, VACCINE
TRIALS WERE PERFORMEP ON SOME APO,POO CHILPREN, WITH STRICT CONTROLS
TO ELIMINATE BIASEP RESULTS. 6OOP STATISTICAL ANALYSIS OF THE RESULTS
FIRMLY ESTABLISHEP THE VACCINE'S EFFECTIVENESS, ANP ТОРАУ POLIO IS
ALMOST UNKNOWN.
3
ТО ACCOMPLISH THEIR FEATS OF MATHEMATICAL
LE6ERPEMAIN, STATISTICIANS RELY ON THREE
RELATEP PISCIPLINES;
Data
analysis
THE 6ATHERIN& PISPLAY, ANP
SUMMARY OF PATA-,
Probability
THE LAWS OF CHANCE, IN
ANP OUT OF THE CASINO;
Statistical
inference
THE SCIENCE OF PRAWIN&
STATISTICAL CONCLUSIONS
FROM SPECIFIC PATA, USINE A
KNOWLEP6E OF PROBABILITY.
IN THIS BOOK, WE’LL LOOK AT ALL THREE, AS APPLIEP TO A WIPE VARIETY OF
SITUATIONS WHERE STATISTICS PLAYS A CRUCIAL ROLE IN THE MOPERN WORLP.
4
IN CHAPTER 2, WE’LL LOOK AT A
SIMPLE FATA SET, THE REPORTEP
WEIGHTS OF A BUNCH OF COLLEGE
STUPENTS.
IN CHAPTER 3, WE STUPY THE LAWS OF
PROBABILITY IN THEIR BIRTHPLACE, THE
6-AMBLIN6 PEN.
CHAPTERS 4 ANP 4 SHOW HOW TO
PESCRIBE THE WORLP WITH
PROBABILITY MOPCL4, USIN6 THE
CONCEPT OF THE RAHPOM VARIABLE
CHAPTER b INTROPUCES ONE OF THE
STATISTICIAN’S ESSENTIAL PRO-
CEPURES, TAKING SAMPLE'S OF A
LAR6E POPULATION.
IN CHAPTER 7 ANP
BEYONP, WE PESCRIBE
HOW TO MAKE
STATISTICAL INFERENCES
IN SUCH COMMON REAL-
WORLP ARENAS AS
ELECTION POLL INC,
MNUFACTURIN6 QUALITY
CONTROL, MEPICAL
TEfTINC,
ENVIRONMENTAL
MONITORINC, RACIAL
BIA4, ANP THE LAW.
5
FlNALLy, IN PI6GU66ING-
6TATI6TIG6, IT’6 HARP TO
AVOIP MENTIONING- ONE
OTHER THING--- THE
WIPE6PREAP MISTRUST OF
6TATI6TIG6 IN THE WORLP
ТОРАУ. EVERyONE KNOW6
ABOUT "LyiN6 WITH
6TATI6TIG6," WHILE 6OOP
6TATI6TIGAL ANALy6l6 16
NEARLy IMPO66IBLE TO FINP
IN РАНУ LIFE. WHAT'6 ONE
TO PO?
OUR HUMBLE OPINION 16 THAT LEARNING A LITTLE MORE ABOUT THE
SUBJECT MI6HT NOT BE 6UGH A BAP IPEA- ANP THAT6 WHy WE WROTE THI6
BOOK/
IN WHAT FOLLOW6, WE TRy TO PRE6ENT THE ELEMENT6 OF 6TATI5TIG6 A6
G-RAPHlGALLy ANP iNTUlTIVELy A6 PO66IBLE. ALL yoU NEEP TO G-ET THROUGH
IT 16 A LITTLE PATI£MO£, THOUGHT, ANP A CERTAIN TOLERANGE FOR
AL6EBRA-OR. IF NOT THAT, THEN МАУВЕ A COURSE REQUIREMENT!!
♦ CHAPTER 2<
DATA DESCRIPTION
< UM...
vlBLl... tfs
TuEy’gE-
«
"Тсооб^.-)
7
РАТА ARE THE STATISTICIANS
RAW MATERIAL, THE NUMBERS WE
USE TO INTERPRET REALITY. ALL
STATISTICAL PROBLEMS INVOLVE
EITHER THE tOLL&TION,
PESCRIPTlON, ANP ANALYSIS OF
PATA, OR ТШЫК1Ы6 ABOUT THE
COLLECTION, PESCRIPTlON, ANP
ANALYSIS OF PATA.
THIS CHAPTER CONCENTRATES ON PATA Р&ОИРТЮН. HOW CAN WE REPRESENT
PATA IN USEFUL WAYS’ HOW CAN WE SEE UNPERLYIN6- PATTERNS IN A HEAP OF
NAKEP NUMBERS? HOW CAN WE SUMMARIZE THE PATA’S BASIC SHAPE?
WELL, TO PESCRIBE PATA, THE FIRST THINE- YOU NEEP IS SOME ACTUAL PATA
TO PESCRIBE... SO LET’S COLLECT SOME PATA.'
HERE & КЖ. REAL РАТА:
A* PART OF A £LA**ROOM
EXPERIMENT, 92 PENN STATE
*TUPENT* REPORTEP THEIR
WEI6-HT, WITH THESE
RESULTS:
FORWT THE S<Au6,
SMITH- JuST TAKE ЛАУ
WORD IT.. >4-
MALES
140 145 160 190 155 165 150 190 195
170 157 130 105 190 155 170 155 215
130 155 150 140 155 150 140 100 190
FEMALES
140 120 130 130 121 125 116 145 150 112 125 130 120 130 131 120 110 125 135 125
110 122 115 102 115 150 110 116 100 95 125 133 110 150 100
160 155 153 145 170 175 175 170 100 135
145 155 155 150 155 150 100 160 135 160
150 164 140 142 136 123 155
6ETTIN6 RI6HT POWN TO BUSINESS, WE PRAW A POT PLOT: ONE POT PER
STUDENT 6OES OVER EAZH STUDENT’S REPORTEP WEIGHT.
I 1 Г I--1--1 I r '"‘T I I I I I I
100 150 200
Weight in Pounds
YOU МАУ SEE A PROBLEM HERE:
THE SLUMPS AT 1*0 ANP 75F
POUNDS. THE STUDENTS TENDED
TO REPORT THEIR WEIGHT IN
FIVE-POUMP IMCREMEHT4. IN
REAL-LIFE SITUATIONS LIKE THIS
ONE, SU£H ROUNDIN6- OFF £AN
OBSCURE GENERAL PATTERN* IN
DATA... BUT FOR NOW, WE'LL JUST
WORK AROUNP IT.
9
' WE CAN SUMMARIZE THE PATA WITH A FREQUENCY TABLE. PIVIPE THE NUMBER
LINE INTO INTERVALS ANP COUNT THE NUMBER OF STUPENT WEISHTS WITHIN
EACH INTERVAL THE FREQUENCY IS THE COUNT IN AN У SIVEN INTERVAL THE
RELATIVE FREQUENCY IS THE PROPORTION OF WEISHTS IN EACH INTERVAL
I.E., IT’S THE FREQUENCY PIVIPEP BY THE TOTAL NUMBER OF STUPENTS.
CLASS INTERVAL MIPPOINT FREQUENCY RELATIVE FREQUENCY
07.5-102.4 95 2 .022
102.5-117.5 110 9 090
117.5-132.4 125 19 .20b
132.5-147.4 140 17 105
147.5-162.4 155 27 .293
162.5-177.4 170 0 .007
177.5-192.4 105 0 .007
192.5-207.5 200 1 .011
2075-222.4 215 1 .011
TOTAL 92 1.000
NOTE: WE KEPT THE INTERVAL BOUNPARIES AWAY FROM THOSE TROUBLESOME
5-POUNP MULTIPLES. THIS SETS AROUNP THE STUPENTS’ REPORTINS BIAS-
SUlPELlNES FOR FORMINS THE CLASS INTERVALS:
| USE INTERVALS OF
I EQUAL LENSTH WITH
MIPPOINTS AT
CONVENIENT ROUNP
NUMBERS.
О | FOR A SMALL PATA
* SET, USE A SMALL
NUMBER OF
INTERVALS.
FOR A LARSE PATA
9 SET, USE MORE
INTERVALS.'
IS 92
LARSE OR
SMALL?
I’M 90% SURE
IT’S SORT OF
LARSE...
10
IN TME FREQUEN6V TABLE, WE ARE *M0WIN6 MOW MANY PATA POINT* ARE
“AROUNP” EMM VALUE. WE 6AN 6-RAPM TMI* INFORMATION, TOO. TME RE5ULTIN6-
BAR 6-RAPM I* 6ALLEP A W4TO6RAM. EMM BAR 6OVER* AN INTERVAL ANP I*
6ENTEREP AT TME MlPPOINT. TME BAR’* MEI6-MT I* TME NUMBER OF PATA
POINT* IN TME INTERVAL
WE 6AN AL*O PRAW A RELATIVE FREQUENCY M4TO6RAM, PLOTTING TME
RELATIVE FREQUENT A6AIN*T TME WEI6-MT. IT LOOK* EXA6TLV TME *AME,
EXCEPT FOR TME VERTICAL *6ALE.
Weight in Pounds
11
THE STATISTICIAN JOW TUKEY
INVENTED A QUICK WAY TO
SUMMARIZE PATA ANP STILL KEEP
THE INPIVIPUAL PATA POINTS. ifS
CALLER THE 5TGM-AMP-LEAF
PIAGRAM.
FOR THE WEIGHT PATA, THE STEM IS A
COLUMN OF NUMBERS, CONSISTING OF
THE WEIGHT PATA COUNTER ВУ TENS
Cl-E., WE LEAVE OFF THE LAST PIGIT9.
NOW APR THE FINAL PIGIT OF EACH
WEIGHT IN THE APPROPRIATE ROW:
FILLER IN, IT LOOKS LIKE THIS:
9 : 5
IP : 200
11 : 620055P6P
12 : P1553PP5525
13 : 05PP05P6PP153
14 : P55P550P5P2
15 : 5P537P55P55P5P5P5PP5PP
16 : P5PPP4
17 P55PPP
10 •- P5PP
19 -- 00500
10:
21 : 5
ANP FINALLY, PUT THE ‘LEAVES’ IN
OR PER.
9 : 5
IP : 200
11 : PP2556600
12 : PPP12355555
13 . PPPPP13555600
14 : PPPP2555550
15 : PPPPPPPPPP355555555557 VP’j
16 : PPPP45 /y\
17 •• PPPP55 /. j | A
10 : 0005 6* ( ' | J
19 : PPPP5
2P: \ |j
21 : 5
ALL THOSE ZEROES ANP FIVES CLEARLY
SHOW THE STUPENTS' REPORTING BIAS!
12
CRUSAPIKG KURSE FLORENCE NIGHTINGALE
COMPILER MORTALITY 4TATI4TIG4 FROM
BRITISH MILITARY HOSPITALS, PRODUCING
SHOCKING HISTOGRAMS LIKE THIS ONE:
THE RAPlAL AXIS
INDICATES PEATHS—IN
HOSPITALS AS WELL AS
ON THE BATTLEFlELP-
Cr BRITISH SOLPIERS
IK THE CRIMEAN WAR.
HER STATISTICAL EFFORTS L£P
PIRECTLY TO IMPROVED HOSPITAL
ey^
Statistics.’
0
SUMMARY STATISTICS
NOW WE MOVE FROM PICTURES TO FORMULAS. OUR OBJECT IS TO &£T SOME
SIMPLE MEASUREMENTS OF THE ^RUPEST ^MARAGTERISTI^S OF A SET OF PATA....
ANV SET OF MEASUREMENTS
HAS TWO IMPORTANT
PROPERTIES-- TME CENTRAL
OR TYPICAL VALUE, ANP
TME 4PREAP ABOUT THAT
VALUE. VOU 6AN SEE TME
IPEA IN THESE
MyPOTM£TJ4AL HISTOGRAMS.
VJIDE raUTER
WE 6AN GO A LONG WAV WITH A LITTLE NOTATION. SUPPOSE WE’RE MAKING A
SERIES OF OBSERVATIONS... n OF THEM, TO BE ЕХАГЕ, THEN WE WRITE
AS THE VALUES WE OBSERVE. THUS, n IS
THE TOTAL NUMBER OF PATA POINTS, ANP
<SAy? IS THE VALUE OF THE FOURTH
PATA POINT.
AN ARRAY IS A TABLE OF PATA:
OBSERVATION 1 2 3 4 .... n
PATA VALUE л;, .... Xn.
14
A SMALL SET OF z? PATA POINTS MAKES THE BOOKKEEPING EASX
SUPPOSE, FOR EXAMPLE, WE ASK FIVE PEOPLE MOW MANX HOURS OF
TELEVISION ТМЕУ WAIZM IN A WEEK... ANP GET TME FOLLOWING ARRAV:
OBSERVATION 1 2 3 4 5
PATA VALUE 5 7 3 30 7
THEN ;Zt = 3, %г » 7, • 3, = 30, ANP -- 7-
VHWVb TME ZENTER” OF
THESE PATA? THERE ARE
A^TUALLV SEVERAL
PIFFERENT WAVS TO
MEASURE IT. WE’LL LOOK AT
JUST TWO OF THEM.
MEAN
COR “AVERAGE"?
TME_/<£4A/ OR AVERAGE VALUE IS REPRESENTEP
ВУ X.. IT’S OBTAINEP ВУ APPING ALL TME PATA ANP
PIVIPING ВУ TME NUMBER OF OBSERVATIONS:
- SUM OF PATA
* = ------n-----
%, +
П
for our example,
S + 7 + 3 + 30 + 7 bO
* - ----------" —
12 HOURS
15
FOR THE SUM X| +X2 + ... +%лг WE
WRITE
WE HAVE A 5 H ORTH AMP FOR THAT
X, + X2 + ... + %n WIN6 THE 6REEK
CAPITAL LETTER $/A*U, FOR SUMMATION:
t'<Z n-
И TEN TIME*
/ Amp you'L-L
& MEVER EOB6ET
П
AMP REAP IT A5
‘THE 5UM OF X/
A5 i GOES FROM
1 TO Л."
50... TO REPEAT, THE AVERAGE, OR ME AM, OF A SET OF PATA X/ 15
OR
i ~ /
П
IN THE £A5E OF OUR 92 PENN 5TATE 5TUPENT5, THE MEAN WEIGHT 15
9г
145.15 PoUNt*
1fe
AJ& Е I А. 1^1 ANOTHER KINP OF CENTER: THE
THE W К IMI W “MlPPOINT” OF THE PATA, LIKE THE
"MEPIAN STRIP" IN A ROAP.
TO ANP THE MEPIAN
VALUE OF A PATA SET,
WE ARRANGE THE PATA
IN ORPER FROM
SMALLEST TO LARGEST.
THE MEPIAN IS THE
VALUE IN THE MIPPLE.
9 7
THE MEPIAN
7 36
IF THE NUMBER OF POINTS IS ГИСА/—IN WHI^H £ASE THERE IS NO MIPPLE. WE
AVERAGE THE TWO VALUES AROUNP THE MIPPLE... SO IF THE PATA ARE
3
WE AVERAGE S
ANP 7 TO 6БТ
2
MIPPLE
SPA^E
THIS OIVES US A GENERAL RULE-- ORPER THE PATA FROM SMALLEST TO LARGEST.
IF THE NUMBER OF PATA
POINTS IS OPP, THE MEPIAN
IS THE MIPPLE PATA POINT.
IF THE NUMBER OF POINTS IS
PVPM THE MEPIAN IS THE
AVERAGE OF THE TWO PATA
POINTS NEAREST THE MIPPLE.
17
FOR TME 7? =92 STUPENT WEIGHTS,
WE OAKJ FINP TME MEPIAN FROM TME
ORPEREP STEM-ANP-LEAF PIA6RAM;
JUST COUNT TO TME 46™
OBSERVATION. TME MEPIAN IS
^46 + ^47 + 149
2'2
-14S POUNPS
9: 5
IP : 200
11 : PP2556600
12 : PPP12355555
13 : PPPPP13555600
14 : PPPP2555 99 0
15 : PPPPPPPPPP355555555557
16 = PPPP45
17 : PPPP55
10 ; PPP5
19 : PPPP5
20:
21 : 5
WHY MORE THAN ONE MEASURE OF TME CENTER? EACH MAS APVANTA6ES. FOR
EXAMPLE, TME MEPIAN IS NOT SENSITIVE TO OUTLIERS, OR EXTREME VALUES
NOT TYPICAL OF TME REST OF TME PATA. SUPPOSE IN OUR SMALL TV-
WATCHING GROUP, ONE PERSON WATCHES 200 HOURS PER WEEK. THEN OUR
PATA ARE 3, 5, 7, 7, 200. TME MEPIAN, 7, IS UNCMAN^EP, BUT TME MEAN IS
NOW % ~ 45-0!
UV
ueifrUT,
TOO’
MV VoO'RE
1/19TO(2W
W MEM
IN 1904 TME UNIVERSITY OF VIRGINIA ANNOUNCEP
TMAT ITS PEPARTMENT OF RHETORIC ANP COM-
MUNICATIONS 6-RAPUATES’ /<£4А/ STARTING SALARY
WAS #9,000. TME OUTLIER, TME SALARY OF N.B.A.
CENTER RALPH SAMPSON, PIP NOT REPRESENT TME
EARNING POWER OF A BA IN SPEECH FROM U. OF V.
СГМЕ MEPIAN SALARY WASN’T PUBLlSMEPJ
IB
MEASURES OF
BESlPES KNOWING THE
CENTRAL POINT OF A PATA
SET, WE’P ALSO LIKE TO
PESCRIBE THE PATA’S
SPREAD, OR MOW FAR
FROM THE CENTER THE
PATA TENP TO RANGE-
FOR INSTANCE, IF THE
STUPENTS ALL WEIGMEP
EXACTLY 145 POUNP5,
THERE WOULP BE NO
SPREAP AT ALL.
NUMERICALLY, THE SPREAP
WOULP BE ZERO, ANP THE
HISTOGRAM WOULP BE
SKINNY.
HI.' WE’RE
IPEKITICAL.'
BUT IF MANY OF THE STUPENTS WERE VERY LIGHT ANP/OR VERY HEAVY,
OBVIOUSLY WE’P SEE SOME SPREAP-SAY, IF THE FOOTBALL TEAM VIA* PART
OF THE SAMPLE...
THE HISTOGRAM WOULP BE WIPER, SOMETHING LIKE THIS-.
19
ГД6А1Ы, THERE’S MORE THAN ONE WAV TO MEASURE A SPREAP. ONE WAV IS
INTERQUARTILE RANGE
THE IPEA IS TO PIVIPE
THE PATA INTO FOUR
EQUAL GROUPS ANP SEE
HOW FAR APART THE
EXTREME GROUPS ARE.
HERE’S THE RECIPE:
f I PUT THE PATA IN NUMERICAL
" ORPER.
* PIVIPE THE PATA INTO TWO
EQUAL HI6H ANP LOW GROUPS
AT THE MEPIAN. (IF THE
MEPIAN IS A PATA POINT,
IN£LUPE IT IN BOTH THE HI6-H
ANP LOW GROUPS.?
FlNP THE MEPIAN OF THE
LOW 6ROUP. THIS IS aLLEP
THE FIRST QUARTILE, OR Qi-
41 THE MEPIAN OF THE НЮН
* ORO UP IS THE THIRP
QUARTILE, OR Q3.
Q or
• О ' MfPIAM OF
1 lows
NOW THE INTERQUARTILE RANOE (IQR? IS THE PISTAN^E (OR PIFFEREN^E)
BETWEEN THEM:
IQR * Q3 — Qj
20
HERE’S THE WEIGHT PATA
WITH THE MIPP0INT5 OF
THE HI6H ANP LOW GROUPS
EMPHA5IZEP:
9: 5
IO: 200
И : 00255ЫЛ9 *
12: 00012555995
13 : 0000015555600
И : 00002555550
15: 00000000005555555555 97
16 : ООООА5 “Ъ
17 : 000055 1
10 : 0005
19 •• 00005
20:
21: 5
JOHN TUKEY INVENTEP ANOTHER KlNP OF
PISPLAY TO SHOW OFF THE IQR, CALLEP A
FOX /WP WHltKERt PLOT. THE BOX’S
ENP5 ARE THE QUARTILES Qt ANP Q3. WE
PRAW THE MEPIAN INSIPE THE BOX.
”T-----1 “1 I " I 1 т ~~
1X0 rto I4C «4*9 (50 iw
IF A POINT 15 MORE THAN 1.5 IQR FROM
AN ENP OF THE BOX, IT’5 AN OUTLIER.
PRAW THE OUTLIERS INPIVIPUALLX
ANP WE SEE THAT
IQR = 156 - 125
= 31 POUNP6
—<----1 -------------------
КГ >55 • • • 200
A6-AIN, THIS 15 THE PIFFERENCE
BETWEEN THE MEPIAN HEAVY
5TVPENT ANP MEPIAN LI6HT ONE
FINALLY, EXTENP "WHISKERS" OUT TO THE
FARTHEST POINTS THAT ARE NOT OUTLIERS
O.E., WITHIN 1.5 IQR OF THE QUARTILES?.
BOX-ANP-
WHI5KER5
PLOTS ARE
ESPECIALLY
6OOP FOR
SHOWING OFF
PIFFERENCE5
BETWEEN
GROUPS.
THE STANDARD MEASURE OF SPREAD IS THE
STANDARD DEVIATION
UNLIKE THE IQR, WHI6H 15
BASED ON MEDIANS, THE
STANDARD DEVIATION MEASURES
THE SPREAD FROM THE MEAN.
YOU 6AN THINK OF IT,
ROUGHLY 5PEAKIN6., A5 THE
AVERAGE DISTANCE OF THE
DATA FROM THE MEAN %-
EXCEPT THAT WE U5E THE SQUARES OF THE_DI5TAN^E5 ]N5TEAD. THAT 15,
IF THE SQUARED DISTANCE OF POINT %, TO X 15 (% - %)2 THEN
n
AVERAGE SQUARED DISTANCE -
i~1
FOR TE^HNI^AL REASONS, WE U5E Л-1 IN
THE DENOMINATOR RATHER THAN n, ANP
DEFINE THE SAMPLE VARIANCE 52 A5
i~1
FOR THE DATA 5ET {3 5 7 7 30}, WITH X = 12 AND n = 5 WE CALCULATE
THE VARIANCE:
THE DATA...
^2 _ <3-12)2 + <5-12}2 + О-12)2 + <7-12)2 + <30-12)2
(9-1)
01 + 49 +• 25 4- 25 + 676
4
= 214
THE LAR6E >
VARIANCE HERE
REFLECTS THE
WIDE SPREAD IN
22
BUT A 5PREAP MEA5URE 5HOULP
HAVE THE 5AME UMIT4 A5 THE
ORIGINAL PATA. IN THE
EXAMPLE OF WEI6-WT5, THE
VARIANCE & 15 MEA5UREP IN
POUNP5 6QUAREP... OOOP51
THE OBVIOU5 THIN6. TO PO 15 TO
TAKE THE 5OUARE ROOT, ANP 50 WE
PO... TO PEFINE-.
5TAMPARP
NATION
i~1
WHICH, FOR OUR 5IMPLE PATA 5ET, 15
5 • >/214 =14.63
EVEN FOR 5MALL PATA 5ET5,
THE ARITHMETIC CAN BE
TEPIOU5! 50 NOWAPAY5, WE
JU5T HIT THE 5 BUTTON ON
THE HANP CALCULATOR, OR
C0N5ULT THE PATA REPORT
6-ENERATEP ВУ A COMPUTER
50FTWARE PACKAGE.
25
THE MEAN ANP STANPARP
PEVIATION ARE VERY GOOV
FOR SUMMARIZING THE
PROPERTIES OF FAIRLY
5УЛИГ7ЖЛ/. /7/5TOGRAM5
without outliers—ie.,
HISTOGRAMS SHAPEP LIKE
MOUNPS.
IT'S OFTEN USEFUL TO KNOW HOW MANY STANPARP PEVIATIONS A PATA POINT
IS FROM THE MEAN. WE PEFlNE Z-SCORES, OR STANPARPIZEP SCORES, AS
PISTAN^E FROM % PER STANPARP PEVIATION.
•. zz —4------- FOR EAAH l.
i 5
A Z-S^ORE OF +2 MEANS THAT AN OBSERVATION IS TWO STANPARP
PEVIATIONS ABOVE THE MEAN. FOR THE WEIGHT PATA OS=14£2ANP
S-23.7), WE AAN PLOT THE PATA ON THE ORIGINAL %-AX IS IN POUNPS ANP
THE z-SZORE AXIS SIMULTANEOUSLY.
175
. : : . . s | j . . .
i • r—:;sf“s r*"1?*'• -i '-i s i i • i-1-1
100 150 200
-2-10 1 2
Z score
1.26
A STUPENT WEIGHING 175 POUNPS HAS A Z-S60RE OF
^M.2b
at
an EMPIRICAL RULE
FOR NEARLY SYMMETRIC MOUNP-SHAPEP PATA SETS, APPROXIMATELY 60%
Of THE PATA IS WITHIN OA/f STANPARP PEVIATION OF THE MEAN ANP 90% Of
TME PATA IS WITHIN TWO STANPARP PEVIATIONS OF TME MEAN.
FOR THE WEIGHTS, OUR EMPIRICAL RULE HOLPS UP PRETTY WELL: 64%
(-59/92) Of THE WEIGHTS ARE WITHIN ONE STANPARP PEVIATION OF THE
MEAN, ANP ₽7% 6= 99/92? OF THE WEIGHTS ARE WITHIN TWO STAN PARP
PEVIATIONS OF THE MEAN.
Weight in pounds
150
59 points
89 points
92 points
Z score
ANP NOW
FOR A REST
FROM NUMBER
CRUNCHING/
Cute Lil
OUTLIER
WE’VE £OME A LON£ WAV IN THIS CHAPTER' STARTING WITH A UN0R6ANIZEP
PILE OF NUMBERS, WE HAVE'-
—
| FOUNP SEVERAL DIFFERENT
1 < WAVS TO PlSPLAy THEM
LOOKUP AT TWO DIFFERENT
2) 6ONZEPT* OF THE CENTER OF
PATA, THE MEPIAN ANP THE
MEAN
3| MEASUREP THE SPREAP OF THE
w>< PATA AROUNP THE CENTER IN
TWO PIFFERENT WAVS
ENCOUNTEREP MOUNP-SHAPEP
4) HISTOGRAMS ANP Z, A VARIABLE
THAT IN PIRATES HOW MANy
STAN PARP PEVIATIONS YOU ARE
FROM THE MEAN.
4_____________________________________________________>
NOW, IN ORPER TO PROBE THE BEHAVIOR OF PATA MORE PEEPLX WE’RE GOING
TO MAKE A LITTLE PETOUR INTO THE REALM OF RAUPOMHE66... A LANP WHERE
THINGS ALWAYS WORK OUT IN THE LON6 RUN, ANP WHERE THE ONLy LAW I*
THE LAW OF THE 6АЮЦМ6 £АЯМО...
♦ Chapter 3<
PROBABILITY
тАП OTHIN6 IN LIFE IS CERTAIN. IN EVERYTHING WE PO, WE
A к \ THE 0F M&ML OUTCOMES, FROM
BUSINESS TO MEPIONE TO THE WEATHER. BUT FOR MOST
OF HUMAN HISTORY, PROBABILITY, THE FORMAL STUPY OF THE
27
Г NOBOPY KNOWS WHEN
£AMBLIN£ BE6-AN. IT
6OES BACK AT LEAST AS
FAR AS ANCIENT Б6УРТ.
WHERE SPORTING MEN
ANP WOMEN USEP FOUR
SIPEP “A*TRA6ALI"
MAPE FROM ANIMAL
HEELBONES.
THE ROMAN EMPEROR CLAUPW* (W BCE-54 CE) WROTE THE FIRST KNOWN
TREATISE ON 6-AMBLIN^. UNFORTUNATELY, THIS BOOK, *HOW TO WIN AT PICE,”
WAS LOST.
MOPERN PICE 64?EW POPULAR IN THE MIPPLE A6ES, IN TIME FOR A RENAIS-
SANCE RAKE, THE CHEVALIER PE &ERE, TO POSE A MATHEMATICAL PUZZLER:
WHAT’S LIKELIER:
ROLLING AT LEAST ONE
s/x in Four throw* of
A *INCLE PIE, OR
ROLLING AT LEAST ONE
POUBLE *!X IN 24
THROW* OF A PAIR OF
PICE?
ZB
THE CHEVALIER REASONEP
THAT THE AVERAGE NUMBER
OF SUCCESSFUL ROLLS WAS
THE SAME FOR BOTH GAMBLES'.
CHANC8 OF OHB SlX -- g
NUMBER
FOUR ROLLS-. <|-
CHAH^b OF ROUBLE A
SIX IN 0NF ROLL-
AVERAGE NUMBER IN «
14 ROUS--24-= 3
WHy, THEN, PIP HE LOSE
MORE OFTEM WITH THE
SECONP GAMBLE???
PE MERE PUT THE QUESTION TO HIS FRIENP, THE 6-ENlUS BLAISE PASCAL
<1623-1666).
AT LAST) A PRoBLFM
That turns oh!
ALTHOUGH PASCAL HAP EARLIER
6-IVEN UP MATHEMATICS AS A FORM
OF SEXUAL INPUL6ENCE 69, HE
A6-REEP TO TACKLE PE MERE’S
PROBLEM.
PASCAL WROTE HIS
FELLOW 6ENIUS PIERRE
PE FERMAT, ANP WITHIN
A FEW LETTERS, THE
TWO HAP WORKER OUT
THE THEORy OF
PROBABILITY IN ITS
MOPERN FORM—EXCEPT,
OF COURSE, FOR THE
CARTOONS.
WORT WECOULJ>
havg, leoHL-v
ОЦ& OF 11*
y^ULPVRAVb-"
LI
BASIC DEFINITIONS
A6 OUR GAMBLER РЕАУ6 A 6AME, WE PLAY
56IENTI5T, OBSERVING THE OUTCOME:
a random experiment
15 THE PROCESS OF OBSERVING THE
OUTCOME OF A CHANGE EVENT.
the elementary
outcomes are all POS-
SIBLE RESULTS OF THE RANPOM EX-
PERIMENT.
the sample space 15
THE SET OR COLLECTION OF ALL THE
ELEMENTARy OUTCOMES.
VJUAT \
GAME9 \
vice 7 1
лемм-ге.
< Feri j
H-et's
FLIP A
CoitV
IF THE EVENT WA5 A £OIN TO55, FOR
EXAMPLE, THE RAWPOM EXPERIMENT
6ON5I5T5 OF RE6ORPIN& IT5
OUTCOME...
ANP THE SAMPLE SPACE 15 THE SET
WRITTEN
{Н.Т}
AMP if \
PICE |$
Voufc GAfAtf
50
ТИБ SAMPLE SPACE OF THE THROW OF A S/A/^-f PIE IS A LITTLE BICKER.
ANP FOR A PAIR OF PICE, THE SAMPLE SPACE LOOKS LIKE THIS (WE MAKE ONE
PIE WHITE ANP ONE BLACK TO TELL THEM APART);
THIS SAMPLE SPACE
HAS 36 (6X6)
ELEMENTARy OUT-
COMES. FOR THREE
PICE, THE SPACE
WOULP HAVE 216
ENTRIES, AS IN THIS
6X6X6 STACK. ANP
FOUR Pl££?
AT SOME POINT, WE HAVE TO STOP
LISTING, ANP START THINKING...
31
NOW LET’S IMAGINE A
RANPOM EXPERIMENT WITH
n ELEMENTARY OUTCOMES
O„ 02, ... On. Ж WANT TO
ASSIGN A NUMERICAL
we&ut or probability,
TO EA£M OUTCOME. WMI£M
MEASURES TME LIKELIMOOP
OF ITS OTCURRIN6. WE
WRITE TME PROBABILITY OF
ot to
FOR EXAMPLE, IN A FAIR 601N
TOSS, MEAPS ANP TAILS ARE
EQUALLY LIKELY, ANP WE
ASSIGN THEM BOTH TME
PROBABILITY .5.
РОО-РГО - .5
ЕА6И OUTCOME 6OMES
UP MALF TME TIME.
ASK ANY FOOTBALL.
PLATER!
IN TME ROLL OF TWO WE, TMERE ARE 36 ELEMENTARY OUTCOMES, ALL
EQUALLY LIKELY, SO TME PROBABILITY OF EA£M IS .
WMI6M MEANS: IF YOU ROLLEP TME
PI6E A VERY LAR6-E NUMBER OF TIMES,
IN TME LON6 RUN TMIS OUTCOME
WOULP O66UR J- OF TME TIME-
QME B1LL10M, -z МиНЪПЕО
MILL ION ... ИА<к- . WHEtZE-
AMP &K
FOR INSTANCE,
P<BLA£K 5, WHITE 2) -
3b
si
WHAT IF OUR RAMBLER
tUSATf ANP THROWS A
LOADED DIE? FOR THE SAKE
OF ARGUMENT, SUPPOSE THAT
NOW A ONE (OMES UP 25%
OF THE TIME (IN THE LON&
RUN).
THE SAMPLE 5PA£E IS THE
SAME AS FOR A FAIR PIC
{1, 2, 3, 4, 5, 6}
.25 -IB ЛБ
•16 .16 ,1$
BUT THE PROBABILITIES ARE
PIFFERENT. NOW P<1) *.25
ANP THE REMAINING
PROBABILTIC5 APP VP TO .75.
IF 2, 3, 4, 5, ANP 6 WERE
ALL EQUALLY LIKELY, THEN
EAZH ONE WOULP HAVE
PROBABILITY .15 = -^(.75)
IN GENERAL, ELEMENTARY OUTCOMES NEEP NOT HAVE EQUAL PROBABILITY
33
—
MOW WHAT CAN WE 5АУ
АБОРТ THE PROBABILITIES
PfOP IN AN ARBITRARY RAN-
POM EXPERIMENT? FIRST OF
ALL,
?(Ot)>O
PROBABILITIES ARE NEVER
NEGATIVE- A PROBABILITY OF
ZERO MEANS AN EVENT CAN’T
HAPPEN. LESS THAN ZERO
WOULP BE MEANINGLESS.
I6N‘T
)
SECONP, IF AN EVENT IS CERTAIN TO HAPPEN, WE ASSIGN IT PROBABILITY 1.
(IN THE LONG RUN, THAT’S THE PROPORTION OF TIMES IT WILL OCCUR.'?
IN PARTICULAR,
THE TOTAL.
PROBABILITY OF
THE SAMPLE
SPACE MUST BE 1. IF WE PO
THE EXPERIMENT, SOMETHING
IS BOUNP TO HAPPEN.'
PUT THESE TWO TOGETHER, ANP YOU HAVE THE CHARACTERISTIC
PROPERTIES OF PROBABILITY:
PCOj)* О
PROBABILITY IS NON-NE&ATIVE
Рбо,)+Рбо2) + ...+Рбол^1
TOTAL PROBABILITY OF ALL
ELEMENTARY OUTCOMES 15 ONE.
M&TAVHY-SICG
VJ/LL 66T Ш
LIKE A CLEVER POLITICIAN, WE
HAVE AVOIPEP CERTAIN
UNPLEASANT QUESTIONS,
SUCH AS a; WHAT POES
PROBABILITY MAN? ANP
B? HOW PO WE ASSI6N
PROBABILITIES TO OUTCOMES?
/ B-PUH, B-PUH... '
I LET’S PISCUSS
SOMETHING EASIER,
LIKE SAYS IN THE
MILITARY...
HERE ARE SOME APPROACHES THAT HAVE BEEN TAKEN-.
Classical PROBABILITY;
BASEP ON 6AMBLIN6- IPEAS, THE
FUNPAMENTAL assumption is that
THE 6-AME IS FAIR ANP ALL
ELEMENTARY OUTCOMES HAVE THE
SAME PROBABILITY.
f c'mou! '
pappy Heeps
a hew
.theory/
Relative Frequency:
WHEN AN EXPERIMENT £AN BE REPEATEP,
THEN AN EVENT’S PROBABILITY IS THE
PROPORTION OF TIMES THE EVENT
OCCURS IN THE LON6 RUN.
Personal PROBABILITY: MOST
OF LIFE’S EVENTS ARE A/OT
RGPGATABLG. PERSONAL PROBABILITY
IS AN INPIVIPUAL’S PERSONAL.
ASSESSMENT OF AN OUTCOME’S
LIKELlHOOP. IF A RAMBLER BELIEVES
THAT A HORSE HAS MORE THAN A SP%
CHANCE OF WINNING, HE'LL TAKE AN
EVEN BET ON THAT HORSE.
DO '/OU
RA WiSPOAA OF
PA TRACK-
AN OBJECTIVIST USES EITHER THE
CLASSICAL OR FREQUENCY PEFINITION
OF PROBABILITY. A SUBJEOTWIST OR
BAYESIAN APPLIES FORMAL LAWS OF
CHANCE TO HIS OWN, OR YOUR,
PERSONAL PROBABILITIES.
WAUNA
ьет?
HOW PO YOU KNOW THE
ELEMENTARY OUTCOMES
ARE EQUALLY LIKELY
WITHOUT ROLLING THE
PICE A BILLION TIMES?
BASIC OPERATIONS
SO FAR, WE HAVE PISCUSSEP ONLY THE
PROBABILITY OF ELEMENTARY OUTCOMES.
IM THEORY, THAT WOULP BE ENOUGH TO
PESCRIBE ANY RANPOM EXPERIMENT, BUT
IM PRACTICE ITS PRETTY UNWlELPY FOR
EXAMPLE, EVEN SUCH AM ORPIMARY
OCCURRENCE AS ROLLING A SEVEN IS NOT
AN ELEMENTARY OUTCOME- SO WE
IMTROPUCE A NEW I PEA;
AN EVENT IS A SET OF ELEMENTARY OUTCOMES. THE PROBABILITY OF AN
EVENT IS THE SUM OF THE PROBABILITIES OF THE ELEMENTARY OUTCOMES IN
THE SET, FOR INSTANCE, SOME EVENTS IN THE LIFE OF A TWO PICEP ROLLER
ARE;
EVENT PESCRIPTlON EVENT'S ELEMENTARY OUTCOMES PROBABILITY
A- PICE APP TO 3 {61,,29, 62,19} PCA9=
В PICE APP TO 6 {61,5), 62,49, 63,39, 64,29, 65,19}
C WHITE PIE SHOWS 1 {61,19, 61,29, 61,39, 61,49, 61,59, 61,69}
P; BLACK PIE SHOWS 1 {61,19, 62,19, 63,19, 64,19, 65,19, 66,19} P6P}= Iz
THE BEAUTY OF USING
EVENTS, RATHER THAN
ELEMENTARY OUTCOMES, IS
THAT WE CAN COMBIMG
EVENTS TO MAKE OTHER
EVENTS, USING LOGICAL
OPERATIONS. THE
RELEVANT WORPS ARE
ANP, OR, ANP NOT.
THAT IS, GIVEN EVENTS E ANP F, WE CAN MAKE NEW EVENTS;
E Cmd F ; THE EVENT E ANP THE EVENT F BOTH O66UR.
E OF F ; THE EVENT E OR THE EVENT F OCCURS (OR BOTH РОЛ
ПО* E ; THE EVENT E POES NOT OCCUR
COMBINING OUR PRIMITIVE
PEFINITIONS OF PROBABILITY WITH
THESE LOGICAL OPERATIONS WILL
GIVE US SOME POWERFUL
FORMULAS FOR MANIPULATING
PROBABILITIES.
GLIM
AKORGE.
< I GAMBLE COMPULSIVELY
AMD I LOST MY SHIRT
ДЫР M- PASCAL IS STILL
WRKINO on му PROBLEM-
vlHAT ARE MY CHANTS
Д/fC TV, CtiERtE R
.\.U M
OR
Э7
LET’5 RETURN TO TME PI6E-TMROWIN6 EXAMPLE. IF C 15 TME EVENT, WMITE
PIE = 1, ANP P 15 TME EVENT, BLA6K PIE ® 1, TMEN;
C OR Р15 TME
ENTIRE 5MAPEP
AREA 6WMERE
ONE PIE OR TME
OTMER 15 0.
C ANP P\5
WMERE TME
5MAPEP AREA5
OVERLAP 6BOTM
Р16Б ARE 0.
TMI5 ILLU5TRATE5 TME АРР1Т10Ы RULE; FOR ANV EVENT5 E, F,
P(E OR F) = P(E) + P(F) - P(E AND F)
APPIN6 P6E9 + P6F; POU8LG tOUMTf TME ELEMENTARy OLHZOME5 5MAREP ВУ
E ANP F, 50 WE MAVE TO 5UBTRA6T TME EXTRA AMOUNT, WUI6M 15 P(E ANp F).
IN TME ABOVE EXAMPLE,
?(C OR P) =
36
A5 YOU 6AN 5EE ВУ
6OUNTIN6 ELEMENTARy
OUT6OME5. LIKEWI5E,
Г(С ДЫР P) = J-
36
ANP WE CONFIRM TME FORMULA--
?(C) + P(P) - ?(C ANP P)
A +A _ ± _ JLL
' 36 36 36 36
= ?(c or p;
36
SOMETIMES, THE OVERLAP E AMP F IS EMPTY, ANP THE TWO EVENTS HAVE
NO ELEMENTARY OUTCOMES IN COMMON. IN THAT 6ASE, WE SAY E ANP F ARE
MUTUALLY EXCLUSIVE, MAKING FYE ANP F9 => 0. HERE WE SEE THE MUTUALLY
EXCLUSIVE EVENTS A, THE PI6E APP TO 3, ANP B, THE PI6E APP TO 6.
Offl Offl Offl н и OB on
Sffl Hffl Sffl Hffl EIB SB В
□ffl ns na nra era on
□ffl Sffl 0® HH SB so
□йагаищииивпп a
□ffl@BHs нга ив □□
FOR MUTUALLY EXCLUSIVE EVENTS, WE 6ET A SPECIAL APPITION RULE' IF E
ANP F ARE MUTUALLY EXCLUSIVE, THEN
P(E OR F) = P(E) + P(F)
ANP WE 6HE6K THAT FYA OR Bj = ^- = = Р/Д'Н ?СВ)
i 36 >6 36
ANP FINALLY, A SUBTRACTION RULE-. FOR ANY EVENT E,
P(E) = 1 - P(NOT E)
THIS IS USEFUL WHEN P(NOT E) IS EASIER TO COMPUTE THAN Р6ЕЛ FOR
INSTANT, LET E BE THE EVENT, A POUBLE-1 IS NOT THROWN. THE EVENT
NOT-E, A POUBLE-1 IS THROWN, HAS PROBABILITY PfNOT E? * i .
5b
so
?(£) = 1-Р6ЫОТ
= 1
=
36
ишввнииаа
ЯП ЯП ЯП sn
ЯП ЯП ЯП ЯП
ЯП ЯП ЯП ЯП
ЯП ЯП ян ян
ян ЯНЕ ЯНЕ ЯНЕ
39
THE FORMULA* WE JU*T PERIVEP
ARE, IN FA£T, APEQUATE FOR
AN*WERIN6 PE MERE’* QUE*TION-
BUT NOT EA*ILYI (YOU MI6-UT TRY
U*lN6> THEM ON A *IMPLER
QUE*TION: WHAT’* THE PROBABILITY
OF ROLLING AT LEA*T ONE *IX IN
TWO ROLL* OF A *IN6>LE PIE?} WE
NEEP MORE MACHINERY!
*O WE INTROPU6E
conditional
probability
(AN E**ENTIAL 6ON6EPT IN
*TATI*TI6*O
LOOK\
*UPPO*E WE ALTER OUR EXPERIMENT *LI6HTLY, ANP THROW THE WHITE PIE
BEFORE THE BLA6K PIE. WHAT’* THE PROBABILITY THAT THE FA£E* *UM TO 3?
BEFORE THE РКБ
ARE THROWN, THE
PROBABILITY I*
FW= Tl
NOW *UPPO*£ THE
WHITE PIE (OME*
UP 1 (EVENT O.
WHAT’* THE
PROBABILITY
OF A NOW?
40
WE/ALL IT TME
tOMPITIOUAL
PROBABILITY THAT EVENT
A WILL O/ZUR, 6-IVEN
TME СОМРГПОМ THAT
EVENT С MAS ALREAPy
O/ZURREP. WE WRITE
P(AIC)
ANP 5АУ “THE
PROBABILITy OF A,
6WW 6.”
BEFORE ANy Pl/E WERE THROWN, TME SAMPLE SPA/E MAP 36 OUT/OMES, BUT
NOW THAT TME EVENT С MAS O/ZURREP, TME OUTCOME MUST BELONG TO TME
REPUtGP 4A&PLG 4PACG £
IN TME REPU/EP SAMPLE SPA/E OF SIX ELEMENTARy OUTCOMES, ONLy ONE
OUTCOME 61,2} SUMS TO 3. ^O TME 6ONPITIONAL PROBABILITy 15 1/6.
\
IN GENERAL, TO FlNP
TME 6ONPITIONAL
PROBABILITy PtelF),
WE LOOK AT TME
EVENT E ANP F A5
PART OF TME REPU^EP
SAMPLE 6PA/E F.
4-1
WE TRANSLATE THIS
INTO A FORMAL
PEFINITION: THE CONDITIONAL
PROBABILITY OF £, CIV£N F, IS
P(EIF) = PtE ond
P(Fj
FROM WHICH YOU CAN PIRECTLY
VERIFY SOME INTUITIVE FACTS:
PfelE) - 1 (ONCE G OCCURS,
IT’S CERTAIN.)
WHEN E ANP F ARE MUTUALLY
EXCLUSIVE,
PftlFJ - 6?
(ONCE F HAS
OCCURREP, G IS
IMPOSSIBLE.)
REARRANGING THE PEFINITION GIVES US THE MULTIPLICATION RUL£=
P(E AND F) = P(EIF)P(F)
WHICH WE WOULP LIKE TO REPUCE TO A “SPECIAL" MULTIPLICATION RULE,
UNPER THE FAVORABLE CIRCUMSTANCES THAT PCElF) » P(E). THAT WOULP BE
EXCELLENT'
/ANP WHILE VOtl'RE 4
WAITING THE
NEXT РД6Е, NOTE THAT
5WAP?IU6> E ANP F
PROVES THAT
?(f)₽(e|F)--P(E)?(F|E)
INDEPENDENCE and the
special multiplication rule.
TWO EVENTS E ANP F ARE INPGPGNPGNT OF EAT OTHER IF THE
OTURRENT OF ONE HAS HO INFLUGKCG ON THE PROBABILITY OF THE
OTHER- FOR INSTANT, THE ROLL OF ONE PIE HAS NO EFFECT ON THE ROLL
OF ANOTHER (UNLESS THEY’RE 6LUEP TOGETHER, MA6NETI6, GTCl).
IN TERMS OF T3NPITIONAL PROBABILITY, THIS AMOUNTS TO SAYING
Pte? • PtelF? OR, EQUIVALENTLY, P<F? * P<FlG?. WHEN E ANP F ARE
INPEPENPENT, WE &ET A 5PGCIAL MULTIPLICATION RULG:
P(E AND F) = P(E)P(F)
LET’S VERIFY THE INPEPENPENT OF PIT, USIN6 THE FORMULAS. C IS THE
EVENT WHITG PIG COMG* UP h P IS THE EVENT BLACK PIG COMG5 UP 1, ANP
WE HAVE: .
ft 1
BUT THE WHITE PIE SHOWING 1 OBVIOUSLY POES AFFECT THE TANTS THAT
THE SUM OF THE TWO PIT IS 3!
±
km .2- -1 4 KM . X
pa) pa) i b 18
SO THESE TWO EVENTS ARE NOT INPGPGNPGNT
43
BEFORE 6OIN& ОМ, LET’S SUMMARIZE ALL THE RULES WEVE ACCUMULATE?:
APPITIOM RULE;
P(E OR F) = P(E) + P(F) - P(E AND F)
SPECIAL APPITIOM RULE: WHEN E AMP F ARE
MUTUALLV EXCLUSIVE,
P(E OR F) = P(E) + P(F)
SUBTRACTION RULE:
P(E) = 1 - P(NOT E)
MULTIPLICATION RULE:
P(E AND F) = P(E I F)P(F)
SPECIAL MULTIPLICATION RULE: WHEN E
AMP F ARE IMPEPEMPEMT,
P(E AND F) = P(E)P(F)
AMP MOW, PE MERE’S PROBLEM AT LAST... LET E BE THE EVENT OF 6ETTIN6
AT LEAST ONE SIX IN FOUR ROLLS OF A SINGLE PIE. WHAT’S РСЕ)? THIS IS
ONE OF THOSE EVENTS WHOSE NEGATIVE IS EASIER TO PESCRIBE'- A/OT C IS
THE EVENT OF ££T77A/£ NO IN FOUR THROWN
IF A, IS THE EVENT, &ETTIN6 NO
SIX ON THE THROW, WE KNOW
THAT PCA,) = . WE ALSO KNOW
THAT ROLLS ARE INPEPENPENT, SO
P(NOT E) =
WULTiPUKATiON
UULt «
SO
PCA, AMP A2 ANp A3 ANp A4)
F(G) 1 - P(NOT E) » .919
44
MOW THE SECONP HALF: LET F BE THE EVENT, 6ETTIN6 AT LEAST ONE
POUBLE SIX IN 24 THROWS. A6AIN, NOT F IS EASIER TO PESCRIBE. IT’S THE
EVENT OF &ETTIN6- NO POUBLE SIXES.
IF В,- IS THE EVENT, NO POUBLE
SIX IS THROWN ON THE
ROLL, THEN NOT F = B, ANP B2
ANP... B24. THE PROBABILITY OF
EACH В IS
P6B/) = 5^ . so
P6NOT F) = (^) = .509
(W THE MULTIPLICATION RULE?
ANP WE CONCLUPE THAT
PCF? » 1 - PCNOT F) -- 1 - .509
* .491
PE MERE TOLP PASCAL HE HAP ACTUALLY OBSERVEP THAT EVENT F OCCURREP
LESS OFTEN THAN EVENT E, BUT HE WAS AT A LOSS TO EXPLAIN WHY... FROM
WHICH WE CONCLUPE THAT PE MERE 6AMBLEP OFTEN ANP KEPT CAREFUL
45
BAYES THEOREM and the
case off the ffalse positives...
FOR A MORE SERIOUS APPLICATION OF
CONDITIONAL PROBABILITY, LET’S ENTER
AN ARENA OF LIFE ANP PEATH...
SUPPOSE A RARE PISEASE INFECTS ONE OUT OF EVERY WOO PEOPLE IN A
POPULATION...
ANP SUPPOSE THAT THERE IS A 6OOP, BUT NOT PERFECT, TEST FOR THIS
PISEASE: IF A PERSON HAS THE PlSEASE, THE TEST COMES BACK POSITIVE 99%
OF THE TIME. ON THE OTHER HANP, THE TEST ALSO PROPUCES SOME FALK
POSITIVE*. ABOUT 2% OF UNINFECTEP PATIENTS ALSO TEST POSITIVE. ANP YOU
JUST TESTEP POSITIVE. WHAT ARE YOUR CHANCES OF HAVING THE PISEASE?
Ab
HAVE TWO EVENTS TO WORK WITH;
A : PATIENT HAS THE PISEASE
В РКТШТ TESTS POSITIVE.
THE INFORMATION ABOUT THE TEST’S
EFFECTIVENESS CAN BE WRITTEN
?(Ю * .001
PftlA) » .99
PfBlNOT A) * .01
ANP WE ASK
» WHAT?
CONE PATIENT IN WOO HAS THE PISEASE)
(PROBABILITY OF A POSITIVE TEST,
GIVEN INFECTION, IS .99)
(PROBABILITY OF A FALSE POSITIVE, GIVEN
NO INFECTION, IS .02)
(PROBABILITY OF HAVING THE PISEASE,
GIVEN A POSITIVE TEST)
SINCE THE TREATMENT FOR THIS PISEASE HAS SERIOUS SIPE EFFECTS, THE
POCTOR, HER LAWYER, ANP HER LAWYER’S LAWYER CALL ON JOE BAYES, CP
(CONSULTING PROBABILIST), FOR AN ANSWER. JOE PERIVES A THEOREM FIRST
PROVEP BY HIS ANCESTOR, THE REV. THOMAS Й4УИ (1744-1009).
\__________________________________________________________У
47
JOE BEGINS WITH A 2X2 TABLE, WHICH PIVIPE* THE SAMPLE SPACE INTO FOUR
MUTUALLV EXCLUSIVE EVENT*. IT PI*PLAV* EVERy POSSIBLE COMBINATION OF
PISEASE STATE ANP TEST RESULT.
A NOT A
В A ANP В NOT A ANP В
NOT В A ANP NOT В NOT A ANP NOT В
LET’S FINP THE PROBABILITIES OF EACH EVENT IN THE TABLE-
A NOT A SUM
В P6A ANP в; PfNOT A ANP B) PCB)
NOT В PCA ANP NOT b) PCNOT A ANP NOT B) PCNOT b)
pca; PCNOT a; 1
THE PROBABILITIES IN THE MARGINS ARE FOUNP ВУ SUMMING ACROSS ROWS
ANP POWN COLUMNS.
\,
V&rlMTtou*.
NOW COMPUTE;
рбд anp в; = p<&ia;p<a; = (.99X001) = .00099
P6NOT A ANP B? = PCBlNOT AMNOT A? = (02)C999) ’ .01990
ALLOWING US TO FILL IN SOME ENTRIES:
A NOT A SUM
В .00099 .01990 .OZO91
NOT В PCA anp NOT b) P(NOT A ANP NOT B) PCNOT b)
.001 .999 1
WE FINP THE REMAINING PROBABILITIES ВУ SUBTRACTING IN THE COLUMNS, THEN
APPING ACROSS THE ROWS.
46
THE FINAL TABLE 19-
В A NOT A P6BP
.00099 .01990 .02097
NOT В .00001 .97902 .979m P6NOT B)
.001 .999 1
pca; P6NOT A)
FROM WHI^H WE PIRE6TLV PERIVE
FYAlB) __ KAWB) = ЛОО99 =
P(B? .02097
PESPITE THE HI6H A^URA^y OF THE TEST, LEM THAN 9% OF THOSE WHO
TEST POSITIVE A^TUALLV HAVE THE PISEASE' THIS IS 6ALLEP THE FAL9E
POSITIVE PARADOX.
ДХ1Р
?д\ед-
LAWVb^-
THIS TABLE. SHOWS
WHAT HAPPENS IN A
6-ROUP OF A THOUSANP
PATIENTS. ON AVERAGE,
ONLy 21 PEOPLE WILL
TEST POSITIVE—ANP
ONLy ONE OF THEM
HAS THE PISEASE' 20
FALSE POSITIVES £OM£
FROM THE #UCH
LARGER UNINFECTED
CROUP.
PISEASE NO PISEASE
TESTS POSITIVE 1 20 21
TESTS NEGATIVE О 979 979
1 999 WOO
49
WHAT'S THE PHYSICIAN TO PO? JOE BAYES APVISES HER NOT TO START
TREATMENT ON THE BASIS OF THIS TEST ALONE. THE TEST POES PROVIPE
INFORMATION, HOWEVER; WITH A POSITIVE TEST THE PATIENT'S CHANCE OF
HAVING THE PISEASE INCREASEP FROM 1 IN WOO TO 1 IN 23. THE POCTOR
FOLLOWS UP WITH MORE TESTS.
JOE BAYES COLLECTS HIS CONSULTING CHECK BEFORE APMITTING THAT ALL
THOSE STEPS HE WENT THROUGH CAN BE COMPRESSEP INTO THE SINGLE
FORMULA CALLEP BAYES THEOREM;
P(A)P(BIA)
P(AIB) =
P(A)P(B I A)+P(NOT A) P( ВI NOT A)
IT COMPUTES P6AIB) FROM P(A) ANP THE TWO CONPITIONAL PROBABILITIES
PCBlA) ANP P6BINOT A?. YOU CAN PERIVE IT BY NOTING THAT THE BIG FRACTION
CAN BE EXPRESSEP AS
P(A and B) _ P(A and В) _ р[д|р«
P(A and B)+P(NOT A and B) ~ P(B) ~ 1
50
IN THIS CHAPTER, WE COVEREP THE
BASICS OF PROBABILITY: ITS PEFINITION,
SAMPLE SPACES ANP ELEMENTARY
OUTCOMES, CONPITIONAL PROBABILITY,
ANP SOME BASIC FORMULAS FOR
COM PUTlNG PROBABILITIES. WE
ILLUSTRATEP THESE IPSAS USING A
2-PICE SAMPLE SPACE. FOR THE MOPERN
6AM BL ER, PROBABILITY IS THE POWER
TOOL OF CHOICE.
DDDDDD
ИИЙИИН
БК QD EE QD EE EE
S3 S3 EK3 S3 EK3 S3
§ИЙ0
ВДЕЗНПН
rm rm rm rm i
уД |Тц |ДД ДДД уд
gSHEQH
ANP FINALLY, IN THE MEPICAL EXAMPLE, WE SHOWEP HOW THESE ABSTRACT
IPSAS COULP HELP TO MAKE GOOP PENSIONS IN THE FACE OF IMPERFECT
INFORMATION ANP REAL RlfK^-THE ULTIMATE COAL OF STATISTICS-
BUT THIS IS JUST THE BEGINNING- FOR US, PROBABILITY IS ONLY A TOOL-AN
ESSENTIAL TOOL, TO BE SURE-IN THE STUPY OF STATISTICS. IN THE CHAPTERS
THAT FOLLOW, WE’LL EXPLORE THE SUBTLE RELATIONSHIP BETWEEN
PROBABILITY, VARIATIONS IN STATISTICAL PATA, ANP OUR CONFIPENCE IN Г"
INTERPRETING THE MEANING OF OUR OBSERVATIONS.
Я
<?2
♦ Chapter 4ф
RANDOM VARIABLES
IN CHAPTER 2, WE SAW THAT OBSERVATIONS OF NUMERICAL
PATA, LIKE STUPENTS’ WEIGHTS, ZAN BE 6RAPMEP ANP
SUMMARIZE? IN TERMS OF MIPPOINTS, SPREAPS, OUTLIERS, ETZ.
IN CHAPTER 3, WE SAW MOW PROBABILITIES ZAN BE ASSI6NEP
TO ТМБ OUTCOMES OF A RANPOM EXPERIMENT.
IF WE IMAGINE A RANPOM EXPERIMENT REPEATEP MANY TIMES,
WE EXPERT THAT THE ACTUAL OUTCOMES OVER TIME WILL BE
6OVERNEP ВУ TMEIR PROBABILITIES. THE PROBABILITIES FORM A
&OPGL FOR REAL-LIFE EXPERIMENTS... SO WHY NOT PO FOR THE
MOPEL WMAT WE'VE ALREAPY PONE FOR TME PATA IT PES6RIBES?
5Э
THE КЕУ (PEA \5 TME RANPOM VARIABLE, WMIdM WE
WRITE AS A LAR6E _________ __
A RANPOM VARIABLE Ъ PEFINEP AS TME NUMERICAL OUTCOME OF A
RANPOM EXPERIMENT.
FOR EXAMPLE, IMAGINE PRAWIN6 ONE STUPENT AT RANPOM FROM TME
STUPENT ВОРУ. THAT'S TME RANPOM EXPERIMENT. TME STUPENTS HEICRT,
WEIERT, FAMILY INCOME, !.A.T. MORE, ANP ERAPE POINT AVERAGE ARE
ALL NUMERICAL VARIABLE! PES£RIBIN6 PROPERTIES OF TME RANPOMLV
SELE6TEP STUPENT. THEY'RE ALL RANPOM VARIABLE!.
ANOTHER EXAMPLE- TOSS TWO 6OINS fTME RANPOM EXPERIMENT? ANP RE^ORP
TME NUMBER OF HEAPS; О, 1, OR 2.
NOTE TME NOTATION! THE VARIABLE IS WRITTEN WITH A CAPITAL X. TME
LOWERCASE REPRESENTS A SINGLE VALUE OF X, FOR EXAMPLE /'-2, IF
HEAPS dOMES UP TWIdE.
Я
ANOTHER EXAMPLE Ъ
BASEP ON THE FAMILIAR
TOSS OF TWO PldE. LET
У REPRESENT THE SUM
OF THE POTS ON THE
TWO P1£E- FOR THIS
RANPOM VARIABLE, У
£AN BE ANY NUMBER
BETWEEN 2 ANP 12.
NOW WE WANT TO LOOK AT THE PROBABILITIES OF THE OUTCOMES- FOR
THE PROBABILITY THAT THE RANPOM VARIABLE X HAS THE VALUE Z, WE
WRITE PrfX - ХЛ OR JUST FOR THE 6OIN-FLIPPIN6 RANPOM
VARIABLE X, WE £AN MAKE THE TABLE*-
Я О 1 THIS TABLE IS 2 £ALLEP THE PROBABILITY
Рг(Х~я) 1 4 1 2 1 DISTRIBUTION OF — THE RANPOM h VARIABLE X-
FOR THE RANPOM VARIABLE У (THE SUM OF TWO PI6EJ, THE PROBABILITY
PISTRIBUTION LOOKS LIKE THIS*.
5S
NOW LET’S PRAW GRAPHS, OR U&TO&WAf, 5Н0Ш1Ш, THESE
PROBABILITY PISTRIBUTIONS. FOR EAZH VALUE OF X, WE PRAW A BAR
EQUAL IN HEIGHT TO pCzY
IT’S EASY TO SEE THAT THE TOTAL AREA OF THESE BOXES IS 1; EACH BOX HAS
BASE 1 ANP HEIGHT р(Я\ 50 THE TOTAL AREA 15 THE SUM OF THE
PROBABILITIES OF ALL OUTCOMES, l.£. L
HERE’S THE PROBABILITY HISTOGRAM OF THE RANPOM VARIABLE У, SHOWING
THE PROBABILITY PISTRIBUTION OF THE SUM OF TWO PI6£;
WHY PO WE £ALL THESE GRAPHS HISTOGRAMS? YOU’LL RECALL THAT IN
CHAPTER 2, A HISTOGRAM WAS A GRAPH THAT PISPLAYEP HOW MANY PATA
POINTS LAY IN EMH OF A SERIES OF INTERVALS:
FROM THIS FREQUENCY HISTOGRAM, WE PERIVEP THE RELATIVE FREQUENCY
HISTOGRAM, SHOWING THE PROPORTION OF PATA IN EMH INTERVAL:
\_________________________________________—--------------------------'
BUT YOU’LL RECALL THAT, BY
ONE PEFINITION, PROBABILITY
IS THE RELATIVE FREQUENCY
OF AN EVENT "IN THE
LONE RUN.” IF WE REPEAT
THE RANPOM EXPERIMENT
MANY TIMES, THE RELATIVE
FREQUENCY HISTOGRAM OF
THE OUTCOMES SHOULP 6OME
TO LOOK VERY MUdH LIKE
THE RANPOM VARIABLE’S
PROBABILITY HISTOGRAM.'
51
WE ILLUSTRATE USIN6 THE RANPOM
VARIABLE X ANP A MAP COIN TOSSER.
WE KNOW X’S PROBABILITY PISTRIBUTION, ANP WE ALSO KNOW THAT THE
ACTUAL COIN FLIPS WILL MATCH THE PROBABILITIES APPROXIMATELY. AFTER
WOO TOSSES, THE MAP TOSSER TALLIES HER PATA:
PROBABILITY MOPEL Я OBSERVEP PATA
пл~NUMBER OF OCCURRENCES RELATIVE FREQUENCY
.25 0 260 .260
.5 1 517 .517
.25 2 223 .223
ANP WE SEE THAT THE PROBABILITY HISTOGRAM OF X LOOKS LIKE THE “PURE
FORM” OR MOPEL OF THE RELATIVE FREQUENCY HISTOGRAM OF THE PATA.
58
ТО EXTENP THE ANALOGY BETWEEN RELATIVE FREQUENCY ANP PATA, WE
DMOULP NOW BE WILLING TO TALK ABOUT THE MEAN ANP VARIANCE (OR
DTANPARP PEVIATION? OF A PROBABILITY PIDTRIBUTION...
ANP JUDT TO REMINP
OURDELVED THAT WE RE IN
THE REALM OF THE
ABDTRAOT WE BREAK OUT
DOME ЛЙЕ0К LETTERS—
MEAN AND VARIANCE OF
RANDOM VARIABLES
WE UDE DPECIAL TERMINOLOGY
ANP DYMBOLD TO PIDTINGUIDM
BETWEEN TME PROPERTIED OF
PATA DETD ANP PROBABILITY
PIDTRIBUTIOND:
PROPERTIED OF PATA ARE 6ALLEP SAMPLE PROPERTIED, WHILE PROPERTIED
OF TME PROBABILITY PIDTRIBUTION ARE ^ALLEP &OPEL OR POPULATION
PROPERTIED WE UDE TME GREEK LETTER JU (MU) FOR TME POPULATION
MEAN, ANP a (LOWER^ADE DIGMA) FOR TME POPULATION DTANPARP
PEVIATION. (FOR PATA, WE UDE TME ROMAN DYMBOLD X ANP D.)
59
MOW SOME OF THESE PATA POINTS я!, МАУ WELL HAVE EQUAL VALUES. THINK
OF THE MAP COIN TOSSER-- THE ОМЕУ AVAILABLE VALUES WERE О, 1, ANP 2, ANP
SHE MAPS WOO TOSSES. THE VALUE О WAS TAKEN ON 26P TIMES, 1 HEAP CAME
UP 917 TIMES, ANP 2 HEAPS, 223 TIMES.
AS WE LET Я RAN6E OVER
ALL VALUES OF X, CALL nx
THE NUMBER OF PATA
POINTS WITH THE VALUE %.
THEN WE CAN REWRITE
THAT FORMULA AS
allz
OR
7 - V
* Xj n
allz
AHI BUT NOW jjr IS THE RELATIVE FREQUENCY.. THE “APPROXIMATE
PROBABILITY.." THE NUMBER THAT APPROACHES p(X)..5O, ВУ ANALOGY WE
FORM THE EXPRESSION
all %
ANP PEFINE THAT AS THE
MSAN OF ТИБ PROBABILITY
P&TRIBUVOU.
PCFIHITIOM: THE
mean of the
RANPOM VARIABLE X
PEFINEP A*
all %
TMI^ 15 ALSO £ALLEP THE EXPECTEP VALUE OF X, OR Е[х]. THINK OF IT AS
THE SUM OF THE POSSIBLE VALUES, EAZH WEI6-HTEP BY ITS PROBABILITY.
THE MAP 6OIN TO59QR'5 EXPERIMENT ALLOWS US TO COMPARE HER SAMPLE
MEAN X WITH OUR MOPEL MEAN /г-
X SAMPLE пл n * n MOPEL
p(^
0 .U 0 0 .25 0
1 .917 .517 1 .5 .5
2 .223 AAb .9ЬЪ = X, 2 .25 .5 1
NOW LET'* PO THE SAME THIN6- TO
THE VARIANCE. MAYBE YOU
REMEMBER THE FORMULA
MORE
MATH!
*2
IT (ALMOST? MEASURES THE AVERAGE
SQUAREP PISTAN6E OF PATA FROM THE
MEAN. A* ABOVE THIS (AN BE REWRITTEN:
61
EXCEPT FOR THAT АК1К1ОУ1К16> PENOMINATOR П-1 IKISTEAP OF n, TH 15 ALSO
LOOKS LIKE A WEl&HTEP SUM OF SQUAREP PlSTANCES... SO WE MAKE ANOTHER
P£FINITIOH:
the variance
OF A RANPOM VARIABLE X
IS THE БХРЕСТБР SQUAREP
PlSTAKlCE FROM THE
POPULATION MEAN:
allz
the standard
deviation cr
IS THE SQUARE ROOT
OF THE VARIANCE.
\________________________________________________________г
WE USE THE TABLE
FROM THE LAST
РА6Б TO FINP THE
VARIANCE OF THE
TWO-COIN TOSS
(FOR WHICH /4 = V.
О .45 (0-})г.45 = .45
1 .5 (1-1)г.5О = О
4 .45 (4-\)г.45 = .45
total 50 ~
ТО SUM UP: р. ANP cr, THE POPULATION Ж£4А/ ANP 5TAUPARP PSVIATIOH^Q
PROPERTIES WE CAN COMPUTE FROM PROBABILITY PlfTRIBUTIOHf. THEY ARE
COMPLETELY ANALOGOUS TO THE SAMPLE MEAN % ANP STANPARP PEVIATION S
COMPUTEP FROM SAMPLE PATA.
62
OUR EXAMPLES SO FAR HAVE BEEN V&tRCTE RANPOM VARIABLES THE R
OUTCOMES ARE A SET OF ISOLATEP rPISCRETE”) VALUES, LIKE THOSE WE SAW
IN CHAPTER 3, BUT THERE ARE ALSO
Continuous
Random
Variables
LET’S IMAGINE A RANPOM EXPERIMENT
IN WHICH ALL OUTZO&E5 HAVE
PROBABILITY ZERO. THAT’S RIC-HT,
fW * 0 FOR EVERY %.
A SIMPLE EXAMPLE IS A BALANCEP SPINNING POINTER IT CAN STOP ANYWHERE
IN THE CIRCLE IF X REPRESENTS THE PROPORTION OF THE TOTAL
CIRCUMFERENCE IT LANPS ON, THE RANPOM VARIABLE X CAN TAKE ON ANY
VALUE BETWEEN О ANP 1-AN INFINITE RANCE OF VALUES.
SOME PROBABILITIES ARE EASY TO
FlNP, LIKE THE PROBABILITY THAT X
FALLS WITHIN A RANCE; FOR
EXAMPLE, Рг(Л5 $ X < .75 ) - .5,
BECAUSE IT’S HALF THE CIRCLE BUT
WHAT ABOUT Pr(X » .5)? SINCE X
CAN TAKE ON AN INFINITE NUMBER
OF VALUES, ANP ALL OF THESE
VALUES ARE EQUALLY LIKELY, THE
PROBABILITY THAT X IS EXACTLY .5
Cor exactly anyth inc.) is
PRECISELY О
HOW CAN WE PRAW A PICTURE OF THIS?
ВУ ANALOG WITH THE CASE OF
PISCRETE PROBABILITIES, WE TRV TO
SEE CONTINUOUS PROBABILITIES AS
ARGAf UHVSR $OM£THIN6. FOR THE
SPINNING POINTER, THE 'SOMETHING*
LOOKS LIKE THIS;
= О WHEN % < О
= 1 WHEN О $ % $ 1
» О WHEN % > 1
THE PROBABILITy THAT THE
POINTER POINTS ANVWHERE
BETWEEN a ANP b IS PREClSELy
THE AREA OF THE SHAPEP REGION
UNPER THE CURVE BETWEEN a ANP
b (IN THIS CASE, b~a).
-
THE PROBABILITV OF AN EXACT
OUTCOME, HOWEVER, IS THE "AREA"
OVER A POINT, WHICH IS ZGRO-
(ANP NOTE THAT THE TOTAL AREA
UNPER THE CURVE IS EXACTLV О
ТИБ *АМЕ PICTURE PE*£RIBE* THE RANDOM NUMGR &GNGRATOR FOUNP ON
MO*T COMPUTER* ANP *OME CALCULATOR*. PRE** THE BUTTON; OUT POP* A
NUMBER BETWEEN о ANP 1-, ANP ALL THE NUMBER* ARE EQUALLY LIKELY, JU*T
A* WITH THE SPINNING POINTER.
BUT *APLY, THEY AREN’T
TRULY RANPOM. THEY’RE
PROPUCEP BY *OME
ALGORITHM, *0, TO BE
ACCURATE, WE CALL THEM
F*EUPO-RANPOM NUMBER*.
THE CURVE y. = IN THI*
EXAMPLE I* CALLEP THE
PROBABILITY Р£НЯТУ Of THE
CONTINUOU* RANPOM VARIABLE X-
EVERY CONTINUOU* RANPOM
VARIABLE HA* IT* OWN P£N*ITY
FUNCTION. THE PROBABILITY
Pr(a $ X $ b) I* THE AREA
UNPER THE ^URVE BETWEEN THE
-VALUE* a ANP b.
6*
IN GENERAL, ТИБ PROBABILITY
PENSITY WON’T BE SO SIMPLE,
ANP COMPUTING THE AREAS CAN
BE FAR FRO/И TRIVIAL
WE HAVE TO USE CALCULUS
NOTATION TO PESCRIBE THE
AREA UNPER THE CURVE /?%).
THIS SYMBOL IS REAP "THE
INTEGRAL OF f FROM a TO b?
•b
bb
ALTHOUGH ТИБ
NOTATION МАУ ВБ
UNFAMILIAR, ALL IT
MEANS IS AN ARSA..
ТИБ INTEGRAL SI6N
ITSELF IS A STRETCHEP
*S," FOR SUM, WHICH
THE INTEGRAL, IN
SOME SENSE, IS.
AS A SUMLIKE SOMETHING, THE INTEGRAL SERVES TO PEFINE THE
MEAN AND VARIANCE of a continuous
random variable.
/U ~ z-F(X)dz
V CD
(гг - (t-ptf-ffadt
BY ANALOGY
WITH THE allx
PISCRETE FORMULAS-- aUZ
ALTHOUGH IT МАУ NOT ВБ OBVIOUS FROM THE FORMULAS, THESE PEFINITIONS
OF MEAN ANP VARIANCE ARE ENTIRELY CONSISTENT WITH THEIR ROLE AS
CENTER ANP AVERAGE SPREAP OF THE PROBABILITIES 6IVEN BY THE PENSITY
/fr). THE PICTURE TO KEEP IN MlNP IS THIS:
67
ADDING
random variables
ONCE you KNOW THE MEAN ANP
VARIANCE OF A RANPOM VARIABLE,
WHAT CAN yOU PO WITH THEM?
WELL, FOR ONE THIN6, УОР CAN
FINP THE MEAN ANP VARIANCE OF
SOME OTUCR RANPOM VARIABLES...
/-----------------------------------------------------------------------\
FOR EXAMPLE, LOOK AT A FAIR COIN TOSS LET X » 1 IF THE COIN COMES UP
HEAPS ANP О IF IT COMES UP TAILS.
X О 1
p<X) 5 .5
ВУ NOW, yOU SHOULP BE ABLE
TO FINP THE MEAN
Б[Х] = О p(O) - 1 -p(l)
= О 4- .5
= .5
ANP THE VARIANCE
(Гг = (o-.tfp(o) 4- (\-jrfptt)
~ .Z5
NOW LET’S PLAy A SIMPLE GAMBLING- 6AME VOU ANTE UP tb.OO TO PLAy-, I
FLIP A COIN; VOU WIN IF THE COIN COMES UP HEAPS, ZERO IF TAILS. THEN
yOUR WINNINGS W ARE
W = 16?X - 6
A NEW RANPOM VARIABLE.'
WHAT ARE ITS MEAN ANP
VARIANCE?
68
A LITTLE THOUGHT SHOULP
CONVINCE /OU THAT E[w]
15 6IVEN ВУ
E[W] = £[юХ-б]
= 10E[X] - b
WHICH WORKS OUT TO
1O(O.5)-b = -1
you CAN CHECK IT US INC,
THIS TABLE:
IN GENERAL, IT IS NOT HARP
TO SHOW THAT
фх+/>] - flt[x] 4-/>
WHEN a ANP b ARE ANy
NUMBERS ANP X IS ANy RANPOM
VARIABLE. FOR THE VARIANCE,
THERE'S ALSO A GENERAL
RESULT:
сгг(аХ+Ь) =• агсгг(Х)
IN THE 6AMBLIN6 6AME ABOVE, THE POSSIBLE OUTCOMES ARE -b ANP 4, SO
IT'S CLEAR THAT THE VARIANCE OF W MUST BE GREATER THAN THE VARIANCE
OF X. IN FACT,
<W? ~ сг2(1ОХ+Ь)
- ЮОа2(Х)
= 29
boUfiPb
LIKE A
ANP
<r(w) ~ 5
6UCKE₽
To
69
you CAN AL*O APP TWO RANPOM VARIABLE* TOGETHER. FOR IN*TANCE, *UP-
PO*E WE TO** A COIN TWltG. THE NUMBER OF HEAP* ON BOTH TO**E* I*
Xj+Xj, WHERE X, ANP X2 ARE THE RANPOM VARIABLE* 6IVIN6- THE RE*ULT*
OF THE FIR*T ANP *ECONP TO**E*.
%l+%2 О 1 2
р(.^+хг)\ .25 .5
A6AIN, IT1* ЕА*У TO *EE THAT
E[X,+X2] = E[X,] + E[X2]
CPON'T A*K ABOUT THE PROBABILITY PI*TRIBUTlON OF X,+X2, BECAU*E IT
PEPENP* IN A COMPLICATE? WAY ON THE TWO ORIGINAL PI*TRIBUTION*. FOR
EXAMPLE, IF X, ANP X2 ARE BOTH THE *PINNIN6 POINTER PI*TRIBUTION, THE
HI*TO6-RAM* ACT LIKE THI*J
70
THE VARIANCE OF ТИБ *UM OF RANPOM VARIABLE* HA* A *IMPLE FORM IN
ТИБ *PE£IAL 4A*E WHEN ТИБ VARIABLE* X ANP У ARE 1НРБРБЫРБНТ. ТИБ
TEZTHNI6AL PEFINITION OF INPEPENPENzTE I* BA*EP ON THE PROBABILITY
PROPERTY P6A ANP B) - P<AMB)... BUT FOR U*. INPEPENPENT JU*T MEAN*
THAT X ANP У ARE 6-ENERATEP ВУ ШРБРБЫРБЫТ ЯБСИАЫЫМ, *U£H A*
FLIP* OF A 6OIN, ROLL* OF A PIE, ЕТЛ
OUT*I PE THE
ZA*INO, IT* HARP
TO FINP COMPLETE
INPEPENPENT...
WHEN X ANP У ARE INPEPENPENT,
ТИБ1Я VARlAMtSf APP:
<r2(*+Y) = crW+crW
IN THE 6A*E OF TWO £OIN TO**E*>
= <r2<Xf)4-<Z2<X2)
- .25+ .25
~ .5
ALL OF THI* 6AN BE 6ENERALIZEP TO THE *UM OF MANY RANPOM VARIABLE*:
s £eK]
i - / i = Z
ANP, WHEN THE Л,- ARE ALL INPEPENPENT,
i~1 i-1
71
THESE CALCULATIONS LIE AT THE
HEART OF MOST SAMPLING THEORV
ANP STATISTICS. МАМУ SUMMARIES
OF. PATA, SUCH AS THE SAMPLE
MEAN, ARE LINEAR COMBINATIONS
OF PATA <!.£., SUMS OF THE ТУРЕ
лХ + ЬУ + cZ + ... )
IN THE NEXT CHAPTER, WE WILL SEE TWO IMPORTANT EXAMPLES OF RANPOM
VARIABLES-- ONE, THE BINOMIAL, IS THE SUM OF MANy REPEATEP INPEPENPENT
RANPOM VARIABLES. THE OTHER, THE NORMAL, IS A CONTINUOUS RANPOM
VARIABLE THAT HAS A SURPRISING RELATIONSHIP TO THE BINOMIAL, ANP ANy
OTHER SUM OF INPEPENPENT RANPOM VARIABLES AS WELL
72
♦ Chapter 5 +
A TALE OF TWO
DISTRIBUTIONS
NOW WE LOOK AT TWO IMPORTANT EXAMPLES OF
RANPOM VARIABLES, ONE PISCRETE ANP ONE CONTINUOUS.
WE BE6-IN WITH THE PISCRETE ONE, CALLEP THE BINOMIAL RANPOM VARIABLE.
SUPPOSE WE HAVE A RANPOM PROCESS WITH JUST TWO PO44IBLE OUTCOMES;
A HEAPS-OR-TAILS COIN ТОМ, A WIN-OR-LOSE FOOTBALL 6AME, A PASS-OR-
FAIL AUTOMOTIVE SMO6- INSPECTION. WE ARBITRARILY CALL ONE OF THESE
OUTCOMES A SUCCESS ANP THE OTHER A FAILURE.
WHAT WE PO IS TO REPEAT THIS EXPERIMENT... WELL, REPEATEPLY. SUCH A
REPEATABLE EXPERIMENT IS CALLEP A
Bernoulli
trial,
PROVIPEP IT HAS THESE CRITICAL
PROPERTIES:
V THE RESULT OF EACH TRIAL
MAY BE EITHER A SUCCESS OR
A FAILURE
2? THE PROBABILITY p Op
SUCCESS IS THE SAME IN
EVERY TRIAL
3) THE TRIALS ARE INDEPENDENT:
THE OUTCOME OF ONE TRIAL HAS
NO INFLUENCE ON LATER OUTCOMES-
74
STARTING WITH A BERNOULLI TRIAL, WITH PROBABILITY OF p, LET’S
BUILP A NEW RANPOM VARIABLE BY REPEATING THE BERNOULLI TRIAL.
The
binomial
random
variable
X IS THE HUMBCR OF
WCC&ttt IN n REPEATEP
BERNOULLI TRIALS WITH
PROBABILITY p OF SU&TESS.
AN EXAMPLE OF A BINOMIAL RANPOM VARIABLE IS THE NUMBER OF HEAPS
(SUCCESSES? IN TWO FLIPS OF A £OIN. HERE n -2 ANP p~.5
k* NUMBER
OF SU^ESSES
РгбХ-ZJ .25 .5 .25
О 1 2
ANOTHER EXAMPLE IS PE MERE’S FIRST GAMBLE’. TOSSING A SINGLE PIE
FOUR TIMES IN A ROW. SU66ESS MEANS ROLLING A 6. THE PISTRIBUTION IS:
UM- THE PtSTRlBOWN
VlMAT 16 TU£
PROBABILITY Op
R0LLIH6 b'**
1Ц 4 ROL15 ?
75
IN GENERAL, WHAT’S ТИС PROB-
ABILITy PISTRIBUTION OF ТИС
BINOMIAL FOR АМУ PROBABILITY
p ANP NUMBER OF TRIALS n? A
PROBABILITY CALCULATION 6IVES
THE ANSWER: THE PROBABILITY
OF OBTAINING k. SUCCESSES IN
П TRIALS, IS
HERE $), REAP *n CHOOSE k? IS THE BINOMIAL COCFFlClCNT. IT COUNTS
ALL POSSIBLE WAYS OF 6£TTIN£ k. SUCCESSES IN n TRIALS. EACH INPIVIPUAL
SEQUENCE OF k SUCCESSES ANP n-k FAILURES HAS PROBABILITY рЧ^-рУ'*,
ВУ THE MULTIPLICATION RULE. THERE ARE (%) OF THESE SEQUENCES.
THE FORMULA FOR IS
WHERE
n! - ... Xi
ANP O! IS TAKEN TO BE 1. FOR INSTANCE,
(*), THE NUMBER OF POSSIBLE WAYS TO
CHOOSE TWO LETTERS FROM A SET OF
FOUR LETTERS, IS
шт
AB Ac AD
BC BD CD
fA} ~ 41 _ 24 , /
" 2'21 " 4 " 6
76
ANOTHER VIEW OF THE BINOMIAL COEFFICIENTS 1$ IN PAMAL'4 TRIAU6LC.
EACH ENTRY 1$ THE SUM OF THE TWO NUMBERS JUST ABOVE IT.
1 10 45 120 210 252 210 120 *45 Ю 1
1 11 55 165 330 462 462 330 165 55 11 1
1 12 66 220 495 792 924 792 495 220 66 12 1
ETC.
TO FINO (%), JUST COUNT POWN TO ROW n ANP OVER TO ENTRY k
(REMEMBERING ALWAYS TO START COUNTING FROM ZERO/
<____________________________________________________,_________________J
WHEN p ~ .4, THE BINOMIAL'S
PROBABILITY PISTRIBUTION IS
PERFECTLY SYMMETRICAL. FOR
6 COIN FLIPS, FOR INSTANCE, ITS
k. * #HEAPS О 1 2 3 4 5 6
(4)fc (ift № at» (ifb (if
WITH THIS
HISTOGRAM:
1
77
THE MEAN ANP VARIANCE OF THE
BINOMIAL PISTRIBUTION ARE
yU - Z7/?
€Гг ~ np(V-p)
NOTE THAT THE MEAN MAKES
INTUITIVE SENSE: IN n BERNOULLI
TRIALS, THE ЕХРЕОГЕР NUMBER OF
SUdZESSES SHOULP BE np. THE
VARIANCE FOLLOWS FROM THE
FA£T THAT THE BINOMIAL IS THE
SUM OF n INPEPENPENT BERNOULLI
TRIALS OF VARIANCE p(y-pY
THE PARAMETERS OF THE BINOMIAL PISTRIBUTION ARE П ANP p. THE
PISTRIBUTION, MEAN, ANP VARIANCE PEPENP OHLY ON THESE TWO NUMBERS.
TABLES OF THE BINOMIAL PISTRIBUTION APPEAR IN MOST TEXTBOOKS ANP
COMPUTER PROGRAMS. HERE IS A TABLE FOR 77 = IP.
VALUE* OF Pr(X~4)
к
01 234 56789 10
.1 0.349 0.387 0.194 0.057 0.011 0.001 0.000 0.000 0.000 0.000 0 000
.25 0.056 0.188 0.282 0.250 0.146 0.058 0.016 0.003 0.000 0.000 0.000
P .50 0.001 0.010 0.044 0.117 0.205 0.246 0.205 0.117 0.044 0.010 0.001
.75 0.000 0.000 0.000 0.003 0.016 0.058 0.146 0.250 0.282 0.188 0.056
.9 0.000 0.000 0.000 0.000 0.000 0.001 0.011 0.057 0.194 0.387 0.349
is
BUT CALCULATING
THESE THINGS FOR
LARGE VALUES OF n
CAN BE A PAIN... OR AT
LEAST, IT WAS BACK IN
THE 10th CENTURY,
WHEN JAMES
BERNOULLI ANP
ABRAHAM PE MOIVRE
WERE TRYING TO PO
IT WITHOUT A
COMPUTER.
DEPLOYING A NEWLY
INVENTEP WEAPON, THE
CALCULUS, PE MOIVRE
SHOWEP THAN WHEN p =.5,
THE BINOMIAL PISTRIBUTION
WAS CLOSELY
APPROXIMATED BY A
CONTINUOUS PENSlTY
FUNCTION WHI^H LOULU BE
PES6RIBEP VERY SIMPLY.
TO SEE HOW THIS WORKS, IMAGINE THE BINOMIAL PISTRIBUTION WITH = .5
ANP Z? VERY LAR6E-A MILLION, SAY...
ттТ1ППТ1ПТП'1ТПТ|Т1Т[Г
7?
NOW, SAIP PEMOIVRC, SLIPS THIS
GRAPH OVER, 50 ITS MEAN IS ZERO.
5QUASH THE CURVE ALONG THE X AXIS
UNTIL THE STANPARP PEVIATION
BECOMES 1, WHILE STRETCHING IT
ALONG THE у AXIS TO KEEP THE AREA
UNPER IT EQUAL TO 1.
ТИБ RESULT IS VERV CLOSE TO A SMOOTH, SYMMGTRItAl., BGLL-SUAPGP
£URV£, WHICH PE^OIVRE SHOWEP WAS GIVEN ВУ THE SIMPLE FORMULA:
THIS FUNCTION IS CALLEP THE
standard normal
distribution.
(б M USEFUL MATHEMATICAL
CON6TAUT APPROXIMATELY
ft^UALTO 2-7'8.)
(CONVINCE VOURSELF THAT THIS FUNCTION REALLV HAS A BELL-SHAPEP
GRAPH. FOR Z FAR FROM ZERO, /(z) IS VERy NEARLy ZERO-IT HAS A BIG
PENOMINATOR-, IT'S SVMMETRICAL, SINCE /?z) « /(-z), ANP IT HAS A
MAXIMUM AT Z = O.)
THE PISTRIBUTION IS CALLEP THE
6TAHPARP NORMAL BECAUSE ALL
THAT SQUASHING ANP STRETCHING
WAS SPECIALLV ARRANGEP TO GIVE
IT THESE SIMPLE PROPERTIES,
WHICH WE PRESENT WITHOUT
PROOF:
/л ~ О
<r 1
Bo
TO SUMMARIZE РЕ MOIVRE,
IF YOU “NORMALIZE" THE
BINOMIAL PISTRIBUTION
WITH p » 1/2—LC., CENTER
IT ON ZERO ANP MAKE ITS
STANPARP PEVIATION » 1,
THEN IT aOSELX FITS
THE STANPARP NORMAL
PISTRIBUTION
OTHER NORMALS, WITH PIFFERENT MEANS ANP VARIANCES, ARE OBTAINEP ВУ
STRETCHING ANP SLIPING THE STANPARP NORMAL IN GENERAL, WE WRITE THE
FORMULA
THIS GIVES A SYMMETRIC
BELL'SHAPEP PISTRIBUTION
6ENTEREP ON THE MEAN jn
WITH THE STANPARP
PEVIATION <r.
HERE ARC TWO PIFFERENT NORMALS WITH THE REGIONS WITHIN THEIR
STANPARP PEVIATIONS SHAPEP.
4 WITH SMALL <r2
4 WITH LARGE <rt
01
pg MOIVRE PROVE? THAT THE STANPARP NORMAL FITS THE (NORMALIZE?)
BINOMIAL WITH p .5, ВОТ, IN FACT, IT WORKS FOR ДЛ/У VALUE OF p.
GENERALLy; FOR АМУ
VALUE OF p, THE
BINOMIAL PISTRIBUTION
OF n TRIALS WITH
PROBABILITY p IS
APPROXIMATEP BY THE
NORMAL CURVE WITH
/< = np ANP
a ~ пр(У - /?).
THIS IS ACTUALLY A
LITTLE STRAN6-E. ALL
NORMALS ARE
SYMMETRICAL ANP
BELL SHAPEP... BUT, AS
WE SAW, BINOMIAL
PlSTRIBUTlONS ARE
HOT ^УMETRICAL
WHEN p*S.
BUT IT TURNS OUT THAT AS n 6-ETS LAR6-E, THE BINOMIAL'S ASYMMETRY IS
OVERWHELMED AS YOU SEE IN THIS EXAMPLE:
82
IN FACT, PEMOIVRE'6 PISCOVERY ABOUT THE BINOMIAL 16 A SPECIAL CASE OF AN
EVEN MORE 6-ENERAL RESULT, WHICH HELPS EXPLAIN WHY THE NORMAL IS SO
IMPORTANT ANP WIPE6PREAP IN NATURE. IT IS THIS:
"Fuzzy
Central Limit
Theorem":
PATA THAT ARE
INFLUENCEP ВУ МАМУ
SMALL AMP UNRELATED
RANDOM EFFECT* ARE
APPROXIMATELY NORMALLY
DISTRIBUTED.
THIS EXPLAINS WHY THE NORMAL IS EVERYWHERE; STOCK MARKET
FLUCTUATIONS, STUPENT WEIGHTS, YEARLY TEMPERATURE AVERAGES, S.A.T.
SCORES: ALL ARE THE RESULT OF MANY PIFFERENT EFFECTS. FOR EXAMPLE,
A STUPENT'S WEIGHT \*> THE RESULT OF GENETICS, NUTRITION, ILLNESS, ANP
LAST NIGHT'S BEER PARTY. WHEN YOU PUT THEM ALL TOGETHER, YOU 6>ET
THE NORMAL/ (REMEMBER, THE BINOMIAL IS THE RESULT OF n INPEPENPENT
BERNOULLI TRIALS.)
BS
THE Z TRANSFORMATION
z “* cr
dHAN6ES A NORMAL
RANPOM VARIABLE WITH
MEAN /< ANP STANPARP
PEVIATION cr INTO A
5TAHPARP NORMAL
RANPOM VARIABLE WITH
MEAN О ANP STANPARP
PEVIATION 1.
THEN ALL WE NEEP TO FINP PROBABILITIES FOR АНУ NORMAL PISTRIBUTION IS
THE SINGLE TABLE FOR THE STANDARD NORMAL f(z).
г -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 -1.6
F(z) 0.006 0.008 0.011 0.014 0.018 0.023 0.029 0.036 0.045 0.055
Z -1.5 “1.4 “1.3 -1.2 “l.1~1.0 -0.9" -0T -0.7 -0.6
F(z) 0.067 0.081 0.097 0.115 0.136 0.159 0.184 0.212 0.242 0.274
Z -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4
F(z) 0.309 0.345 0.382 0.421 0.460 0.500 0.540 0.579 0.618 0.655
Z (15 0.6 0.7 — 0?8 “ 6.9 1.0 1.1 1”2 13 1.4
F(z) 0.691 0.726 0.758 0.788 0.816 0.841 0.864 0.885 0.903 0.919
Z 1.5 1.6 1.7 18 1.9 2.0 2.1 2.2 2.3 2.4
F(z) 0.933 0.945 0.955 0.964 0.971 0.977 0.982 0.986 0.989 0.992
z 2.5~
F(z) 0.994
HERE F(a) = Pr(z $ л), THE AREA UNPER THE PENSlTY dURVE TO THE LEFT
OF Z — Л.
(WE dAN ALSO
6-RAPH THE
dURVE
у
THE
CUMULATIVE
PROBABILITY.
IT LOOKS
LIKE THIS.;
и
THE TABLE ALLOWS W> TO FINP THE
PROBABILITY OF Z BEIN6- IN ANY INTERVAL
a £ z $ b. IT 15 JUST THE PIFFERENCE
BETWEEN THE AREAS F(b~) ANP F(a).
SO, FOR EXAMPLE,
Prf-1<z<1) = F(V-F(-f)
- -6413-1567
PrO^2)^ l-f(2)
U5IN0 THE SUBSTITUTION
Z = , WE 6AM USE
THE SAME TABLE TO FINP
PROBABILITIES FOR OTHER
NORMAL PlSTRIBUTlONS.
NOW IT’S -JUST” ALGEBRA.
Pv(X>|70^ =
Prfx-M > I7o-I9g\ .
' <r j,o ' ~
Mz>i)
FOR EXAMPLE, SUPPOSE STUPENT WEIGHTS ARE
NORMALLY PISTRIBUTEP WITH A MEAN yU- Ko
POUNPS ANP STANPARP PEVIATION СТ. 2P;
THAT'S t-ffl), WHI6H WE 6AN REAP FROM THE
TABLE AS I - .«413 « 1567
AREA- .\^Q1
\90 V70
A LITTLE LESS THAN ONE STUPENT IN SIX TIPS
THE SCALES ABOVE I7P POUNPS.
THE GENERAL RULE FOR £OMPUTIN6 NORMAL PROBABILITIES IS THEREFORE:
Pr(a € X« />) =
HOW BAZK TO PE MOIVRE
ANP HlS BINOMIAL
APPROXIMATION... LET’S
LOOK AT A BINOMIAL
PISTRIBUTION WITH П ^25
TRIALS ANP p^5 (25
(OIN FLIPS, SAY). WE (AN
COMPUTE (OR LOOK UP IN
A TABLE? ANY PROBABILITY,
FOR EXAMPLE, Pr(* $ 14).
IT IS -7970 EXACTLY.
NOW CALCULATE A NORMAL RANPOM VARIABLE X* WITH THE SAME MEAN
p~np » (25)65) - 125 ANP STANPARP PEVIATION cr ~ np(\-p} ~ 25.
115 U
AH, BUT WE (AN PO BETTER/
IF YOU LOOK (LOSELY AT THE
FIRST HISTOGRAM, YOU SEE
THE BARS ARE СБЫТБИБР ON
THE NUMBERS. THIS MEANS
Pr(X*$ 14) IS ACTUALLY THE
AREA UNPER THE BARS LESS
THAN % = 14.9- WE NEEP TO
AMOUNT FOR THAT EXTRA 5,
ANP IN FA(T,
Pr(X*$ 145) = Pr(Z^ .0)
.7001
A VERY 6OOP APPROXIMATION
TO .7670 IN PEEP/
THAT LITTLE EXTRA 5 WE
APPEP 15 £ALLEP THE
continuity
correction.
WE HAVE TO INdLUPE IT
TO 6>ET A 6OOP
6ONTINUOU5
APPROXIMATION TO OUR
Pl5dRETE BINOMIAL
RANPOM VARIABLE X. ГГ5
5UMMARIZEP ВУ THI5 ONE
HIPEOU5 FORMULA-
v Vftpa-p) Vnp(i-p) /
WHEN 15 THI5 APPROXIMATION *6OOP ENOUGH?” FOR 5TATI5TIOAN5, THE
RULE OF THUMB 15; WHENEVER n 15 BI6- ENOUGH TO MAKE THE NUMBER OF
EXPE^TEP 5U66E55E5 ANP FAILURE5 BOTH 6-REATER THAN FIV&
Пр > 5 and n(t-p) 5
YOU dAN 5EE FROM THE5E HI5TO6-RAM5 THAT THE FIT WHEN p Л1 15
MEPIO^RE OR WOR5E UNTIL n REMHE5 40, MAKING np^5
n-2t p~ 0.1
n^5ot p= Qi
ИМО, 0.1
H
WHAT'S 50 6-REAT ABOUT THIS NORMAL APPROXIMATION? THE BINOMIAL
PISTRIBUTION OCCURS COMMONLY IN NATURE, ANP IT ISN'T HARP TO UNPER-
STANP, BUT IT CAN BE TIRESOME TO CALCULATE.
THE NORMAL WHICH APPROXIMATES IT МАУ BE LESS INTUITIVE, BUT IT'S VERY
EASY TO USE. THE ^'TRANSFORM CONVERTS ЛА/У NORMAL TO THE 6TANPARP
MORTAL, ALLOWING US TO REAP PROBABILITIES STRAIGHT OUT OF A SINGLE
NUMERICAL TABLE.
♦Chapter 6*
SAMPLING
ВУ NOW, AFTER А 6ТЕАРУ PIET OF COINI PICE, ANP ABSTRACT
IPEA1 УСУ МАУ BE WONPERIN& WHAT ALL THIS STATISTICAL
EQUIPMENT WEVE BEEN BUILPIN6- HAS TO PO WITH THE R£AL
WORLD- WELL, NOW WE’RE FlNALLy 6OIN^ TO FINP OUT...
IN THIS CHAPTER, WE BE6-IN LOOKING AT THE REAL BUSINESS OF STATISTICS,
WHICH IS, AFTER ALL, TO SAVE PEOPLE Т1МБ ANP MOAtfX PEOPLE HATE TO
WASTE TIME POIN6- UNNEtSMARY WORK, ANP ONE THIN6- 5TATI6TlC^ CAN PO
TELL U5 ЕХАСТЕУ HOW LA/У WE CAN AFFORP TO BE.
THE PROBLEM WITH THE WORLP & THAT THE COLLECTIONS OF STUFF IN IT
ARE SO LARGE, IT’S HARP TO GET THE INFORMATION WE WANT:
PICKLES: WHAT’S THEIR
AVERAGE LENGTH?
VOTING POPULATIONS:
WHAT PERCENTAGE
FAVORS EACH CANPIPATE?
THE PICKLE-JAR MAKERS
NEEP TO KNOW!
THE INPUSTRIOUS,
HARP-WORKING,
SIMPLE-MINPEP
BEAVERLIKE WAY TO
ANSWER THESE
QUESTIONS WOULP
BE TO MEASURE
EVERY SINGLE
PICKLE IN THE
WORLP 6SAY? ANP
PO SOME
ARITHMETIC.
BUT WE AREN’T BEAVERS—WE’RE
STATIfnOANf! WE’RE LOOKING
90
OUR METHOP 15 TO TAKE
A 6AMPL5... a
RELATIVELY SMALL
SUBSET OF THE TOTAL
POPULATION, THE WAY
POLLSTERS PO AT
ELECTION TIME.
AN OBVIOUS QUESTION IS: HOW BIG A SAMPLE PO WE HAVE TO TAKE TO GET
EVEN KNOW IT
WAS ON THE
BALLOT!
ANP THE ANSWER,
WHI^H YOU SHOULP
INSCRIBE IN YOUR
BRAIN FOREVERMORE,
WILL TURN OUT TO
BE: IF n IS THE
NUMBER OF ITEMS IN
THE SAMPLE, THEN
EVERYTHING IS
GOVERNEP BY
GOVERNEP ВУ
-7=? PIPNT
91
SAMPLING
DESIGN
BEFORE POIN6- THE NUMBERS, WE
WOULP POINT OUT THAT THE
QUALITY OF THE SAMPLE IS AS
IMPORTANT AS ITS $IZ£. HOW PO
WE ASSURE OURSELVES THAT
WE’RE CHOOSING A
REPRESENTATIVE SAMPLE?
THE KLCCTIOM PROtCH
IT4CLF IS CRITICAL FOR
EXAMPLE, A VOTER SURVEY THAT
SYSTEMATICALLY EXCLUPEP BLACK
PEOPLE WOULP BE WORTHLESS,
ANP THERE ARE A HOST OF
OTHER WAYS TO RUIN, OR BIAS, A
SAMPLE.
NOT TO PROLONG THE MYSTERY, THE WAY TO £ET STATISTICALLY PEPENPABLE
RESULTS IS TO CHOOSE THE SAMPLE AT rOndolYI.
92
THE SIMPLE RANDOM SAMPLE
SUPPOSE WE HAVE A LAR6-E
POPULATION OF OBJECT* ANP A
PROCEPURE FOR SELECTING П OF
THEM. IF THE PROCEPURE
ENSURE* THAT ALL PO44IBLC
4A&PLS4 OF n OBJECT* ARE
SQUALLY LIKSLY, THEN WE CALL
THE PROCEPURE A Simple
random sample.
THE *IMPLE RANPOM *AMPLE HA* TWO PROPERTIE* THAT MAKE IT THE
*TANPARP AGAINST WHICH WE MEA*URE ALL OTHER METHOP*:
)UNBIA*EP: EACH UNIT HA* THE *AME
CHANCE OF BEIN6- CHO*EN-
At INPEPENPENC& SELECTION OF ONE
UNIT HA* NO INFLUENCE ON THE
*EL£CTION OF OTHER UNIT*.
—
UNFORTUNATELY, IN THE REAL WORLP, COMPLETELY UNBIA*EP, INPEPENPENT
*AMPLE* ARE HARP TO FINP. FOR IN*TANCE, *URVEYIN6> VOTER* BY RANPOMLY
PIALIN6 TELEPHONE NUMBER* I* BIA*EP- IT IGNORE* VOTER* WITHOUT A
TELEPHONE ANP OVER*AMPLE* PEOPLE WITH MORE THAN ONE NUMBER.
л» Л
EQUIVALENTLY, WE CAN PUT ALL THE
NAME* ON CARP* ANP PULL n OF
THEM OUT OF A PRUM.
IT’* theoretically po**ible
TO 6>ET A RANPOM *AMPLE BY
BUILPIN6 А 5A/APUM6
FRAME: A LI*T OF EVERY
UNIT IN ТИБ POPULATION. BY
U*IN6 A RANPOM NUMBER
GENERATOR, WE CAN PICK n
OBJECT* AT RANPOM.
BUT THI* I* NOT ALWAY* ЕА*У. MAKING TME FRAME МАУ BE PROHIBITIVELY
CO*TLY, CONTROVER*IAL, OR EVEN IMPO**IBL£. FOR EXAMPLE, AN E.PA. WATER
QUALITY *TUPY NEEPEP A * AM PLINC, FRAME OF LAKE* IN THE U.*., *0 THEN
*ОМ£ВОРУ MA* TO PECIPE-.
ARE THERE OTHER WAY* TO *AMPLE THAT ARE MORE CFFICICMT ANP CO$T-
CFFCtTIVC THAN A *IMPLE RANPOM *AMPLE? Y£*-IF YOU ALREAPY KNOW
*0METHIN6, ABOUT TME POPULATION. FOR IN*TANCE...
*4
Stratified
SAMPLING; PIVIPE THE
POPULATION UNITS INTO
HOMOGENEOUS GROUPS
STRATA) ANP PRAW A
SIMPLE RANPOM SAMPLE
FROM EACH GROUP.
6>arlic Luis
pickled
peppers
6UCRKi(dS
PGLIGH
UAMPUP^EI?
LETTUCE
G1AUT fe
Dills
KfiGUEft Dills
FOR EXAMPLE, THE POPULATION OF ALL PIOO.& CAN BE STRATIFIEP ВУ
ТУРС OF PICKLC. WITHIN EACH ТУРЕ OR STRATUM, THE SIZE SHOULP BE
LESS VARIABLE.
<____________________________________________________________________>
Cluster
SAMPLING GROUPS THE POPULATION INTO SMALL
CLUSTERS, PRAWS A SIMPLE RANPOM SAMPLE OF
CLUSTERS, ANP OBSERVES EVERYTHING IN THE SAMPLEP CLUSTERS. THIS CAN BE
COST-EFFECTIVE IF TRAVEL COSTS BETWEEN RANPOMLV SAMPLEP UNITS IS HIGH.
AN EXAMPLE IS А С1ТУ
HOUSING SURVEy WHICH
PIVIPES А С1ТУ INTO
BLOCKS, RANPOMLy
SAMPLES THE BLOCKS,
ANP LOOKS AT EVERy
HOUSING UNIT IN EACH
SAMPLEP BLOCK.
SAMPLING STARTS WITH A RANPOMLY
** J *1 * CHOSEN UNIT ANP THEN SELECTS EVERY k™
UNIT THEREAFTER. FOR INSTANCE, A HIGHWAY TRAFFIC 4TUPY MIGHT CHECK
EVERY HUNPREPTH CAR AT A TOLL BOOTH. THIS PLAN IS EASY TO IMPLEMENT
ANP CAN BE MORE EFFICIENT IF TRAFFIC PATTERNS VARY SMOOTHLY OVER TIME.
excuse ме... vJooLD you
MIKP AFT/
OR «1КТУ 6?U^TIOsl^>?
Word of warning #1
MOST STATISTICAL METHOPS PEPENP ON
TME INPEPENPENCE ANP LACK OF BIAS OF
TME SIMPLE RANPOM SAMPLE. TME RESULTS
AHEAP APPLY TO THE SIMPLE RANPOM
SAMPLE ONLY. FOR OTHER SAMPLING
PROCEPURES, TME RESULTS MUST BE
MOPIFIEP. TME PETAILS APPEAR IN
SPECIALIZEP SAMPLING TEXTBOOKS ANP
COMPUTER ALGORITHMS.
9A
Word of warning #2:
WITHOUT RANPOMIZEP
PESIGN, THERE CAN BE NO
PEPENPABLE STATISTICAL
ANALYSIS, NO MATTER
HOW IT IS MOPIFIEP. THE
BEAUTY OF RANPOM
SAMPLING IS THAT IT
‘STATISTICALLY
GUARANTEES" THE
ACCURACY OF THE SURVEY.
DoH’T VJORfW’
A COMMONLY USEP METHOP IS ESPECIALLY PRONE TO BIAS: IT'S CALLEP AN
Opportunity SAMPLE. AVOIPING ALL
THE BOTHER OF PESIGNING A
PROCEPURE, THE OPPORTUNITY
SAMPLER JUST GRABS THE
FIRST n POPULATION UNITS
TO COME ALONG
WE VOLUUTEEQEbf
A CLASSIC EXAMPLE IS SHERE HITE'S BOOK, WOMGA/ ЛЛ/Р LOV£. WO ООО
QUESTIONNAIRES WENT TO WOMEN’S ORGANIZATIONS CAN OPPORTlfMITY
fMPLC), ONLY WERE FILLEP OUT ANP RETURNEP (R&POHK FMS).
SO HER "RESULTS" WERE BASEP ON A SAMPLE OF WOMEN WHO WERE HIGHLY
MOTIVATEP TO ANSWER THE SURVEY'S QUESTIONS, FOR WHATEVER REASON.
Q1
SAMPLE SIZE
& standard error
HOW LET 6 6CT POWN TO
BRA4S TACKB... REAL BRASS
TAZKS, THAT IS- SUPPOSE THE
BERNOULLI TACK FACTORY IS
6HURNIN6 OUT BRASS TAZKS,
SOME OF WHIZH, INEVITABLY,
ARE PEFECTIVE-
THE ASTUTE REAPER WILL REZO^NIZE THIS AS A BERNOULLI SYSTEM- EAZH
NEW TAZK IS THE OUTZOME OF A BERNOULLI TRIAL WITH SOME PROBABILITY p
OF SUZZESS (!.£., BEIN6. PEFEZT-FREE? ANP PROBABILITY \-p OF FAILURE
6I.E-, BEIN6- PEFEZTIVEZ
WE THINK OF THIS SITUATION AS IF THERE WERE A RIPPEN BUT REAL
“BERNOULLI MACHINE" WHOSE PROBABILITY p bCNQRM THE OUTCOMES WE
OBSERVE IN THE SO-6ALLEP “REAL WORLP."
9B
SINZE ТИБ BERNOULLI
MACHINE 15 INVISIBLE, WE
PONT KNOW WHAT p 15,
BUT WE’P LIKE TO FINP
OUT. 50 WE TAKE A
Я4А/РОМ SAMFAf OF n
TAZK5, ANP FINP THAT
X OP THEM ARE O.K.
NOW THE PROPORTION OF 5U66E55E5 IN THE SAMPLE 5H0ULP BE SOMEWHERE
AROUNP p... 50 WE £ALL IT p, PRONOUNCE? ‘P-HAT."
p 15 THE NUMBER OF 5U&X55E5 % IN THE SAMPLE, PIVIPEP ВУ THE
SAMPLE SIZE П. FOR EXAMPLE, IF p WAS .05, ANP WE 5AMPLEP n -WOO
TAZK5, МАУВЕ WE FOUNP 032 6O0P ONE5, MAKING p • .032.
WE A5K-. MOW 6OOP
15 ТИ15 ESTIMATE?
ANP WE AN5WER WITH
ANOTHER QUESTION: WHAT
P0E5 THE FIRST
QUESTION MEAN?
№ CAN'T KNOW ТИБ PRECISE PlFFERENCE BETWEEN p ANP p, BECAUSE WE
PONT KNOW THE VALUE OF p. TME REAL QUESTION IS THIS: IF WE TOOK AAW
fAMPLtt OF 10OO TA0K4 ANP OBSERVEP p FOR EACH SAMPLE, MOW WOULP
THOSE VALUES OF p BE PI5TRIBUTEP NROVHV p?
IN FACT, THESE p VALUES ARE LOOKING MORE ANP MORE LIKE А RAA/POM
KAR/AW-E; TME SELECTION OF TME Л-UNIT SAMPLE IS A RANPOM EXPERIMENT,
ANP TME OBSERVATION p IS A NUMERICAL OUTCOME!
I AlA B£COIAI№
fNUC?HTEN£P NOV»—
I KNEVl \f yJODUV
NOT e& PMHLE^...
TO BE PRECISE, IF X IS
TME NUMBER OF
SUCCESSES IN TME SAMPLE,
THEN X IS NOTHING BUT
OUR OLP FRIENP TME
BINOMIAL RANPOM
VARIABLE СП TRIALS,
PROBABILITY p)... ANP WE
PEFINE TME OBSERVEP
PROPORTION TO BE TME
RANPOM VARIABLE
X
n
RANPCM VARIABLE,
LITTLE -p, IT^
MAUX FOR A PARTICULAR
^AM?L6!
516 P THE-
^KNOWING ALL ABOUT X, WE QUICKLY COHO-UPC A FEW FATTS ABOUT P:
1) THE MEAN OF P & £[ P] = p
2? THE STANPARP PEVIATION OF P IS
ж vs
V7T
3? FOR LARGE П, P IS
APPROXIMATELY NORMAL.
ANP THERE YOU WAVE IT ALLI THE OBSERVEP VALUES OF P WILL BE <XNT£REP
ON p (NOT SURPRISINGLY?, ANP THEIR STANPARP PEVIATION, OR SPREAP, IS
PROPORTIONAL TO TWAT MAGI6 HUMBER WE MEHTIONEP AT TWE BEGINNING OF
THE CHAPTER:
ANP, SINCE P IS NEARLY NORMAL, WE CAN USE OUR RULE OF THUMB TO
CONCLUPE THAT APPROXIMATELY 60% OF ALL ESTIMATES WILL FALL WITHIN ONE
STANPARP PEVIATION OF THE TRUE VALUE p.
t'lA MEARuy
7 W-
GOING BACK ТО THE TACKS,
WITH п * WOO ANP р =.95,
WE GET A STANPARP
PEVIATION OF
= .0113
\ooo
50 WE EXPECT ABOUT 59%
OF OUR ESTIMATES TO FALL
IN THE NARROW INTERVAL
.0307 « Д £ .0613
THE STANPARP PEVIATION OF P IS A MEASURE
of the sampling error.
AS WEVE SEEN, FOR THE BINOMIAL P, THIS
SAMPLING ERROR IS INVERSELY PROPORTIONAL
TO V7F. INCREASING THE SAMPLE SIZE ВУ A
FACTOR OF 4 REPUCES THE SPREAP <r(P) ВУ A
FACTOR OF 2.
ALR^APV
AT Л-ЮО,
W SL^T(-p)
\S> PCW
SAMPLE SIZES FOR TACKS, p * 0.95
n 1 4 U 25 \OO wpoo
1 г 4 5 \O 100
<r(.p) .357 .1195 .099 .071 .0357 .0035
<r(P)^
LINGUISTIC NOTE; AN С5ТШТС IS A SINGLE MEASURE OR OBSERVATION. AN
ESTIMATOR IS A RULE FOR GETTING ESTIMATES. IN THIS CASE, THE ESTIMATOR
IS THE RANPOM VARIABLE P- — .
n
юг
MOST OF STATISTICS INVOLVES THE 4-STEP PROCESS WE'VE JUST WALKEP
THROUGH;
PEFJNE POPULATION WITH UNKNOWN
PARAMETER
FINP AN ESTIMATOR, IT* THEORETICAL
SAMPLING PISTRIBUTION ANP
STANPARP PEVIATION.
REPORT THE RESULT ANP ITS
STATISTICAL OR SAMPLING ERROR.
ACTUALLY PRAW A RANPOM SAMPLE
ANP FINP THE ESTIMATE.
IOS
Sampling Distribution
ef the MEAN
JAR MANUFACTURERS WOULP LIKE TO KNOW THE AVCRA6S LENGTH OF A
PICKLE WITHOUT EXAMINING EVERY CUCUMBER IN CALIFORNIA. THEY RANPOMLY
SELECT n PICKLES ANP MEASURE THEIR LENGTHS Xn.
BY NOW YOU MAY BE
USEP TO THE IPEA
THAT EACH Л, IS A
RANPOM VARIABLE:
THE NUMERICAL
OUTCOME OF A
RANPOM EXPERIMENT.
IF jj. IS THE (UNKNOWN)
MEAN PICKLE LENGTH, ANP
cr IS THE STANPARP
PEVIATION OF THE PIEKLC
LENGTH PISTRIBUTION,
THEN
В[Х/] 'Л4
<r(X() = cr
FOR EVERY i (BECAUSE %,
COULP HAVE BEEN THE
LENGTH OF ANY PICKLE).
IOA
6TRANkE, wow
MUCH VIE KNOW ABOUT
RWOfA VARIABLE
WE PlPrt'T EVEN KNOW
WERE РДНРОМ VARIABLE*
A MINUTE Д6Ю-
/-------------------------------
NOW WE LOOK AT THE SAMPLE
MEAN: THE AVERAGE LENGTH OF
THE SELEOTEP PICKLES. IT'S A
NEW RANPOM VARIABLE &IVEN
ВУ:
A” i 4- 4- ... 4- К n
* n
to BEFORE, WE'P LIKE TO KNOW ’MOW FLOSS’ THIS IS TO /г, MEANING, IF
THIS SAMPLING WERE PONE MANY TIMES, WMATS THE PISTRIBUTION OF X?
BECAUSE WE KNOW ABOUT Xv X2....ANP X„, WE ALSO KNOW THAT
IO?
ГГ TURNS OUT THAT X IS ALSO APPROXIMATELY NORMAL' THIS FAMOUS
RESULT IS CALLEP THE
CENTRAL LIMIT
THEOREM
IT SAYS: IF ONE TAKES RANPOM SAMPLES
OF SIZE Z7 FROM A POPULATION OF MEAN
jbi ANP STANPARP PEVIATION cr, THEN, AS
n 6-ETS LAR6-E, X APPROACHES THE
NORMAL PISTRIBUTION WITH MEAN /z
ANP STANPARP PEVIATION . THEN
WHAT IS REMARKABLE ABOUT THIS? IT SAYS THAT RE6ARPLESS OF THE SHAPE
OF THE ORIGINAL PISTRIBUTION <|N THIS CASE, OF PICKLE LENGTHS?, THE
TAKING OF AVEWffS RESULTS IN A NORMAL. TO FINP THE PISTRIBUTION OF
X, WE NEEP KNOW ONLY THE POPULATION MEAN ANP STANPARP PEVIATION.
THE THREE PROBABILITY PENSITIES ABOVE ALL HAVE THE SAME MEAN ANP
STANPARP PEVIATION. PESPITE THEIR PIFFERENT SHAPES WHEN n*W, THE
SAMPLING PISTRIBUTIONS OF THE MEAN, X, ARE NEARLY IPENTICAL.
The t-distribution
AMAZING AS THE CENTRAL LIMIT THEOREM IS, IT HAS AT LEAST TWO PROBLEMS.
BUT SAMPLE SIZES ARE OFTEN
SMALL, ANP cr IS USUALLY
UNKNOWN. CERTAINLY, IN THE CASE
OF THE PICKLES, WE HAVE НО IPEA
HOW WIPELy THEIR LENGTHS VARY
AROUNP THE AVERAGE.
\___________________________________
ONE: ГГ PEPENPS ON A LAR6-E
SAMPLE SIZE
TWO- TO USE IT, WE NEEP TO
KNOW cr. THE STANPARP
PEVIATION.
WHAT WE CAN PO IN THIS CASE IS TO Г5Т/А4ТГ cr BY TAKING THE 4TAHPARP
PEVIATIOH OF THE SAMPLE, WHICH, VOU'LL RECALL, IS 6IVEN BY THE FORMULA
i-1
THEN, IN PLACE OF THE RANPOM
VARIABLE
WE SUBSTITUTE S FOR cr,
ANP PEFINE A HEW RAHPOM
VARIABLE t ВУ
107
you CAN THINK OF THE RANPOM VARIABLE t AS TH£ BEST WC ZAN PO UNPCR
TNC CIRCUMSTANCES. ITS PISTRIBUTION IS CALLEP STUPCNT'S t BECAUSE ITS
INVENTOR, WILLIAM COSSET, PUBLISHER UNPER THE PSEUPONYM "STUPENT"
MAKING THE ASSUMPTION THAT THE
ORIGINAL POPULATION PISTRIBUTION
WAS NORMAL, OR NEARLY NORMAL,
"STUPENT WAS ABLE TO CONCLUPE:
t IS MORE SPREAP OUT THAN Z. ITS
“FLATTER" THAN NORMAL. THIS IS
BECAUSE THE USE OF S INTROPUCES
MORE UNCERTAINTY, MAKING t
"SLOPPIER' THAN Z.
THE AMOUNT OF SPREAP PEPENPS ON
THE SAMPLE SIZE. THE GREATER THE
SAMPLE SIZE, THE MORE CONRPENT WE
CAN BE THAT S IS NEAR cr, ANP THE
CLOSER t GETS TO z, THE NORMAL.
GOSSET WAS ABLE TO COMPUTE
TABLES OF t FOR VARIOUS SAMPLE
SIZES, WHICH WE WILL SEE HOW TO
USE IN THE FOLLOWING CHAPTER.
Л IM THE
ЗИ6ТТНЩК
OF Wff 'Ml£
ALREtxVy f
L J
tob
IN THIS CHAPTER, WE CONS I PER EP A CENTRAL PROBLEM OF REAL-WORLP
5TATI5TI&: MOW TO SELECT A 5AMPLE FROM A LAR6-E POPULATION 50 THAT
STATISTICAL ANALYSIS CAN BE VALIP. BESIDES THE "6OLP STANPARP" OF TME
SIMPLE RANPOM SAMPLE, WE ALSO PESCRIBEP SOME OTHER SAMPLING SCHEMES
THAT ARE USEP IN THE INTERESTS OF EFFICIENCY, COST, ANP PRACTICALITY.
THEN, ASSUMING A SIMPLE RANPOM SAMPLE, WE CONSlPEREP MOW VARIOUS
SAMPLE STATISTICS WERE PI5TRIWTEP. THAT IS, WE RE6-ARPEP TME ACT OF
TAKING TME SAMPLE AS A RANPOM EXPERIMENT, 50 THAT ITS STATISTICS
BECAME RANPOM VARIABLE*.
WE FOUNP THAT SAMPLE
PROPORTION* p WERE
APPROXIMATELY NORMALLY
PISTRIBUTEP, WHILE TME
PISTRIBUTION OF TME
SAMPLE MEAN X PEPENPEP
ON TME SAMPLE SIZE. FOR
LAR&E SAMPLES, TME
PISTRIBUTION WAS
APPROXIMATELY NORMAL,
WHILE FOR SMALL SAMPLES,
WE USE TME STUPENTS t
PISTRIBUTION.
109
IN ТИБ NEXT TWO CHAPTERS, WE LOOK
AT HOW TO USE THESE PISTRIBUTIONS TO
MAKE ST4T/ST/^4A IMFCRCMC4; 6IVEN A
SINGLE OBSERVATION, LIKE A POLITICAL
POLL, HOW PO WE USE OUR KNOWLEP6-E
OF p ANP X TO EVALUATE IT?
110
♦ Chapter 7<
CONFIDENCE
INTERVALS
111
РСЬ±)
IM THE LAST CHAPTER WE
LOOKEP AT SAMPLING-
STARTING WITH A LARGE
POPULATION, WE IMAGINE?
TAKING- MANY SAMPLES, ANP
WE PEPUCEP HOW SOME
SAMPLE ESTIMATORS WERE
PlSTRIBUTEP.
IM THIS CHAPTER, WE PO THE REVERSE. GIVEN ONG SAMPLE, WE ASK THE
QUESTION, WHAT WAS THE RANPOM SYSTEM THAT GENERATE? ITS STATISTICS?
TUAT >k.
well a ^iK^Le
0OK OF TAG KG, ANP
THE RE6ULT6» of
THE LAS-Т гЦА?Т£₽,
WT^AM \d£
112.
TMI5 SHIFT REPRESENTS A CHANGE
IN OUR MOPE OF THINKING- FROM
PEPUCTIVE REASONING TO
IN P£PUfTIV£ R£A4ON!H&, № REASON
FROM А HVPOTHESlS TO A CONCLUSION;
•IF LORP FASTBACK COMMITTEP MURPER,
THEN HE WOULP WIPE THE FINGER-
PRINTS OFF THE GUN "
ILJPUCTIV£ REA4ONIH6, ВУ
CONTRAST, ARGUES MCKWARP
FROM A SET OF OBSERVATIONS
TO A REASONABLE HyPOTHESl*
IN MANy WAVS, SCIENCE, INCLUPING STATISTICS, IS LIKE PETECTIVE WORK
BEGINNING WITH A SET OF OBSERVATIONS, WE ASK WHAT CAN BE SAIP ABOUT
THE SySTEMS THAT GENERATEP THEM.
10
ESTIMATING
CONFIDENCE INTERVALS
IS ONE OF THE MOST
EFFECTIVE FORMS OF
STATISTICAL INFERENCE,
ANP ONE YOU SEE EVERY
РАУ BEFORE ELECTION
TIME...
BUSINESS 4
HOLP М'У
\Ut, Wrsoir
I’M GOING into
THE POLL IM6
IN A RECENT ELECTION SOMEWHERE, INCUMBENT SENATOR ASTUTE (ACCENT
ON THE LAST SYLLABLE, PLEASE.') COMMlSSlONEP A POLL BY BETTER HOLMES
RESEARCH. POLLSTER HOLMES PRAWS A SIMPLE RANPOM SAMPLE OF WOO
ЫСК&& ANP ASKS THEM WHAT THEY THINK OF ASTUTE.
A) HE’S SOP’S SIFT
TO HUMANITY
B) HE’S THE PElTY’S
SPECIAL BLCSSINS
ON MOST OF
HUMANITY
AFTER CENSORINS THE REMARKS OF A FEW SRUMPY OUTLIERS, HOLMES FlNPS
THAT 990 VOTERS FAVOR HIS CLIENT, SENATOR ASTUTE.
THIS IS THE SINSLE
OBSERVATION.
114
AFTER A*TUTE £ALM*
POWN, HOLME* EXPLAIN*
WHAT HE MEAN* ВУ 99%
tOUFIPCHCC; HE KNOW*
THAT HI* E*TIMATION
PROCEPURE HA* A 99%
PROBABILITY OF
PROPU6IN6 AN INTERVAL
6ONTAININ6 p, I.E., IN HI*
МАЫУ VEAR* OF POLLING,
p MA* FALLEN WITHIN THE
6OHFIPEN6E INTERVAL
AROUNP THE OB*ERVEP
VALUE, p, 99% OF THE
TIME.
115
ЯТПМ6 BEMlNP THE TARGET 15 A BRAVE
PETE6TIVE, WHO £AN’T 5EE TME BULL'Y-
ЕУЕ. TME AR^MER YMOOTY A YIN6LE
ARROW.
KNOWING TME AR^MER’Y YKILL LEVEL,
TME PETE6TIVE PRAWY A ORILE WITH
W CM RAPlUY AROUNP TME ARROW.
HE NOW MAY CQNFIPCMX THAT
MI5 OR^LE IN^LUPEY THE CENTER OF
TME BULL'5-ЕУЕ/
ME REAYONEP THAT IF ME PREW IP ОИ RAPlUY ORTLEY
AROVNP Ж4Л/У ARROW*, MlY OR6LE5 WOULP IN^LUPE
TME CENTER 95% OF TME TIME.
(PROBABILITY
U5E TME TERM
$TO£MA$TI£
TO PESCRIBE
RANPOM
MOPEL5. IT’Y
PERIVEP FROM
TME 6REEK
fTO6MZCf-
TMAI, MEANING
TO AIM AT A
TAR6ET, OR
6UE55. FROM
5ТОЛ/О5, A
target.;
116
HOLMES NOW TRANSLATES
THE ARCHERY LESSON INTO
THE LAN6UA6E WE
PEVELOPEP LAST CHAPTER.
Step One: SHOOT a LOT OF ARROWS
A PROBABILITY CALCULATION FlNPS
THE WIPTH OF THE “BULL'S-EYE."
THE ESTIMATES p ARE OUR ARROWS.
WE SAW THAT THE SAMPLING
PISTRIBUTION OF p IS NEARLY
NORMAL WITH MEAN p ANP
STANPARP PEVIATION
П
SINCE THE CURVE IS NORMAL, WE USE THE Z-TRANSFORM ANP A STANPARP
TABLE TO FINP THE WIPTH OF THE INTERVAL WITHIN WHICH 95% OF THE
‘ARROWS* WT. (WE’LL SEE EXACTLY HOW TO PO THIS IN A FEW PA^ES.) WE
FINP THIS WIPTH TO BE 1-96 STANPARP PEVIATIONS*-
117
crCp")
WHI6H BECOMES
NOW WE PO SOME ALGEBRA. ВУ
PEFINITION OF THE Z-TRANSFORM,
WHl£H IS JUST ANOTHER WAX OF SAVINS- THAT 95% OF THE p "ARROWS” LANP
BETWEEN p - 1.96 <r(p) ANP p + 1.96cr(p).
NOW WE’RE IN A POSITION TO VIEW THE TARGET FROM BEHINPf ONE MORE
TURN OF THE AL4EBRA 6RANK MAKES IT
.97 = Pr(p-l-9*<pH ? ‘ p + l 9Wpi)
HERE WE ARE PRAWINS
6IR6LES AROUNP A LOT
OF ARROWS (IE,
MAKING INTERVALS
AROUNP p) ANP
SAVING THAT 95% OF
THEM ZOVER p.
AR& Ml PiFFLRWt
ЦОИ, But
1тЧ> оКДУ,
\FB-ALLy...
BUT THERE IS ONE TlNy PROBLEM... Wf POA/’T ACTUALLY KHOW THE SIZE
OF THE BULL'S-EYE, BECAUSE WE PON’T KNOW p, ANP THE WIPTH IS A
MULTIPLE OF cr(p').
50 WE FUP6-E A LITTLE ANPJJSE
THE STANPARP ERROR OF P:
^)sW
Ул
IN ITS PLA£E... IT’S 6LOSE
ENOUGH- IT’S THE BEST WE
ZAN PO... ANP IT ZAN EVEN BE
THEORETlZALLy JU STI Fl EP'
не
NOW THE FORMULA IS
Ср - I96$£(p), p + t9t>^(pY),
LETS STARE
Al THIS A
Minute...
NOW OUR PROBABILITY CALCULATION IS PONE, ANP IT’S TIME FOR...
A6AIN, THIS EQUATION PESCRIBES THE
PROBABILITY THAT THE TRUE, FlXEP
POPULATION PROPORTION FALLS
WITHIN THE Л4А/РО/И INTERVAL
IF WE SAMPLEP REPEATEPLY, THESE
INTERVALS WOULP CCNQR p 95% OF
THE TIME.
Step Two:
THE PETECTIVE WORK. IN A REAL POLL,
HOLMES TAKES JUST ONE SIMPLE
RANPOM SAMPLE OF WOO VOTES, FlNPS
p .550, ANP WANTS TO INFER p.
HE MAKES USE OF STEP ONE TO
COMPUTE
HE CONCLUPES THAT WE CAN HAVE
95% CONFIPENCE THAT p IS WITHIN
THE RAN6E
p * i.9t
= 5?o ± .031
THIS IS WHAT POLLS MEAN
WHEN THEY REFER TO THEIR
"MARGIN OF ERROR." IN THIS
CASE, HOLMES FOUNP THAT
519 p 561,
IN OTHER WORPS THAT
p 55% WITH A 3% MARGIN OF
ERROR. (POLLS TYPICALLY USE A
95% CONFIPENCE LEVEL.)
119
THIS РА6-Е SHOWS ТИБ RESULTS OF A COMPUTER SIMULATION OF TWENTY
SAMPLES OF SIZE n » Woo. № ASSUMEP THAT THE TRUE VALUE OF p .5. AT
THE TOP YOU SEE THE SAMPLING PISTRIBUTION OF p (NORMAL, WITH MEAN p
ANP BELOW ARE THE 95% 6ONFlPEN(E INTERVALS FROM EA£H
SAMPLE. ON AVERAGE, ONE OUT OF TWENTY (OR 5%) OF THESE INTERVALS WILL
NOT (OVER THE POINT p ~ 5.
I l I I I I I
0.44 0.46 0.48 0.50 0.52 0.54 0 56
95% Confidence Intervals for p
120
ALTHOUGH 95%
6ONFIPEN6E 15
GOOP ENOUGH FOR
NEW5PAPER POLL5,
IT WT GOOP
ENOUGH FOR
SENATOR ASTUTE.
HE WANT5 99%'
HOW TO INZREA5E ZONFlPENZE? USIUG
THE ARZHCRY TARGET, WE ZAN 5EE TWO
WAY5; ONE 15 TO INCREASE THE SIZE
OF THE CIRCLE YOU PRAW...
ANP ANOTHER WOULP BE TO IMPROVE
THE AIM OF TME ARCHER IN TME FIR5T
PLAZE, 50 HER ARROW5 LANP ZLO5ER
TO THE BULL'5-ЕУЕ-
THE FIR5T METHOP 15 EQUIVALENT TO WIPENIN6 THE CONFIDENCE INTERVAL
THE GREATER THE MARGIN OF ERROR, THE MORE CERTAIN YOU ARE THE TRUE
VALUE OF p LIE5 IN THE INTERVAL.
МАУВЕ IT’5 TIME TO 5EE EXACTLY
HOW WE FINP THE ENP5 OF
THE5E ZONFIPENZE INTERVALS.
121
THE RELEVANT NUMBER
HERE WE U5UALLY CALL a.
TT MEA5URE5 THE
PlFFERENCE BETWEEN THE
PE5IREP CONFIPENCE
LEVEL ANP CERTAINTY. FOR
EXAMPLE, WHEN THE
CONFIPENCE LEVEL 15 95%,
OR 0.95, a 15 05. 50 WE
5PEAK OF THE (1-а)Ю0%
CONFIPENCE LEVEL
FlNPINfr THE (l-a)'1OO% CONFIPENCE
INTERVAL MEAN*; LOOK AT A 5TANPARP
NORMAL CURVE, ANP FINP THE POINT* ±Z
BETWEEN WHICH THE AREA I* 1-Л.
WE CAN FINP х.ал 5TRAI6HT
FROM THE 5TANPARP NORMAL
TABLE 6РАС-Е В4Л IT’5 THE
POINT WITH THE PROPERTY
Pr(z »z ) = f
/2
IN PARTICULAR,
Pr(z - .025
-2.5 -2.4 -2.3 -2.2 -2.1
0.006 0.008 0.011 0.014 0.018
F(z)
F(z)
F(z)
112
Jv£r THE
AH5(tep,
PCEASe...
HERE'S A LITTLE TABLE OF THE ERITI^AL
VALUES FOR VARIOUS LEVELS OF
£ONFIPEN£E-
fbR THl*> LEVEL OF
6OMFIPENEE. Go OUT
THIS MAH'/ -stahdarv
P£VlATtOM6/
1-а
Л/2
.90 .90 .95 .99
.го .10 .05 .01
.10 .05 .025 .005
1.29 Ш 1.96 2.59
л
2
TO MAKE A 99% £ONFIPEN£E INTERVAL, WE USE THAT TABLE TO WRITE
.99 = Pr(p - 256 SE(/?) $ p « p + 250SEC/?)}
WHI^H WE SLOPPILV ABBREVIATE AS
p-p± г.59
WIPENIN6 ТИБ INTERVAL IS ONE WAV' TO INCREASE OUR £ONFIPEN£E IN TME
RESULT. AS WE MENTIONS?, ANOTHER WAV' WOULP BE TO SMOOT OUR ARROWS
MORE ACCUfUmy. IF WE KNEW THAT THE ARCHER 6OT 95% OF HER ARROWS
WITHIN 1 OF THE BULL'S-БУЕ, OUR ESTIMATES £OULP BE A LOT SHARPER/
1Z4
TAKING A CONSERVATIVE GUESS
OF = .5, HOLMES FlNPS
П-
COI)1
= (6-65)(.25)
• 0001
= 16,641
WOO VOTERS GAVE A 3%
ERROR WITH 95% CONFIPENCE.
TO GET A 1% ERROR WITH 99%
CONFIPENCE, HOLMES HAS TO
SAMPLE 16,641 VOTERS!
ON THE OTHER
HANP, WHO CAN
PLACE A VALUE ON
PEACE OF MINP’
SO THEY PO THE POLL,
ANP GO INTO THE
ELECTION WITH 99%
EOKFIPCWG.
BUT... ALL THIS PROBABILITY STUFF IS ONLY GOOP BEFORE AN ELECTION.
AFTER THE ELECTION, THE SENATOR IS EITHER 100% IN OR 100% OUT! ANP
PESPITE EVERYTHING, SENATOR ASTUTE LOSES THE ELECTION...
125
WHAT HAPPENS? IS THAT POLITICIANS ARE NOT ELECTS? ВУ POLLS'
f OUTRA^ous!
Vp PASS A LAwl
ASAiNST TH|S,|F
I V16RE STILL
<IN THE SENATE1
SOME PROBLEMS WITH POLLS, AS OPPOSE? TO ELECTIONS;
RESPONSE BIAS;
VOTERS МАУ LIE TO
THE INTERVIEWER
OR CHANGE THEIR MINPS
BEFORE ELECTION РАУ.
ALTHOUGH THE
POLL IS AN
UNBIASEV SAMPLE
OF POTENTIAL VOTERS,
THE VOTING BOOTH
COUNTS ONLy ACTUAL
I LOVE BOTH MAJOR
PARTIES ДЫр ONLY VliSH
I coulp Vote for both
VOTERS.
THERE IS NO WAY FOR A
POLLSTER TO SET INSIPE
A POTENTIAL VOTER’S
HEAP ANP KNOW IF SHE’S
6OIN6- TO VOTE, IF SUE’S
LXINS, OR IF SHE’S SOINS
TO CHANSE HER MINP
BEFORE ELECTION РАУ.
LARSE SAMPLE SICES
CANNOT REPUCE THESE
KINPS OF ERRORS.
12b
ЯН6Е THESE ERRORS CAN BE
LARGE, IT SELPOM PAYS TO TAKE sa
A VERY LARGE RANPOM SAMPLE.
IN THE LAST FIVE PR ESI PE MT! AL ELECTIONS, THE GALLUP POLL HAS INTER-
VlEWEP FEWER THAN 4,DOO VOTERS FOR EACH ELECTION. УЕТ IN ALL FIVE
ELECTIONS, THE GALLUP ORGANIZATION’S ERRORS IN PREPICTING THE
PRESIPENTIAL ELECTION OUTCOME HAVE BEEN LESS THAN 2%.
THEIR SUCCESS IS PUE TO THEIR USE OF ESTIMATORS THAT ACCOUNT FOR
NON-RESPONSE, ANP ТНЕУ SCREEN OUT ELIGIBLE VOTERS WHO ARE NOT
LIKELY TO VOTE.
TO SUMMARIZE, ESTlMATEP
PROPORTION = TRUE PROPORTION 4-
BIAS -I- RANPOM SAMPLING ERROR.
EVEN POLLSTERS HAVE LIMITEP
FUNPS. THEY WISELY CHOOSE TO
SPENP THEIR MONEY REDUCING
BIAS, RATHER THAN INCREASING THE
SAMPLES BEYONP 4,000 VOTERS.
127
Confidence Intervals
for y,
UP TO NOW, WE’VE BEEN
LOOKING AT LONFlPENLE
INTERVALS FOR A PROPOR-
TION p OF A POPULATION.
EXACTLY THE SAME
REASONING WORKS FOR
THE POPULATION MEAN p.
IN THE LAST CHAPTER 6R W5), WE SAW THAT THE PISTRIBUTION OF SAMPLE
MEANS Л IS APPROXIMATELY NORMAL, LENTEREP ON THE ALTUAL POPULATION
MEAN p, WITH STANPARP PEVIATION WHERE <r IS THE POPULATION
STANPARP PEVIATION. SO, FOR LAR6E n,
.99 = PrC-1,96 $ г $ 1,96?
- Рг6-1.96 $ 1.96?
f TURUlM THF
SAME AL#₽RA
LRAuK AS
MAIN, NOT KNOWING c, WE REPLALE <r
WITH S, THE SAMPLE STANPARP PEVIATION;
X —
.95 - Pr(-1.96 -r-^- 1.96?
THE_T£RM IS LALLEP THE SAMPLE' STANPARP ERROR, ANP WRITTEN
5E6U WE LONLLUPE THAT
.95 = PrCX- 1.965E6X) « jj. s Л+ 1.965EW)
JUST AS BEFORE, WE HAVE
FOUNP THAT THE RANPOM
INTERVAL
Л ± 1.96
CCNG& THE TRUE MEAN, /z, WITH
PROBABILITY .95... 60 NOW WE CAN
CALL IN SHERLOCK HOLMES TO
MAKE A STATISTICAL INFERENCE
BASEP ON A SIN6L£_6AMPLE OF
SIZE n WITH MEAN
LET 5 REVISIT THE STUPENT WEIGHT PATA
FROM CHAPTER 2, ASSUMING THAT THE
П « 92 5TUPENT5 WERE A SIMPLE
RANPOM SAMPLE OF ALL PENN STATE
THE SAMPLE MEAN % WAS 145.2
LPS. ANP SAMPLE STANPARP
PEVIATION 5 WA5 23.7. SO THE
STANPARP ERROR IS
^ = ^=- = 2.47
ANP WE NOW HAVE 95% ZONFIPENZE
THAT TME MEAN WEIGHT OF ALL
PENN STATE STVPENTS FALLS IN THE
INTERVAL
Z ± 1.96SE<A?
= 145.2 ± (1.96X2.47)
= 149.2 ± 4.9 POUMP4
TO SUMMARIZE: FOR A SIMPLE RANPOM SAMPLE (SRS) OF LAR6E SIZE, THE
ZONFlPENZE INTERVAL IS:
POPULATION WAN p
p. ~ X ± Zg.SE^)
2
WHERE SEC#) »
POPULATION PROPORTION, p
P ~ P ± ^a^(p)
2
WHERE SE(^) -
THE SIZE OF BOTH
INTERVALS IS
6ONTROLLEP ВУ
THE LEVEL OF
6ONFIPEN^E
(1-а)-Юа% ANP
THE SAMPLE SIZE, n.
130
Student's t (again!)
AS WE SAW IN CHAPTER 6, THE STATISTIC
HAS AN APPROXIMATELV NORMAU
DISTRIBUTION ONLY WHEN IT IS
COMPUTED USING A LARGG 4AMPLG
FOR SMALL SAMPLES Cz?=5, W,
THIS IS NO LONGER THE CASE, AND
WE HAVE TO USE THE STUDENT’S t
LET’S LOOK AT t A LITTLE MORE CLOSELY. WE MENTIONED THAT THE t
DISTRIBUTION IS MORE SPREAD OUT THAN THE NORMAL, AND THAT THE
AMOUNT OF SPREAD DEPENDS ON THE SAMPLE SIZE.
О
WHAT ITS DISCOVERER
6OSSET DID WAS TO
QUANTIFY THIS
RELATIONSHIP. IF n IS
THE SAMPLE SIZE, HE
SAID, THEN CALL Л-1
THE NUMBER OF
degrees of
freedom
OF THE SAMPLE.
THE GENERAL IDEA; GIVEN n
PIECES OF PATA %(1 %2,
YOU USE UP ONE “DEGREE
OF FREEDOM” WHEN YOU
COMPUTE % , LEAVING Л-1
INDEPENDENT PIECES OF
INFORMATION.
6OSSET tOMPUTGP TABLES OF
THE t PISTRIBUTION FOR
PIFFERENT SAMPLE SIZES—I.E.,
PE6-REES OF FREEPOM. WE
REPEAT, THE MORS PG6RSS5 OF
FRGGPOM, THE ZLOSER t
BECOMES TO THE STANPARP
NORMAL.
KNOWING THE SAMPLE SIZE 77, WE ZHOOSE THE t PISTRIBUTION WITH 77-1
PE6-REES OF FREEPOM.
AS WITH THE z
PISTRIBUTION O.E.,
THE STANPARP
NORMAL), WE 6ET A
95% ZONFlPENZE
LEVEL ВУ FINPIN6»
THE ZRITIZAL VALUE
BEYONP
WHIZH THE AREA
UNPER THE ZURVE
IS .025.
AREA _
t-olS
THE ZURVE \£,
PLANTER ТНМ NORMN-,
t.cnS father FKo/A
\^O THAH ’Z’.pif,,
FOR A (1-a)-1OO% ZONFlPENZE INTERVAL, WE FINP THE ZRITIZAL VALUE ta
SUZH THAT Pr(t£ ta) - v • HERE IS A SHORT TABLE OF ZRITIZAL VALUES
FOR THE t PISTRIBUTION:
1-a .eo .90 .95 .99
a .20 .10 .05 .01
a/2 .10 .05 .025 .005
PE6-REES OF 1 3.09 6.31 12.71 63.66
FREEPOM 10 137 1.01 2.23 4.14
30 1.31 1.70 2.OA 2.75
loo 1.29 1.66 1.90 2.63
GO 1.20 1.65 1.96 2.50
EACH COLUMN REPRESENTS A FlXEP LEVEL OF CONFIPENCE, WITH INCREASING
NUMBERS OF PEGREES OF FREEPOM. THE HIGHER THE PEGREES OF FREEPOM,
THE CLOSER THE CRITICAL VALUE GETS TO za/, THE CRITICAL VALUE OF THE
NORMAL PISTRIBUTION
WE PERIVE THE WIPTH OF OUR
CONFIPENCE INTERVAL PIRECTLX
FROM THE PEFINITION OF t-
t - -'tL
THEN, FOR CONFIPENCE LEVEL
(1-a) = Pr(^-'ta$Q(X) $ /z
FROM WHICH WE INFER GIVEN A
SlNGLE_SAMPLE OF SIZE n ANP
MEAN X, WE CAN BE
CONFIPENT THAT THE POPULATION
MEAN /z FALLS IN THE RANGE
r You- \
Will- '
MEMOplZE-
4TH(G- >
/z = ШаК№
'г
WHERE SE6CJ = %, ANP ta IS THE
CRITICAL VALUE OF THE t PISTRIBUTION
WITH n-\ PEGREES OF FREEPOM.
IIAVE* STRICTLY SPEAKING,
IlV I E« THE PERIVATlON OF
THE t PISTRIBUTION PEPENPEP ON
THE ASSUMPTION THAT THE SAMPLE
WAS FROM A NORMAL POPULATION IN
PRACTICE, CONFIPENCE INTERVALS
BASEP ON THE t WORK REASONABLY
WELL, EVEN WHEN THE POPULATION
PISTRIBUTION IS ONLY APPROXIMATELY
MOUNP-SHAPEP
133
example:
SUPPOSE OiMCLCON ЮТОМ HAS TO TRASH TEST
ITS TARS TO PETERMINE THE AVERAGE REPAIR C&>T OF A \0 M.P.H. HEAP-ON
TOLLlSlON. THIS IS EXPENSIVE! THEY PEOPE TO TRY IT ON JUST FIVE
THEY FINP THE PAMA6E PATA TO BE $150, }4O0, $720, $500, ANP $930.
THE SAMPLE MEAN:
X - *540
THE STANPARP PEVIATION:
5 * 1299
YOU TAN ШК S WITH A
HANP TALTULATOR. IT’S
А/ +(720 -9^o'jL + (5оо-6ЦоТН93о-^%Уь)
SO WHERE TAN WE PLATE THE MEAN WITH 95% TONFlPENTE? WE FINP OUR
CRITICAL VALUE t.M5 WITH 4 PE6-REES OF FREEPOM--
1-a .00 .90 .95 .99
a .20 .10 .05 .01
a/2 .10 .05 .025 005
PETREES OF 1 3.09 6.31 12.71 63.66
FREEPOM 2 1.09 2.92 4.30 9-92
3 1.64 2.35 3.10 5.04
4 1.53 2.13 2.70 4.60
5 1.40 2.01 2.57 4.03
1M
ANP PLU6 IT IN:
- 540 * 372
50 THE BE5T WE CAN 5АУ WITH 95% CONFIPENCE 15 THAT THE AVERAGE
PAMA6E WILL LIE BETWEEN $160 ANP $912.
01УТ O/t
CobtflDENT TtUT
It'Ll C№T
tXALTLV »3Co, •
THE COMPANY CAN EITHER
BE 5ATI5FIEP WITH THAT,
OR PO FURTHER TE5T5-
TO COMPUTE THI5 CONFIPENCE INTERVAL U5IN6 5TUPENT’5 t, WE HAVE MAPE
AN UNtTATCP AHUftPTlON: № A55UMEP THAT CRA5H REPAIR CO5T5 ARE
APPROXIMATELY AW/ULLX PltTRlBUTCP, I.E., IF WE CRA5HEP 1000
CHAMELE0N5, THE HI5TO6RAM OF REPAIR CO5T5 WOULP BE 5YMMETRICAL ANP
M0UNP-5HAPEP- WE CAN NOT KNOW TN!$ FROM 5 PATA P0INT5 ALONE- BUT
MAYBE YEAR5 OF EXPERIENCE WITH EARLIER M0PEL5 PROVIPE NORMALLY
PI5TRIBUTEP CO5T HI5TC&RAM5 FOR FRONT ENP REPAIR5: INFORMATION WHICH
WOULP TENP TO 5UPPORT OUR U5E OF 5TUPENT’5 t
THt TAIL GRoviS
BAC-K SV IT5CIT.
1ТЧ A FEATURE
OH C'HAfAELEOkS-.
135
Г ТО 4UM UP (I), WE
NOW HAVE THREE
SIMPLE REOPES FOR
FINPIN6 £ONFIPEN£C
INTERVALS. FOR
PROPORTIONS, OR
MEANS WITH LAR6E
SAMPLE SIZES, WE
LOOK UP za IN A
NORMAL TABLE. FOR
MEANS OF SMALL
SAMPLE SIZES (fSAV
№ FINP tft
IN THE t TABLE. 2
IN ALL 6ASES, THE WIPTH OF THE INTERVAL IS THAT ^RITIdAL VALUE TIMES
THE STANPARP ERROR:
Z«$E(X)
2 2
г
ANP EA^H OF THOSE STANPARP ERRORS IS PROPORTIONAL TO THAT MA6l£
NUMBER:
♦ Chapter 8 +
HYPOTHESIS TESTING
NOW WE ENTER A NEW AREA... GOVERNMENT,
BUSINESS, ANP THE HARP ANP SOFT SCIENCES ALL
USE ANP OFTEN ABUSE THESE TESTS OF
SIGNIFICANCE. IT'S ALL ABOUT ANSWERING THE
QUESTION, “6OULP TRE4E OBSERVATION
REALLY RAVE O&URREP BY FRANCE?”
e <3
137
WE BE6IN WITH AM EXAMPLE
FROM THE LAW-- A COMPOSITE
OF SEVERAL CASES AR6-UEP IM
THE SOUTH BETWEEN I960
AMP 1900, IM WHICH EXPERT
WITNESSES PRESENTER THE
CASE FOR RAM ЫА6 IN
JURY f£L££TtoN.
PANELS OF JURORS ARE THEORETICALLY PRAWN AT RANPOM FROM A LIST OF
ELIGIBLE CITIZENS. HOWEVER, IM SOUTHERN STATES IN THE '5oS ANP '60S, FEW
AFRICAN AMERICANS WERE FOUNP ON JURY PANELS, SO SOME PEFENPANTS
CHALLENGER THE VERPlCTS. ON APPEAL, AN EXPERT STATISTICAL WITNESS GAVE
THIS EVIPENCE-
1 50% OF ELIGIBLE CITIZENS Л &(*)£) (в AHj?
f WERE AFRICAN AMERICAN R А К X X к X Д Щ
COULP THIS BE THE RESULT OF
PURE CHANCE?
1ЭЙ
FOR THE SAKE OF ARGUMENT,
SUPPOSE THAT THE SELECTION OF
POTENTIAL JURORS WAS RANPOM.
THEN THE NUMBER OF AFRICAN
AMERICANS ON THE 06?-PERSON
PANEL WOULP BE THE BINOMIAL
RANPOM VARIABLE X WITH
П *40 TRIALS ANP p *.5.
W>&BR№ULL(
TRIALS, EA^H
WITH p 11
THUS, THE CHANCES OF 6ETTIN6 A JURY
WITH ONLY 4 AFRICAN AMERICANS IS
PrO<4/ WHICH WORKS OUT TO ABOUT
.0000000000000000014 (I).
SINCE THE PROBABILITY IS SO SMALL,
THE PARTICULAR PANEL WITH ONLY FOUR
BLACK MEMBERS IS 4TRON& CVIPENEC
A6AINST THE ИУРОТНМН OF RANPOM
TO PRIVE THE POINT HOME, THE
STATISTICIAN NOTES THAT THIS
SO THE JUP&E RCJSET4 THE
ИУРОТИЕЯ* OF RANPOM SELECTION.
PROBABILITY IS LESS THAN THE CHANCES
139
LET'S FOLLOW ТИС PROCESS A6A1N TO
SORT OUT THE FOUR FORMAL 4TRP4 OF
STATISTICAL HYPOTHESIS TESTING.
Step 1. FORMULATE ALL
НУРОТНЕ5Е5.
Ho, THE NULL ЦУРОТНЕШ, IS
USUALLY THAT THE
OBSERVATIONS ARE THE RESULT
PURELY OF CHANCE.
Ha, THE ALTERNATE НУРОТИ&14,
IS THAT THERE IS A REAL
EFFECT, THAT THE
OBSERVATIONS ARE THE
RESULT OF THIS REAL EFFECT,
PLUS CHANCE VARIATION.
Step 2. the test statistic.
IPENTlFY A STATISTIC THAT WILL ASSESS
THE EVIDENCE AGAINST THE NULL
HYPOTHESIS.
IN THE COURT CASE, Ho SAYS THE
JURY WAS RANPO^LT CHOSEN
ГКОН THE WHOLE POPULATION.
AFRICAN AMERICANS HAVE
PROBABILITY p- .50 OF BEIN6
CHOSEN..
Ha SAYS THAT AFRICAN AMERICANS
ARE LESS LIKELY THAN THEIR
PROPORTION IN THE POPULATION
TO BE SELECTEP FOR A JURY
PANEL p <50.
IN THE COURT CASE, THE TEST
STATISTIC IS THE BINOMIAL RANPOM
VARIABLE X WITH p~.5O ANP
ns- eo.
1*fr0
Step 3. P-VALUE:
A PROBABILITY STATEMENT WHICH
ANSWERS THE QUESTION: IF THE
NULL HYPOTHESIS WERE TRUE, THEN
WHAT IS THE PROBABILITY OF
OBSERVING A TEST STATISTIC AT
LEAST AS EXTREME AS THE ONE WE
OBSERVEP?
IN THE EXAMPLE, THE P-VALUE WAS
Pr(X£4 | ANP г?»0Р)
- 1.4 x IO'10
WE LOMPUTEP THIS P-VALUE THE
MOPERN WAY, USING A STATISTICAL
SOFTWARE PACKAGE.
THE SMALLER
THE P-VALUE,
THE STRONGER
THE EVIPCNCE
ASAINST Mo.
HORSE-
PRAWN ,
COIWOteRSL
Step 4. COMPARE THE
P-VALUE TO A FlXEP flEMFltANCE
LEVEL, a.
a MV AS A CUT-OFF POINT
BELOW WHICH WE AGREE THAT AN
EFFECT IS STATISTICALLY SIGNIFI-
CANT. THAT IS, IF
P-VALUE $ a
IN THE JURY CASE, THE STATISTICIAN
TOOK a TO BE 3.6 x IO'10, THE
CHANCES OF BEING PEALT THREE
ROYAL FLUSHES IN A ROW.
A P-N/ALlie
EVEN A jUpfrE
CAM
THEN WE RULE OUT THE NULL
HYPOTHEC H<> ANP AGREE THAT
SOMETHING ELSE IS GOING ON.
HI
IN SOENTlFl^ WORK, A FlXEP «-LEVEL OF .05 OR 01 IS OFTEN USEP. THESE
FlXEP LEVELS ARE A HOLPOVER ARTIFACT FROM THE PRE COMPUTER ERA,
WHEN WE HAP TO REFER TO TABLES, WHI6H WERE PRINTEP ONLY FOR
SELECTEP CRITICAL VALUES. STILL, MANY SOENTIRE JOURNALS CONTINUE TO
PUBLISH RESULTS ON LX WHEN THE P-VALUE $ 05.
IN LE6AL PRO^EEPIN^S, THE
STANPARP IS MORE FLEXIBLE-
A Royal Flo^H—
M 2
LARGE SAMPLE
SIGNIFICANCE TEST FOR
PROPORTIONS
TME JURY EXAMPLE WA* A *PEOAL GA*£
OF A GENERAL PROBLEM TME KULL
МУРОТМЕ*!* MAP TME FORM p ~ p0,
WMERE p0 WA* 5OME PROBABILITY <IN
TMI* GA*E, -5Л NOW LET’* LOOK AT
*UGM PROBLEM* GENERALLY; LET’*
7E*7 7WE НУРОТИБШ p ~ p0.
LARGE
A4 U*UAL, WE IMAGINE WE MAVE A BIG POPULATION... WE OB*ERVE A
*AMPLE... ANP WE FINP TMAT *OME GMARAGTERI*TIG OGGUR* WITM
PROBABILITY p.
4
BA*EP ON TMI*
OB*ERVATION, WE WANT
TO KNOW IF TME TRUE
POPULATION PROBABILITY I*
6FOR IN*TAN6E? LARGER TMAN *OME OTMER VALUE p0. FOR EXAMPLE,
*ENATOR A*TUTE, MAYING FOUNP A p OF 54, WOULP LIKE TO KNOW TMAT
p > .5, A WINNING MAJORITY.
143
Step 1.
THE HULL HyPOTHESIS Ъ
Hp ’• p - po
THE ALTGRMATK HyPOTHESIS PEPENPS
ON THE PIRECTION OF THE EFFECT
WE ARE LOOKING FOR. IN SENATOR
ASTUTE’S CASE,
p > Po
BUT IN OTHER CASES, THE ALTERNATE
HyPOTHESIS MI6HT WELL BE
•• p < po
OR
Ия: p*p0
FOR EXAMPLE, IN THE JURy SELEC-
TION EXAMPLE, THE ALTERNATIVE
HyPOTHESIS WAS
p < 05
ANP AT OTHER TIMES, WE ARE
INTERESTEP IN KNOWING THAT p IS
PIFFERENT FROM SOME VALUE po-
FOR INSTANCE, IN TESTING FOR A
FAIR COIN, WE HAVE AN ALTERNATE
HyPOTHESIS OF
Ия : p*O5
Step 2 • THE TEST STATISTIC IS
P~f*
{рь(1-рьУ47Г
WHICH MEASURES HOW FAR p
FROM po. UNPER THE NULL
HyPOTHESIS, Zow HAS THE STANPARP
NORMAL PISTRIBUTION.
Step 3 •THE P-VALUE PEPENPS
ON WHICH ALTERNATE HyPOTHESIS IS
RELEVANT;
d) "RI^HT-HANPEP" Цл : p> po
USES P-VALUE Pr(Z > zO6i)
b) "LEFT-HANPEP" p < p0
USES P-VALUE PrCZ < zOB5)
BUT HAVE NO A PRIORI OPINION
ABOUT WHETHER HEAPS OR TAILS
WILL COME UP MORE OFTEN.
o
c) TWO-SIPEP" Ил : p*po
USES P-VALUE Pr(lzl > \zO6il)
14^1
IN THE CASE OF SENATOR ASTUTE;
1 ) THE HyPOTHESES ARE
Ня ; ^>.5
3) HIS TEST STATISTIC IS
THE SENATOR THUS REJECTS
THE NULL HyPOTHESIS, ANP
HE CANP HIS BACKERS) NOW
FEEL CERTAIN HE’S IN THE
LEAP.
ZOBS *
.55-50
- 3.16
3) HIS P-VALUE IS
Рг(2>гм$) = Pr(z 2 3.16) -.0009
(FROM THE NORMAL TABLE).
4) ASTUTE. BEING FAIRLY CONSERVATIVE,
TAKES A SIGNIFICANCE LEVEL a OF O\
ANP OBSERVES THAT
Pr(Z> z^) = .0009 < a
145"
LARGE SAMPLE
TEST FOR THE
POPULATION MEAN
HERE IS HOW A SIGNIFICANCE TEST
MIGHT BE USEP IN IMPGCTION
SAMPLING, AN IMPORTANT INPUSTRIAL
APPLICATION.
FfbL^
LIGHT,
MAU-
h£aw,
/ЛАК-
О MG GRANOLA INC. CLAIMS THAT
THE AVERAGE WEIGHT OF ITS CEREAL
BOXES IS AT LEAST 16 OZ. THE GGNUING
GROCGRT CORPORATION WILL SENP BACK
A SHIPMENT IF THE AVERAGE WEIGHT IS
ANY LESS.
BUT OF tOURtE GENUINE GROCERY HAS
NO INTGNTION OF WEIGHING EVERY BOX
IK A SHIPMENT. THEY’RE GOING TO USE
STATISTICS'
SUTISTU* 16
THE EA-sy UAY,
REME/ABEI??
146
FIRST, ТИБУ ОЮ&Е THEIR
HYPOTHESES.
H^: A * U OZ.
Ня: /4 <16 OZ.
REJECTING THE HOLL
НУ?Р1И^16 MEAN*
K^06lU6 THE 6RAUOLA-
NEXT, THEY GHOOSE A TEST STATISTIG. BY NOW, IT SHOULP BE PRETTY MUGH A
KNEE-JERK REACTION TO KNOW THAT THE SAMPLE SPREAP FROM THE MEAN ft
g — M-p
WHERE b & THE SAMPLE
STANPARP PEVIATION. UN PER
THE NULL HYPOTHESIS, THIS
APPROXIMATES THE
STANPARP NORMAL WHEN
THE SAMPLE IS LARGE, BY
THE CENTRAL LIMIT
THEOREM.
SKIPPING- OVER STEP 3 FOR A MOMENT, THEY SET A SlGNlFlGANGE LEVEL. BEING- A
BUNGH OF PROPPEP-OUT SGIENGE MAJORS, THE GROGERS THINK a*.O5 SOUNPS
ABOUT RIG-HT.
JUST THEN, A BOXGAR
LOAPEP WITH WftOO
BOXES OF G-RANOLA
ARRIVES AT THE POOR
\ RgMCMgep the
Number 5 yeah-
147
ТИБУ PULL OUT A
SIMPLE RANPOM
SAMPLE OF 49 BOXES,
WEIGH EACH ONE, ANP
PETERMINE THE
SAMPLE’S SUMMARY
STATISTICS:
% * 15.90 07.
5 ~ 35 oz..
h LITTLE LIGHT—BUT
SIGNIFICANTLY SO?
THEY PLUG THE VALUES INTO THE TEST STATISTIC TO FINP
15.9-16
Zoess
-2
NOW THEY COMPUTE TME P-VALUE'-
Prcz < -г I Hp? ® . 0227
SEND IT
THIS BEING LESS THAN THE .04
SIGNIFICANCE LEVEL, GENUINE GROCERY
REJOtTTt THE NULL HYPOTHESIS, ANP
THE SHIPMENT.
" FLAKY ftOPFRo/1
Got w munchies,
MAM-- I PIPN'T
T^lMK АНУОЦБ WOUUP
NOTICE IF I ATE A
LITTLE FROM EVER/
14B
SMALL SAMPLE
TEST FOR THE POPULATION
MEAN
WE RETURN TO CHAMELEON MOTORS, ANP ITS to M.RH. CRASH TEST. ТИБ
RI6UT£OU5 INSURANCE COMPANY WILL INSURE AN AUTO ONLY IF ТИБ MEAN
REPAIR COST AFTER A to M-P.H. COLLISION IS LESS THAN $1ORO. THE COMPANY
USES A STANPARP a = .05 AS ITS SIGNIFICANCE LEVEL. SO...
Htf: A > iWOO
Ня: JLL<tWOO
MEAN COST IS TOO HIGH
MEAN COST IS O.K.
THE TEST STATISTIC IS THE t PISTRIBUTION
SEC*)
WHERE jn0 IS THE
HYPOTHETICAL MEAN
OF $1000
ANP WE WANT OUR OBSERVEP
t VALUE TO LIE TO THE LEFT
OF -t.05 ^BECAUSE LOW X IS
PESIRABLE, X-yUp SHOULP BE
NEGATIVE, TO SUPPORT Н«Л
MS
FROM ТИБ TABLE OF CRITICAL
a
.05 .015 .005
1 6.31 12.71 (Л.ЬЬ
2 2.92 4.30 9.92
LU U _ LU LU ? 2.35 3.16 9.94
LU 4 2.13 2.70 4.60
5 2.01 2.57 4.03
t VALUES, WE SEE THAT
t.05 » 2.1?, SO WE PECIPE TO
REJECT Ho IF
ioBf ~ "2.13
FROM CHAPTER 0, WE HAVE
X = i5AO ANP * * $299
THE CAR PASSES THE TEST... Ho IS REJECTEP- ANP THE INSURANCE POLlCy I*
ISSUEP.
THIS IS AN EXAMPLE OF A€€£PTAN£E 5AMPLIH& THE NULL НУРОТНЕ*!* IS
THAT REPAIR COSTS ARE UNACCEPTABLE, ANP THE MOTOR COMPANY I*
ASSUMEP 6-UlLTy UNTIL IT PRESENT* SUFFICIENT EVIPENCE OF IT*
INNOCENCE—!•£., THAT IT* PROPUCT I* WITHIN SPECIFICATION*.
150
DECISION THEORY
Ж £AN THINK OF HyPOTHESIS TESTIN6 ANP
SI6NIFI<AN4E TESTS IN TERMS OF A UOlHEHOLP
5MOKE-PETEETOR. IF YOU HAVE ONE OF THESE
WHERE УОи LIVE, yoUVE PROBABLy NOTI^EP HOW IT
TENPS TO 60 OFF EVERY TIME УОи MAKE THE TOAST
TOO PARK/
THIS IS WHAT IS £ALLEP А 7УЯС I ERROR; AN ALARM WITHOUT A FIRE.
CONVERSELY А ТУРЕ II ERROR IS A FIRE WITHOUT AN ALARM. EVERY COOK
KNOWS HOW TO AVOIP А ТУРЕ I ERROR: JUST REMOVE THE BATTERIES
UNFORTUNATELY THIS INCREASES THE INCIPENCE OF ТУРЕ П ERRORS'
SIMILARLY REPUCIN6 THE CHANCES OF ТУРЕ П ERROR, FOR EXAMPLE ВУ MAKING
THE ALARM HYPERSENSITIVE, CAN INCREASE THE NUMBER OF FALSE ALARMS.
151
WE CAN SUMMARIZE THIS IN A TWO-BY-TWO РБДЯОН TABLE.
NO ALARM
ALARM
NO FIRE FIRE
NO ERROR ТУРЕ II
ТУРЕ I NO ERROR
NOW THINK’ OF THE NULL HYPOTHESIS AS THE CONDITION OF KO FIRE, WHILE
THE ALTERNATE HYPOTHESIS IS THAT A FIRE IS BURNING. THE ALARM
CORRESPONDS TO REJECTION OF THE NULL HYPOTHESIS;
TRUE STATE
Hp Ид
ACCEPT He
REJECT
NO ERROR ТУРЕ П
ТУРЕ I NO ERROR
ALL THE SIGNIFICANCE TESTS WE PIP EARLIER IN THIS CHAPTER EMPHASIZED
THE PROBABILITY OF COMMITTING А ТУРЕ I ERROR-I.E-, THE PROBABILITY OF
OUR OBSERVATIONS OCCURRING IF WAS TRUE. WE DEMANDED THAT
^REJECTING Hj Htf? - Рг(ТУР£ I ERROR I H#) = a
1-a MEASURES OUR CONFIPENCE THAT ANY ALARM BELLS WE HEAR ARE
GENUINE. HIGH CONFIPENCE MEANS RARELY SETTING OFF FALSE ALARMS.
BUT 6OMETIME6 WHAT WE REALLV WANT TO KNOW 16 TME CHANCE OF MAKING
А TYPE // ERROR/ IN OTHER WORP6, MOW 6EN6ITIVE 16 OUR ‘ALARM 6У6ТЕМ"
WHEN TME ALTERNATE НУРОТНЕ616 16 TRUE?
IN THE PA6T, FACTORIE6 PI6CHARGING CHEMICAL6 INTO WATERWAV6 WERE
REOUIREP TO 6MOW THAT THE PI6CHARGE HAP NO EFFECT ON THE POWN-
6TREAM WILPLIFE. THAT’6 THE POLLUTER COULP CONTINUE A6 LONG A6
THE NULL НУРОТИЕ616 WA6 NOT REJECTEP AT THE .05 6IGNIFICANCE LEVEL
60 A POLLUTER, 6U6PECTIN6- THAT HE WA6 IN VIOLATION OF EPA 6TANPARP6,
WOULP PEVI6E AN INEFFECTIVE POLLUTION MONITORING PROGRAM.
153
THE POLLUTER 15 PELIGHTER 5lN£E, LIKE OUR SMOKE ALARM WITHOUT A
BATTERY, Hl5 TEST HAS LITTLE OR NO £HAN£E OF SETTING OFF AN ALARM.
wRiye this cowN >
'"THE ОМСК RESPONDED
ENTHOSlASTKALLY" >
LET $ FORMALIZE THIS
IPEA. TO PE5ZRIBE THE
PROBABILITY OF А ТУРЕ
II ERROR, № BREAK OUT
ANOTHER GREEK LETTER:
BETA, OR Д
fi * Р/YAOTPTlNG Hp |ИЯ?
« РгСГУРБ II ERROR |ИЯ?
THE POWER OF А TEST
IS PEFlNEP AS \-fi- IT’S
Prf REJECTING Нр |НЙ ).
YOU'LL BE HAPPY TO KNOW THE
ENVIRONMENTAL RE6ULATOR5 HAVE
MOVEP IN THE PIREZTION OF REQUIRING
POLLUTION MONITORING- PROG-RAM5 TO
5HOW THAT THEY HAVE A HIOH
PROBABILITY OF PETEGTING- 5ERIOV5
POLLUTION EVENT5. THE REQUlREP
POWER АКА1.УЯ4 OFTEN REVEAL5
HIPPEN FLAW6 IN THE MONITORING
PROGRAM.
ONE WAV TO VISUALIZE THE EFFECT OF A TEST'S POWER IS ВУ GRAPHING THE
PROBABILITy OF REJECTING Hp AGAINST THE ACTUAL STATE OF THE SVSTEM IN
THE CASE OF A SMOKE ALARM, THE PROBABILITy CLIMBS TOWARP 1 AS THE
SMOKE GETS THICKER.
FOR THE E.P.A. WATER QUALITy EXAMPLE, THE HORIZONTAL AXIS IS THE TRUE
CONCENTRATION OF POLLUTANT IN THE WATER.
HERE ARE THE POWER CURVES FOR THREE MONITORING PROGRAMS. THE SAVE
EVERY LAfT &Um С6ОЖ $9 MILLION), THE EOLPEH &EAN (CCW
i9OO,OOO), ANP PON'T ROCK THE BOAT (ALSO COSTS i9OO,OOO, BUT ТНЕУ PUT
ON A GOOP SHOW/). THE HIGHER THE TEST'S POWER, THE STEEPER THE CURVE.
155
WHY THEN PO you HAVE 6U£H AN EMPTY FEELINS IK YOUR ЯО/ШМ?
BE£AU5E, TO USE THESE IPEAS IN ANY PRA£Tl£AL WAY, WE HAVE TO BE ABLE
TO APPLY THEM TO A VARIETY OF SITUATIONS WE HAVEN’T EVEN TOU^HEP ON
УБТ. TMAT ft WHERE WE ARE £OIK6 NEXT, WITH THE €OMPARI^ON OF TWO
POPULATIONS
\____________________________________________________________>
1%
♦ Chapter
COMPARING
TWO POPULATIONS
IN WHI6H WE LEARN SOME NEW RECIPES U5IN6-
OLP IN&REPIENTS...
157
гтпв
TME LAST TWO CHAPTERS EXPLAINEP
CONFIPENCE INTERVALS ANP
HYPOTHESIS TESTING WITH TME
4TCAK AW POTATO& OF RANPOM
MOPELS-. TME NORMAL ANP TME
BINOMIAL PlSTRIBUTIONS.
VJITIA IMG NORMAL
TkE RotE
of w Potatoes'
BUT WMAT MAKES STATISTICS ALMOST AS CHALLENGING AS COOKING IS TME
VARIETY. LIKE AN EXPERT COOK, TME STATISTICIAN CAN “TASTE” TME
INGREPIENTS IN A PROBLEM ANP THEN FINP TME MOST EFFECTIVE WAY TO
COMBINE THEM INTO A STATISTICAL RECIPE-
(TME REASON COOKBOOKS ANP STATISTICAL METMOPS TEXTS ARE SO HEAVY IS
THAT THEY BOTH PROVIPE SOLUTIONS IN A GREAT VARIETY OF SITUATIONS')
1S8
IN THIS CHAPTER, WE’LL USE OUR MEAT
ANP POTATOES METHOPS IN SOME NEW
RECIPES THAT WILL HELP US ANSWER
THE FOLLOWING QUESTIONS;
POES TAKING ASPIRIN REGULARLY
REPUCE THE RISK OF HEART ATTACK?
POES A PARTICULAR PESTICIPE
INCREASE THE YlELP OF CORN PER
ACRE?
PO MEN ANP WOMEN IN THE SAME
OCCUPATION HAVE PIFFERENT SALARIES?
THE COMMON IN6REPIENT IN THESE
QUESTIONS IS THIS; ТНЕУ CAN BE
ANSWEREP ВУ COMPAR/Mtf TWO
ЩРСРСЫРСМТ RANPOM 5AMPLZ5,
ONE FROM EACH OF TWO
POPULATIONS.
ANP, AT THE ENP OF THE CHAPTER,
WE’LL LOOK AT A PIFFERENT WAV TO
COMPARE TWO MEANS THAT POESN’T
INVOLVE TAKING TWO SIMPLE
HO
169
Comparing SUCCESS RATES
(or failure rates) for two populations.
WE BEGIN WITH AN EXPERIMENT, PART OF A HARVARP STUPY, TMAT SOUGHT TO
PEC! PE TME EFFECTIVENESS OF ASPIRIN IN REPUClNG HEART ATTACKS. AS IN
MOST CLINICAL TRIALS, TME CHANCES THAT ANY ONE INPIVIPUAL GETS TME
PISEASE—IN THIS CASE. A HEART ATTACK—IS VERY SMALL IN ANY GIVEN YEAR.
PUT WE WANT ANSWERS QUICKLY.' WHAT PO WE PO?
TAKE
20,000
<111 I ---------------------1—
THE SIMPLE, BUT EXPENSIVE, SOLUTION IS TO TEST A LARGE NUMBER OF
INPIVIPUALS IN A SHORT TIME. IN THIS STUPX, 22,071 SUBJECTS CALL
VOLUNTEER POCTORS) WERE RANDOMLY ASSIGNEP TO TWO fROUPf.
&ROUP ONE TOOK A PLACCBO-A
PILL IPENTICAL TO ASPIRIN, BUT
CONTAINING NO ASPIRIN.
GROUP TWO RECEIVER ONE
ASPIRIN А PAY.
OVER A PERIOP AVERAGING
NEARLY FIVE YEARS*, TME
INVESTIGATORS RECORPEP
TME RESPONSES: HEART
ATTACK OR NO HEART ATTACK.
TME RESULT: (IN TME
NUMBERS THAT FOLLOW, WE
HAVE COMBINEP FATAL ANP
NONFATAL HEART ATTACKS.?
ATTACK NO ATTACK П ATTACK RATE
PLACEBO 239 10,795 11,034 p, = = .0217 r’ 11/734
ASPIRIN 139 10,090 11,037 p- = ’39 = .0126 гг 11,037
THE OBSERVEP PIFFERENCE
IN SUCCESS RATE IS
А £?£?91 • 11
SMALL UNTIL YOU LOOK AT
THE RELATIVE RISK,
Py
— =1-72.
,0126
MEMBERS OF TME PLACEBO
GROUP WERE 1.72 TIMES
LIKELIER TO SUFFER A HEART
ATTACK THAN THOSE IN THE
ASPIRIN GROUP.
“THE 5TUPY WA5 5TOPPEP EARLY BECAUSE OF ITS POSITIVE OUTCOME. IT WOULP
HAVE BEEN UNWISE ANP IMPRACTICAL TO PENY THE RESULTS TO THE GROUP
TAKING TME PLACEBO.
161
The Model: THE PLACEBO ANP ASPIRIN 6-ROUP OBSERVATIONS
ARE INPEPENPENT SAMPLES FROM TWO BINOMIAL POPULATIONS. FOR
CONSISTENCY, WE REFER TO A HEART ATTACK AS A (I).
PLACEBO
POPULATION ONE
CHANCE OF SUCCESS = P
ASPIRIN
POPULATION TWO
CHANCE OF SUCCESS = p
THE OBJECTIVE IS TO ESTIMATE THE TRUE PIFFERENCE, р}-рг-
к__________________________________.___________________________
FOR EACH POPULATION (ACTUALLY LAR6-E SAMPLES OF THE GENERAL POPU-
LATIONZ WE HAVE THE FAMILIAR RANPOM VARIABLES;
у NUMBER OF SUCCESSES v
1 IN POPULATION ONE 2
л X< PROPORTION OF л X„
P s. —L SUCCESSES IN p -=. —±-
1 Z?1 POPULATION ONE 2 77
ANP AN ESTIMATOR OF PIFFERENCE IN RATE; P^ P2
NUMBER OF SU^ZESSES
IN POPULATION TWO
PROPORTION OF
SU<ZESSES IN
POPULATION TWO
162
Sampling distribution for Р,- P2
FOR LAR6-E SAMPLES, P,-P2
IS APPROXIMATELY
NORMALLY PISTRIBUTEP,
MUCH AS IN THE CASE OF
ONLY ONE SAMPLE. WE CAN
MAKE THE USUAL Z-
TRANSFORM TO 6ET A
STANPARP NORMAL RANPOM
VARIABLE (APPROXIMATELY;
. ^К'(РгРг>
cr^-P,')
BUT HOW PO WE FINP
THAT STANPARP PEVIATION
IN THE PENOMINATOR?
SINCE THE TWO SAMPLES ARE INPEPENPENT, SO ARE THE RANPOM VARIABLES
Pt ANP Рг, ANP THE TWO VARIANCES APP.
a-(?, -Я) *
I РЕСО/ШМР
AH ASPIRIN To
6FT THROUGH
THIS*..
ANP NOW, KNOWING
THE PISTRIBUTION
OF THE TEST
STATISTICS, WE CAN
PROCEEP TO
ESTIMATE
БОЫР1РБЫББ
INTERVALS ANP
ТБ4Т ТИБ
НУРОТНБШ THAT
ASPIRIN REPUCES
HEART ATTACKS.
Confidence
Intervals for
—
TO &ET THE 95% CONFIPENCE
INTERVAL FOR THE ASPIRIN
STUPy, WE JUST PLU6- IN THE
OBSERVEP VALUES:
P1-P2
AS USUAL. THE CONFIPENCE INTERVALS
FOR OUR ESTIMATE LOOK LIKE THIS:
рх~рг* .РР91±б1.94Х.РР175>
^.OQW + .ООЪА
STANPARP
ERROR
TRUE
PlFFERENCE
OF POPULATION
PROPORTIONS
OBSCRVEP
PIFFERENCE
CRITICAL
VALUE
Р\~Рг
THE VARIANCES OF P, ANP P2 APP, SO
THE STANPARP ERROR BECOMES
IN THE ASPIRIN STUPy, THE STANPARP
ERROR IS
WE ARE AT LEAST 95%
CONFIPENT THAT THE
DIFFERENCE IN HEART ATTACK
RATES IS BETWEEN .0047 ANP
.0125 DEFINITELY A POSITIVE
NUMBER' WE ARE NOW AT
LEAST 95% CONFIPENT THAT
ASPIRIN REALLY POES LOWER
HEART ATTACK RATES-
hypothesis
testing
THE FORMAL HYPOTHE5I5-TE5TIN6.
QUE5TION 15
/ IF ASPIRIN HAP >
/ NO EFFECT, WHAT
[ IS THE PROBABILITY
THAT THIS RESULT
\ O<^URREP BY »
\ 6HAN^E? J
Ho, THE NULL НУРОТНЕ515,15 THAT
ASPIRIN HAP NO EFFECT: px = pT
He, THE ALTERNATIVE, 15 THAT
ASPIRIN POE5 REPUTE THE HEART
ATTACK RATE1 p\ >рг.
HOW WE NEEP A TE5T STATISTIC WITH
A NORMAL PISTRIBUTION WHEN Ho 15
TRUE...
NOTE THAT UNPER Ho, THE TWO
PROPORTIONS ARE THE 5AME,
P- LET’5 POOL ALL THE
PATA TO 6>£T THE PROPORTION OF
HEART ATTACKS IN BOTH SAMPLES
TOGETHER:
л
~ Щ + Л2
WHEN THE NULL НУРОТНЕ515 15
TRUE, THE 5TANPARP ERROR
PEPENPS ONLY ON THI5 POOLEP
ESTIMATE:
SEp6P,-P2; -
ANP WE CAN WRITE A TEST
STATISTIC:
Z* SEp(P,-P2)
(THE NUMERATOR WOULP
ORPINARILY BE ^~^г~^Р~Р-2)'
BUT Ho A55UME5 р~рг O.)
FOR THE ASPIRIN 5TUPY, WE FINP
л 370
22,071
5E0('Pt-P29 = -00175
so
.0091
^0B5= D0\15 =
165
Z0V> 15 MORE THAN FIVE 5TANPARP PEVIATIOW FROM ZERO, A STRONG-
POSITIVE EFFECT. U5ING A TABLE OR A COMPUTER, WE FINP THE P-VALUE;
P-VALUE = PR(Z> toei) ~ PR(Z> 5.2) * .OOOOOO]
О £2_
IF THE NULL НУРОТНЕ515 WERE TRUE, THE PROBABILITY OF OBSERVING- AN
EFFECT THI5 LARG-E 15 ONE /Л/ TEN MILLION-N&RY STRONG- EVIPENZE
AGAINST H0H!
The
general
recipe:
THE RELEVANT P-VALUE PEPENP5 ON
THE ALTERNATE НУРОТНЕ515:
A? TWO-5IPEP Нл * рг
TO TE5T THE NULL HYPOTHESIS
H<2: - рг
COMPUTE THE TEST 5TATI5TIG
- Ру 'Pz
P-VALUE - PrClZI >IZOB$D
в; RI6HT Нл -р>рг
(WHERE 5E0 15 COMPUTE? U5IN6-
THE POOLEP PROBABILITY
OBTAINEP ВУ 6OMBININ6 BOTH
P-VALUE = Pr(Z > lo^
LEFT Нл :p<p2
P-VALUE = Pr( Z < ZOB$)
16b
THE ANALYSIS OF THE ASPIRIN STUPY PEPENPEP ON CERTAIN FEATURES OF THE
EXPERIMENT PESI6NEP TO ENSURE RANPOMNESS ANP TO ELIMINATE BIAS:
^^T^THE EXPERIMENT WAS BLIMP;
SUBJECTS PIPN’T KNOW IF
THEY WERE TAKING ASPIRIN
V OR PLACEBO.
POINTS 1 ANP 2 ARE ESSENTIAL
PARTS OF MOST HUMAN £LINl6AL
TRIAL PESI6-NS, BUT POINT 3 IS
NOT ESSENTIAL 6OOP SMALL-
SAMPLE TESTS PO EXIST ANP
ARE AVAILABLE IN COMPUTER
SOFTWARE PACKAGES. THESE
MOMPARAMCTRlt PROCEDURES
PEPENP ON SIMPLE, BUT
LENGTHY, PROBABILITY
CALCULATIONS SIMILAR TO THE
&AMBLIN6- COMPUTATIONS WE
ENCOUNTEREP IN CHAPTER 4...
uoyj >
1NWLT/N6-
UM...WB MAO
A^UMBP THAT POCTOHS
kRE REPRESENTATIVE
op THE 6EUERXL
Population...
167
Comparing the
MEANS of two populations
SUPPOSE WE WANTEP TO COMPARE THE
AVERAGE SALARy OF MALE ANP FEMALE
EMPLOVEE5 IN THE SAME JOB AT SOME
COMPANY
POPULATION ONE 15 THE WOMEN, ANP POPULATION TWO 15 THE MEN.
POPULATION ONE HA5 MEAN
SALARy /z, ANP STANPARP
PEVIATION <7]
POPULATION TWO HA5 MEAN
SALARy ANP 5TANPARP
PEVIATION <Гг
к RANPOM SAMPLE OF SIZE /7, FROM 6ROUP 1 ANP n2 FROM GROUP 2 6IVE5
SAMPLE MEANS OF %, ANP %2 ANP 5TANPARP PEVIATIONS 5, ANP S,,
RESPECTIVELY THE ESTIMATOR OF ЛгЛг &
л,-л2
160
HOW 6OOP AN ESTIMATOR IS Л,-Лг?
FOR LAR6-E SAMPLE SIZES, IT’S
APPROXIMATELY NORMAL (ВУ THE
CENTRAL LIMIT THEOREM), ANP ITS
STANPARP ERROR IS
я(хгхг) s
(THE VARIANCES АРР, SIN6E
SAMPLES ARE INPEPENPENT.) NOW
WE £AN PRO6EEP PIREOTLY TO:
confidence
нет(бич$-_
Look!
Formula I1
intervals: FOR
LAROE SAMPLE SIZES, THE (A-a)WO%
£ONFlPEN£E INTERVAL FOR THE
PIFFEREN6E BETWEEN MEANS IS
А'Л - %t-%2 ± Za $E(X-X2)
(?l64AT \U THE
п,
hypothesis testing: WE ASSESS
THE NULL HYPOTHESIS THAT THE TWO POPULATION MEANS ARE EQUAL
: Al =
THE TEST STATISTIC IS
_ Л,-Лг
ANP THE P-VALUES WORK IN
THE USUAL WAX
and how about comparing
SMALL SAMPLE
MEANS?
REMEMBER 6HAM£L£ON №TOR4? THEIR COMPETITOR, l&UAMA AUTO, CLAIMS
THAT ITS STYROFOAM HOOP ORNAMENT &IVES BETTER FRONT ENP CRASH
PROTECTION, ANP THEY’VE CRASHEP t£V£M 16UAW& TO PROVE IT'
X----------------------------------------------------------------------
170
THE t DISTRIBUTION CAN BE USED
IF BOTH POPULATIONS ARE MOUND
SHAPED AND HAVE THE SAME
STANDARD DEVIATION <7=<7, =cr2.
THE ONLV WRINKLE IS THAT WE
HAVE TO POOL THE SUM OF
SQUARES ABOUT THE MEANS TO
FORM A SINGLE ESTIMATE OF cr:
Ъ _ бм)*'?' т %,
Pw?L
THE STANDARD ERROR IS THE SAME AS
FOR LARGE SAMPLES, EXCEPT THAT
Sp^L REPLACES S, AND S2=
я(х,-х,) =
$₽•«. ,
— — 4- - “
HJ-
THE G-a>lDD% CONFIDENCE
INTERVAL IS
”* i~'^2 ~
2
WHERE ta IS A CRITICAL VALUE OF t
WITH Щ-Пг-2 DEGREES OF FREEDOM.
THE REPTILIAN CARMAKERS AGREE THAT THEIR STANDARD DEVIATIONS ARE
CLOSE AND THEIR REPAIR HISTOGRAMS ARE MOUND-SHAPED. ТНЕУ COMPUTE;
^POOL ~
4 7992 4- 6-32g2
W
= wy/l^ = 194
THE 95% CONFIDENCE INTERVAL IS
^-^2 -5A0-300 + t^CKA)
= 24D ± (2ЛЗХ154)
240 ± ЗАО
SIN^E THIS INCLUDES THE VALUE O,
IGUANA AUTOS HAS HOT SHOWN A
SIGNIFICANT IMPROVEMENT IN
REPAIR COSTS.
171
HERE’S AN EXAMPLE TMAT SHOWS THE
PITFALLS OF MINPLESSLY FOLLOWING
TME COOKBOOK A LAR6-E TAXI FLEET
OWNER WANTS TO COMPARE TME 6AS
MILEAGE USIN6 6A5 A ANP 6A5 9-
< i'M A A'"">
LAP6E OWNER
VIITM A
la(?6»& fleet.'
\_______________________________________________________________________7
STARTING WITM WO U&5. ME RAN POM LY ASSIGNS 50 TO EA£M 6ASOLINE, ANP,
AFTER A PAY’S PRIVIN6-, PETERMINES
SAMPLE SIZE MEAN MILEAGE STANPARP PEVIATION
A 50 25 5.00
В 50 26 A.00
172
OWING TO THE LARGE STANPARP
PEVIATIONS, THE STANPARP ERROR IS
PRETTY SUBSTANTIAL:
^бл-л2;- 4k Ль
1 a,
- J 2^ Ik ~
' V чю ‘So
« 305
AT THE 95% CONFIPENCE LEVEL, WE HAVE
Z*i—= Я?, ~~ i
» -1 ± 6.96X.9P5)
. -1 ± 1.774
THIS INCLUPES THE VALUE O,
CORRESPONPING TO
THE P-VALUE FOR THE ALTERNATE
HYPOTHESIS, Ня: A*/U2,,$
Pr6lzl>lzoj? = Pr(lzl>
® Prdzl^l.O = 261357?
- .2714
/'"'X TOTAL SHAPED
/ \ AREA x .1714
THIS ЕХ6ББР6 THE a » .P5
SIGNIFICANCE LEVEL, SO WE
CONCLUPE THAT THE EVIPENCE
IN FAVOR OF EITHER GAS IS
VERY WEAK.
- /./ /•/
113
PAIRED COMPARISONS
a better way to compare gasolines
TME TAXI OWNER FOLLOWS? TME
COOKBOOK EXACTLY. HIS SAMPLES
WERE RANPOM, ANP MIS SAMPLE
SIZE WAS LAR6-E ENOUGH- HE
JUST FAILEP TO THINK WHEN
NE^ESSARX'
ALTHOUGH 6AS В APPEARS TO BE SLIGHTLY BETTER THAN 6AS A, TME
^ONFIPEN^E INTERVAL WAS WIPE BECAUSE OF TME LAR6-E STANPARP
PEVIATIONS—I.E., TME VARIEP WIPELY FROM ONE OAB TO THE
NEXT. WMY SU6M MI6-M VARIABILITY? BECAUSE ^ABS-ANP CABBIES—HAVE
PIFFERENT PERSONALITIES!
174
A FAR BETTER WAY TO PO THIS STUPY 15 TO ASSIGN CAS A ANP CAS В TO THE
MMG CA9 ON PIFFFRFNT РАУ5-
WE STILL RANPOMIZE THE TREATMENT ВУ FLIPPIN6 A COIN TO PE6IPE
WHETHER TO USE CAS A ON TUESDAY OR WEPNESPAY. WE CAN ALSO 6UT THE
EXPERIMENT POWN TO 10 SAVINC THE OWNER A LOT OF TIME ANP
MONEY.'
my л CAB CAS A 6AS В DIFFERENCE
To ToSSI J 1 27.01 26.95 0.06
2 20.00 2P.44 -0.44
3 23.41 25.P5 - 1.64
4 25.22 26.32 -1.10
5 30.11 2956 0.55
b 25.55 26.6P - 1.05
7 22.23 22.93 -0.70
6 19.76 2P.23 -0.45
9 33.45 33.95 -0.50
10 25.22 26.01 -0.79
MEAN 25.20 25.60 - 0.60
STANPARP PEVIATION 4.27 4.10 0.61
NOTE THAT THE MEANS ANP STANPARP PEVIATIONS OF CAS A ANP CAS В ARE
ABOUT THE SAME. THAT’S TO BE EXPECTED, SINCE ТНЕУ HAVE THE SAME SOURCE
OF VARIABILITY AS IN THE UNPAIREP EXPERIMENT. BUT NOW THE PIFFFRFNCF
COLUMN HAS A VERY MALL STANPARP PEVIATION. THE DIFFERENCE COLUMN,
ВУ COMPARINC CAS PERFORMANCE WITHIN A SINCLE CAR, ELIMINATES
VARIABILITY BETWEEN TAXIS.
\_____________________________________________________________________________7
П5
THE PlFFEREN^ES di PROVIPE A
ЯЫЫ.Б MEASURE OF
PlFFFRFHCF FOR £А£Ц TAXI,
ANP WE £AN USE IT TO MAKE A
SMALL-SAMPLE t TEST STATISTIC
THE 95% dONFIPENdE INTERVAL AROUNP d IS
А/ ~ — ^.029 (^d/]/7r)
SAMPLE £RITI£AL STANPARP
MEAN VALUE ERROR
= -.60 ±.44 -104
SO WE HAVE -1.04 < p.d < -.16 WITH 95% dONFlPENdE, 6OOP EVIPENСE THAT
6-AS В REALLY IS BETTER.
THE MVPOTMESIS-TESTIN6- P-VALUE <JAN BE FOUNP USIN6- A SOFTWARE
PA<JKA6-E-.
Нл-л^о
P-VALUE = Prdtl >
= PrCItl >£)
Prdtl > 3.15)
= .012 < .05
A6-AIN, 6.AS В PASSES THE TEST.
176
THE PREDOMINATE OF RI6MT-
LEANIN6- LINES IS A STRONG
HINT THAT 6A6 В GIVES
BETTER MILEAGE.
VlHAT
RlMTLfcMWk
LIHE^*? >
177
A PAIREP COMPARISON EXPERIMENT IS ONE OF TME MOST EFFECTIVE WAYS TO
REPUCE NATURAL VARIABILITY WHILE COMPARING TREATMENTS. FOR EXAMPLE, IN
COMPARING MANP CREAMS, TME TWO BRANPS ARE RANPOMLY ASSI6-NEP TO
EACM SUBJECT’S RI6-MT OR LEFT MANPS. THIS ELIMINATES VARIABILITY PUE TO
SKIN PIFFERENCES.
OR, IN COMPARING TWO BREAKFAST CEREALS, EACM TASTER RATES BOTM
CEREALS ON RANPOM ORPER). TME PAIREP COMPARISON REMOVES TME
NATURAL BIAS OF TME TASTER FOR OR ASAINST CEREAL IN GENERAL-
178
IN THIS CHAPTER, WE APPLIEP THE
BASIC I PEAS ABOUT CONFIPENCE
INTERVALS ANP HYPOTHESIS
TESTING TO THE COMPARISON OF
TWO POPULATIONS. THERE ARE
INNUMERABLE FURTHER POSSI-
BILITIES. WE COULP HAVE 6ONE ON
TO PESCRIBE COMPARISONS OR
ф THE STANPARP
PEVIATIONS OF TWO
POPULATIONS WHEN
SAMPLE SIZE IS
SMALL,
ф THE MEANS OF
MORE THAN TWO
POPULATIONS WHEN
SAMPLE SIZE IS
LAR6-E,
ф THE MEANS OF
MORE THAN TWO
POPULATIONS WHEN
SAMPLE SIZE IS
SMALL,
ETC!
IN PRACTICE, STATISTICIANS PETERMINE THE GENERAL NATURE OF THE
PROBLEM, ANP THEN CONSULT THE RI6-HT REFERENCE BOOK.
I_______________________________——----------------------------------'
THE ONLY THIN6- REALLY NEW
IN THE CHAPTER WAS THE IPSA
OF THE PAIRZP tOMPARMOU
TE5T. IN THE NEXT CHAPTER,
WE'LL LOOK AT SOME OTHER
KINPS OF EXPERIMENTAL
PESI6NS.
IT?
180
♦ Chapter 10f
EXPERIMENTAL
DESIGN
THE PESION OF AN EXPERIMENT OFTEN SPELLS SUCCESS OR FAILURE-
IN THE PAIREP COMPARISONS EXAMPLE, OUR STATISTICIAN CHANOEP
ROLES FROM PASSIVE NUMBER OATH ER I NO ANP ANALYSIS TO ACTIVE
PARTICIPATION IN THE PESlON OF THE EXPERIMENT.
you
Ride FREE
. МУ TIME-
isi
IN THIS CHAPTER, WE
INTROPUCE THE BASIC
IPEAS OF EXPERI-
MENTAL PESIGN,
WHILE LEAVING THE
PETAILEP NUMERICAL
ANALySlS TO VOUR
HANPy STATISTICAL
SOFTWARE PACK.
NO WMXMIN
IMS СнМж-
c\
THE ELEMENTS OF A PESIGN ARE THE EXPERIMENTAL UNIT* ANP THE
TREATMENT* THAT ARE TO BE ASSIGNEP TO THE UNITS. THE OBJECTIVE OF
ANY PESIGN IS TO COMPARE THE TREATMENTS-
шюпшлпк
FOR MEPICAL TRIALS, THE PATIENT*
ARE THE UNITS, ANP THE PRU6* ARE
THE TREATMENTS. IN THE MILEAGE
EXAMPLE, THE EXPERIMENTAL UNITS
ARE TAXICABS, ANP THE TREATMENTS
TO BE COMPAREP ARE GAS A
ANP GAS B.
IN AGRICULTURAL EXPERIMENTS, THE EXPERIMENTAL UNITS ARE OFTEN PLOTS
IN A HELP, ANP THE TREATMENTS MIGHT BE APPLICATION OF PIFFERENT
WHEAT VARIETIES, PESTICIPES, FERTILIZERS, ETC.
ТОРАУ, EXPERIMENTAL PESlGN I PEAS
ARE USEP EXTENSIVELY IM INPUSTRIAL
PROZES5 OPTIMIZATION, MEPIZINE
AMP SOZIAL SZIENZE- АМУ EXPERI-
MENTAL PESlGN USES THREE BASIZ
PRINZIPLES, WHICH ARE CLEARLY
ILLUSTRATE? IM OUR CAB EXAMPLE'
Replication: the same
TREATMENTS ARE ASSIGN EP TO
PIFFERENT EXPERIMENTAL UNITS.
WITHOUT REPLICATION, IT’S
IMPOSSIBLE TO ASSESS NATURAL
VARIABILITY ANP MEASUREMENT
ERROR.
Local control refers
TO ANY METHOP THAT ACCOUNTS FOR
ANP REPUCES NATURAL VARIABILITY.
ONE WAY IS TO GROUP SIMILAR
EXPERIMENTAL UNITS INTO BLOZKS.
IN THE CAB EXAMPLE, BOTH GASO-
LINES WERE USEP IN EACH CAR, ANP
WE SAY THAT THE CAB IS A BLOCK.
Randomization:
THE ESSENTIAL STEP IN ALL
STATISTICS! TREATMENTS MUST BE
ASSIGNEP RANPOMLY TO EXPERI-
MENTAL UNITS. FOR EACH TAXI, WE
ASSIGNEP GAS A TO TUESPAY OR
WEPNESPAY ВУ FLIPPING A COIN. IF
WE HAPNT, THE RESULTS COULP HAVE
BEEN RUINEP BY PIFFERENCES
BETWEEN TUESPAY ANP WEPNESPAY.'
183
MOW SUPPOSE WE WANT TO INVESTIGATE THE EFFEGT OF TWO BRAN PS OF
TIRES AS WELL AS TWO GASOLINES WE HAVE FOUR POSSIBLE TREATMENTS,
WHIGH WE aN LAV OUT IN A TWO-BY-TWO FACTORIAL PESIG-N. THE TWO
FAGTORS ARE GAS ANP TIRE MAKE.
WE GAN ASSIGN THE FOUR TREATMENTS AT RANPOM TO FOUR PIFFERENT PAYS
FOR EAGH CM. ALU FOUR TREATMENTS (a, b, C, ANP d) ARE REPEATEP
WITHIN EAGH BLOGK GGABZ THIS IS GALLEP A COMPLETE RAKPOMIZEP BLOCK
v PESlGN.,
SO FAR, WE HAVE
ASSUMEP THAT EVERY
PAY OF THE WEEK IS
THE SAME, BUT WE GAN
CONTROL FOR THIS,
TOO, IN THE
FOLLOWING WAY; USE
ONLY FOUR CAM, ANP
ASSIGN THE
TREATMENT AGGORPING
TO THE TABLE AT
RIGHT:
PAY
1 2 3 4
GAB 1 a b c d
2 b C d a
3 c d a b
4 d a b c
194
A FOUR-BY-FOUR TABLE
W/ГГИ FOUR PIFFERENT
ELEMENTS, EAZH APPEARING
ONCE IN EVERY COLUMN
ANP ROW, ft CALLEP A
Latin square.
IN THIS EXPERIMENT, THE
FOUR PAYS ANP FOUR CABS
6>£T ALL FOUR TREATMENTS
EXACTLY ONCE.
THE RANPOMIZATION STEP
PICKS A SIN6LE LATIN SQUARE
PESl&N AT RANPOM FROM A
LIST OF ALL POSSIBLE FOUR-
WAY LATIN SQUARES.
IF FOUR UNITS ISN’T ENOUS-H, WE CAN INCREASE THE NUMBER OF
EXPERIMENTAL UNITS BY REPEATING THE EXPERIMENTAL PESIE-N. STARTIN6-
WITH EIS-HT CABS, WE COULP PIVIPE THEM INTO TWO GROUPS OF FOUR ANP
THEN REPEAT THE PESI6N WITHIN EACH 6-ROUP.
)B5
'w PROMI*EP HOT TO GO INTO THE PATA ANALY*I* IN ANY PETAIL, BUT HERE Л
I* ROUGHLY HOW A GOMPLEX PE*IGN LIKE THI* I* HANPLEP.
EXPERIMENTAL PE*IGN* ARE ANALYZEP ВУ ALLOCATING TOTAL VARIABILITY
AMONG- PIFFERENT *OURGE*. IN THE GAB EXAMPLE, THE *OURGE* OF
VARIABILITY ARE THE GAB, THE TIRE MAKE, GA* ТУРЕ, РАУ—ANP RANPOM
ERROR ANALY*I* OF VARIANGE, АА/ОИ4 FOR *HORT PARTITION* THE TOTAL
VARIATION, ALLOGATING- PORTION* TO EAGH *OURGE-
IN THE NEXT GHAPTER, WE EXPLAIN IN
PETAIL ONE MOPEL FOR ANALYZING
GOMPLEX PE*IGN* THE LINEAR
REGRESSION ANOPEL. IN LINEAR
REGRE**ION, YOU’LL BE ABLE TO *EE
AA/OVA VP GLO*E ANP NUMERIGAL...
166
♦ Chapter 11 ♦
REGRESSION
50 FAR, WE’VE POME STATISTICS ON ONE VARIABLE AT A TIME, WHETHER IT
CAME FROM A POPULATION OF PILL TAKERS, PICKLES, OR CRASHEP CARS. IN
THIS CHAPTER, WE’LL SEE HOW TO RELATE TWO VARIABLE* 6IVEN THE
WE&HTf OF THE 92 STUPENTS |N CHAPTER 2, WE ASK HOW THEY ARE RELATEP
TO THE STUPENTS’ HEI6HT*
THIS 15 AN EXAMPLE OF A BROAP CLASS OF IMPORTANT QUESTIONS; POES
BLOOP PRESSURE LEVEL PREPICT LIFE EXPECTANCY? PO M.T. SCORES
PREPICT COLLEGE PERFORMANCE? P0E5 REAPING 5TATI5TIC5 BOOKS MAKE
YOU A BETTER PERSON?
IN MATH /OU PROBABLY
LEARNEP TO SEE RELATIONSHIPS
PISPLAYEP AS 6RAPM. 6IVEN %,
YOU CAN PREPICT v.
BUT IN STATISTICS, THIN6-S
ARE NEVER SO CLEAN' WE
KNOW (OR SUPPOSE WE
KNOW? THAT HEI6-HT HAS
AN INFLUENCE ON WEIGHT—
BUT IT'S NOT THE SOLE
INFLUENCE. THERE ARE
OTHER FACTORS, TOO, LIKE
SEX, АСЕ, ВОРУ ТУРЕ, ANP
Я4А/РО/И VARIATION.
FOR THIS CHAPTER, LET'S LABEL THE WEI6-HT PATA AS у ANP THE HEIGHT PATA
AS X. THUS (%,, y,J IS THE HEI6-HT ANP WEI6-HT OF STUPENT i. WE PISPLAY
THE POINTS (X,, y,) IN A 2-PIMENSlONAL POT PLOT CALLEP A ^ATTERPLOT.
(SOME OF THE POTS ARE BI6^ER, BECAUSE THEY REPRESENT TWO OR THREE
STUPENTS WITH THE SAME HEIGHT ANP WEIGHT.?
100
6AM WE PREPI6T A STUPENTS WElG-HT у FROM HIS OR HER HEIGHT %?
Regression analysis
FITS A STRAIGHT LIME TO
THIS MESS/ SGATTERPLOT.
% IS 6ALLEP THE
INPEPENPENT OR
PREPICTOR VARIABLE, AMP
у IS THE PEPENPENT OR
RESPONSE VARIABLE. THE
REGRESSION OR PREPIGTION
LIME HAS THE FORM
у - a+b%
TO ILLUSTRATE THE FITTING PROCESS, LETS USE A SMALLER, RIGGEP PATA SET
WITH ONLY А//А/Г STUPENT HEIGHT-WEIGHT PAIRS;
HEIGHT WEIGHT
bO 64
bl 95
64 140
66 155
66 119
70 175
71 145
74 197
76 150
200 -
О)
'со
5
150
.0
250 П
О
ф
О
О
О
о
100
о
о
50
60 65 70 75
height
NOW HOW PO WE 6-ET THE BEST-FITTING- LINE?
169
THE IPCA IS ТО MINIMIZE
ТИС TOTAL *PRCAP OF THE
у VALUE* FROM ТИС LING.
JU*T A* WHEN WE PEFINCP
ТИС VARIANCE, WE LOOK AT
ALL ТИС SQUAREP у
PISTANOES FROM ТИС LING,
ANP APP THEM UP TO 6GT
ТИС OF SQUAREP
ERRORS:
n
- Yjtyi-fa)2
i-1
IT’* AN A66RG&ATG MEASURE OF HOW MUCH ТИС LINE'* "PREPICTEP y,/
OR J,, PIFFCR FROM ТИС ACTUAL PATA VALUE* y,.
Bifr **E
The regression or
least squares line
I* THC LING WITH THC SMALLEST
HI*TORlCAL NOTE'- WHY PO WE CALL THI* PROCEPURE REGRESSION
ANALYSIS? AROUNP THE TURN OF THE CENTURA GENETICIST FRANKS
GALTON PI*COVEREP A PHENOMENON CALLEP REGRESSION TOWARP
THE MEAN. SEEKING LAW* OF INHERITANCE, HE FOUNP THAT SONS'
HEIGHTS TENPEP TO REGRESS TOWARP THE MEAN HEIGHT OF THE
POPULATION, COMPAREP TO THEIR FATHER*’ НБ16-НТ* TALL FATHER*
TENPEP TO HAVE 5OMEWUAT *MORTCR SONS, ANP VICE VER*A 6ALTON
PEVELOPEP REGRESSION ANALYSIS TO *TUPY THl* EFFECT, WHICH HE
OPTlMI*TlCALLy REFERREP TO A* "REGRESSION TOWARP MEPIOCRITX'
МОТ ТО BEAT AROUMP THE BUSH, WE
б-IVE WITHOUT PROOF THE REORE55IOM
LIME'* FORMULA: ГГ$ МЕ55У BUT
COMPUTABLE.
WHERE
АМР
У(Х-%)2
(HERE % AMP ARE THE MEAMS OF
{%,} AMP {yj RESPECTIVELY J
ЬО we’ll
$«P IT?
SOU CAH ACTUALLY
WE THI* MATH
IMTUITIVE- PUT you
HAVE TP GO IMTO
H-PIAAbK^OHAL
WE TO t>0 IT...
у •=• а+Ь%
а -
BECAUSE SOME OF THESE EXPRESSIONS WILL SHOW UP A6-AIM, WE ABBREVIATE
THEM:
- Z^(^i ^UM OF %>UARE^ AROUMP
Cs THE MEAM, ТНБ5Б MEASURE
-i. Cf THE 5PREAP OF %, AMP
= E<yrP2 J
i^l
= У ^-Я)С2/-5) PR0PU6T
(with ььлл) тн£ comtm b.
w
FOR TUG RI66EP PATA, HERE'* THE WHOLE COMPUTATION-.
Zj ¥i (Zj-X) (хГх')г (¥г^
bo 04 -0 -5b 64 3136 440
62 95 -b -45 36 2025 270
b4 140 -4 0 16 О 0
bb 155 -2 15 4 225 -30
b0 119 0 -21 О 441 0
70 175 г 35 4 1225 70
n 145 4 5 16 25 20
74 197 b 57 36 3249 342
7b 150 0 10 64 100 00
5UM-612 \uo •МО * 10426 9Ь^12ОО
%^b0 ^140
192
ANOVA
IM TEZHHICKL
-TERMS, HOW
pAP IS THE
SLOP?
<AS PROMlSEP, OR THREATENEP/)
NOW WE ASK: IF THIS IS THE
BEST FIT, HOW 6OOP IS IT?
AS УОи aN IMAGINE, THE ANSWER TO THIS QUESTION PEPENPS ON HOW
SLOPPlLy THE PATA POINTS ARE SPREAP OUT, I.E., HOW Bl£ SS£ IS, RELATIVE
TO THE TOTAL SPREAP OF THE PATA SOME EXAMPLES:
193
LET'S QUANTIFY THIS BY
APPORTIONING THE VARIABILITY
IN U. REFER TO THE PICTURE AT
RIGHT FOR GUlPANCE. WE LET
THUS, y.i ARE THE PREPIGTEP
WEIGHTS PETERMINEP ВУ ТИБ
REGRESSION LINE.
ANOVA table
SOURCE OF VARIABILITY SUM OF SQUARES VALUE FOR RlGGEP PATA
REGRESSION ERROR i-1 «‘ЁН’’1 i = f 6000 4426
TOTAL IP,426
(BY THE WAY, IT IS NOT OBVIOUS THAT SS^ = SSR + SSE-BUT IT'S TRUE.'?
ANYWAY, HERE IS HOW THE REGRESSION AMP ERROR SUMS OF SQUARES ARE
^ALZULATEP FOR THE RlGGEP PATA SET. WITH у = -2PP + SX.
REGRESSION ERROR
Ъ Ф-Р2
6P 94 100 -40 1600 -16 296
62 95 110 -30 900 -15 229
64 140 120 -20 400 20 400
66 155 130 -10 100 29 629
69 119 140 0 0 -21 441
70 175 190 10 100 25 629
72 145 160 20 400 -15 229
74 197 170 30 900 27 729
76 190 100 40 1600 -30 900
X-69 f^14O 56R • 6000 56E-4426
55/? MEASURES THE TOTAL
VARIABILITY PUE TO THE
REGRESSION, I.E., THE
PREPIOTEP VALUES OF
55Г WE’VE ALREAPY MET.
NOTE THAT
5SE
44
К ШМШ
EXPRESSION
fcPTUE "SLoV!
15 THE PROPORTION OF
ERROR, RELATIVE TO
TME TOTAL SPREAP.
The squared correlation
IS THE PROPORTION OF THE TOTAL SSyy
AZZOUNTEP FOR BY TME REGRESSION:
Rl.^. .1-
(BECAUSE SSR = 55yy-55E). R2 IS
ALWAYS LESS THAN 1. TME CLOSER IT
IS TO 1, THE TIGHTER TME FIT OF
TME f^URVE- R2 » 1 TORRES PON PS
TO PERFECT FIT.
aUULATlNG R2 FOR TME
RIGGEP PATA SRT, WE GET
R2 . -^22- = .90
50% OF TME VARIATION IN
WEIGHT IS EXPLAINER BY
HEIGHT. TME OTHER 42%
IS ’’ERROR."
ALTERNATELY, THE
correlation
coefficient
15 THE 5QUARE ROOT OF R2 WITH
THE 5I6N OF b.
г = 65I6N OF b) 'vtf1
МЕДК5 that % i$
THU5, Г 15 + IF THE LINE 6OE5
UP TO THE RI6HT ANP - IF IT
6OE5 POWN TO THE RIOHT.
HE6ATNE r
RELATE? TO 4.’
Г MEA5URE5 THE TI6-HTNE55 OF FIT, A5 WELL A5 5AYIN6- WHETHER IN£REA5IN6-
% MAKE5 GO UP OR POWN.
196
NOW LET'S BE
HONEST; NOBOPy-
WELL, ALMOST
NOBOPy-POES
THESE CALCULATIONS
ВУ HANP ANyMORE.
WITH A COMPUTER,
ALL THIS WORK CAN
BE PONE IN ОПС
Line of cove...
IN FACT, THIS ENTIRE
ROOK CAN BE СолА-
PRES^EP IHT£> THE
HEAP OF A
WllSTlUM..
USINO THE М1П1ТАВ STATISTICAL SOFTWARE SySTEM, PEVELOPEP AT PENN
STATE, THE SINGLE COMMANP LOOKS LIKE THIS-.
ПТВ > regress 'weight' on 1 independent uariable 'height'
ANP THE RESULTS ARE
The regression equation is
HEIGHT = - 200 + 5.00 height
Pred i ctor Coe f St deu t-rat i о
Constant -200.0 Г10.7 -1 .81
he i ght 5.000 1 .623 3.08
s * 25.15 R-sq = 57.5X R-sq(adj)
Analysis of Variance
SOURCE OF SS П5
Regression 1 6000.0 6000.0
Error
7 4426.0 632.3
Total 8 10426.0
197
NOW LET'S PO IT TO THE REAL
PATA OF 92 STUPENTS-.
ПТВ > regress 'weight' on 1 independent uariable 'height'
ANP THE RESULTS
The regression equation is
HEIGHT - - 205 + 5 .09 HEIGHT
Pred i ct or Coef St deu t-rat 1 о p
Constant -204,74 29.16 -7.02 0.000
he i ght 5.0918 0.4237 12.02 0.000
s * 14.79 R-sq - 61 .6X R- sq(adj) 61 . 2X
Analysis of Variance
SOURCE OF SS ns F P
Regress i on I 31592 31592 144.38 0.000
Error 90 19692 219
Total 91 51284
HERE IS THE SCATTERPLOT WITH THE FlTTEP REGRESSION LINE. THE CORRELATION COEFFICIENT FOR THIS 2 PATA SET IS | Г = +1Г616 =.70 riO ibi u i о 200 - „ "08 8 °o8J> I»-... ° о аЛтГю оp ° ° Jo и i о 100 - 0 ° 50 । т 1—i 1 60 65 70 75 height 190
STATISTICAL
INFERENCE
UP TO NOW, WE HAVE BEEN
POING PATA AHALYW,
PESCRIBING THE NEAREST LINEAR
RELATIONSHIP BETWEEN THE
OBSERVE? PATA % ANP NOW
LET'S SHIFT OUR POINT OF VIEW,
ANP REGARP THE 92 STUPENTS
AS A SAMPLE OF THE
POPULATION OF STUPENTS AT
LARGE. WHAT CAN WE INFER?
A RGGRCMoM ЯОРО. FOR THE WHOLE POPULATION IS A LINEAR
RELATIONSHIP __________
cr+ + g
У IS THE PEPENPENT RANPOM VARIABLE; % IS THE INPEPENPENT VARIABLE
(WHICH МАУ OR МАУ NOT BE RANPOM); a ANP 0 ARE THE UNKNOWN
PARAMETERS WE SEEK TO ESTIMATE; ANP 6 REPRESENTS RANPOM ERROR
FLUCTUATIONS.
</
FOR THE HEIGHT
VS. WEIGHT MOPEL,
У IS WEIGHT, % IS
HEIGHT, a ANP £
ARE UNKNOWN, ANP
yOU CAN THINK OF
e AS THE RAMPOH
COflPOUGKT OF
THE WEIGHTS У
FOR EACH VALUE
OF HEIGHT %.
199
THE PISTRIBUTION OF 6 IS IN FAOT MFFOfNT FOR AFFERENT VALUES OF
5-FOOTERS VARY LESS IN THEIR WEIGHT THAN 6-FOOTERS. NEVERTHELESS, WE
NOW MAKE A SIMPLIFYING ASSUMPTION-. LET'S SUPPOSE THAT FOR ALL VALUES
OF %, THE £’S ARE 1ЫРБРБМРБНТ, NORMAL, ANP HAVE THE SAME’ 5TAMPARP
Ю... MAYtt THE WEIGHT
MOPEL MIGHT BE
У * -125+4^ + e
G IS NORMAL WITH A - О
ANP 15 POUNPS (SAY).
THEN, A^ORPING TO THIS
MOPEL, STUPENTS WHO ARE
6'A* (76 INCHES) HAVE THE
PISTRIBUTION OF
У - -125 + 4<7б?+6
= 175+6
SO, FOR % = 76, У IS NORMAL
WITH MEAN 175 ANP STANPARP
PEVIATION 15 POUNPS.
•zoo
HOW, GIVEN THE MOPEL Y Л + + G, WE WANT TO PO AS WE’VE PONE
REPEATEPLV IN THE LAST FEW CHAPTERS: TAKE A fAflPLC ANP USE IT TO
С5ПМАТС a ANP /?.
ONE CAN SHOW THAT THE
a ANP b WE GOT ВУ THE
LEAST-SQUARES METHOP
are BLUE, the Best
Linear Unbiasep
Estimators of a anp p
(WHATEVER THAT MEANS/).
AS USUAL, PIFFERENT SAMPLES VIELP PIFFERENT COLLECTIONS OF PATA,
WHICH GENERATE PIFFERENT REGRESSION LINES. THESE LINES ARE
D&TRIVUTCP AROUNP THE LINE У - Л+ + €. OUR QUESTION BECOMES:
HOW ARE a ANP b PlSTRIBUTEP AROUNP a ANP Д RESPECTIVELY ANP HOW
PO WE CONSTRUCT COMFIPCMCC INTERVAL# ANP T&T НУРОТЦ&&?
2/71
FOR EACH PATA POINT у, Л
WE HAVE
JG - a + />%,+ e,
WHERE е{ = y.i - 1^
THE ^-PlSTAHCE OF
FROM THE REGRESSION
LINE- THE e, ARE SAMPLE
VALUE* OF e, ANP THEY
GIVE V* AN ESTIMATOR S
FOR trfe)-.
(WHY Z?-2 IN THE PENOMINATOR? BECAUSE WE HAVE USEP UP TWO PEGREES
OF FREEPOM TO COMPUTE a ANP b, LEAVING Z?-2 INPEPENPENT PIECES OF
INFORMATION TO ESTIMATE cr.)
ALTHOUGH IT ISN’T OBVIOUS,
WE CAN ALSO WRITE S AS
П-2.
A FORMULA WHICH ALLOWS
US TO COMPUTE S
PIRECTLY FROM THE
SAMPLE STATISTICS.
TO REPEAT, S IS AN ESTIMATOR OF //OW И7РД.У
ТИБ DATA POINT* WILL EE SCATTERS^
AROUND THE LINE
2Z72
confidence intervals
ТИБ 95% £ONFIPEN6E INTERVALS
FOR a ANP fi HAVE THAT OLP,
FAMILIAR FORM:
Р=Ь±Ьп,ЪЕ(Ь)
a ~a ±
WHERE WE USE THE t PISTRIBUTION
WITH z?-2 PETREES OF FREEPOM
6FOR THE SAME REASON AS ABOVE?.
У£^.. UDPKS LtK
ТИ₽ С/МПрЕ LAC0?
ALMONP ToRrt
троил тне wrtfy
op тле
THE STANPARP ERRORS, HOWEVER, LOOK RATHER UNFAMILIAR. THEY ARE
(WITHOUT PERIVATIONZ
*'«- vta
SE"”'T <
WHAT HAPPENEP TO OUR PREVIOUS ^=? IT WAS REPLA£EP ВУ SS„. LIKE n,
$$xx INCREASES AS WE APP MORE PATA POINTS, BUT IT ALSO REFLECTS THE
TOTAL. 5PRCAD OF THC % DATA. FOR EXAMPLE, IF ALL 6TUDCUT4 MflPLCD
HAD THC 4A&C HC16HT, WOULP BE UNJUSTIFIEP IN PRAWIN6- АМУ
(ON6LUSION ABOUT THE PEPENP£N(£ OF WEI6HT ON HEIGHT. IN THAT £ASE,
SS^» O, 6MN& /> = <» ANP IHFIMTCLP WIPC £ONFIPEN(£ INTERVALS.
MORE QUE5TION5
HOW WELL 6АЫ WE PREPlOT
THE MEAN RE4PON4E У AT
A FlXEP VALUE W FOR
INSTANCE, WHAT 15 THE
MEAN WEI6HT OF 5TUPEMT5
OF HEIGHT 76 IN6HE5? THE
95% 6ONFIPEN4E INTERVAL
FOR У = a + 0ZD 15
0^(3%q —* Я + ЬуС/Q i t 2^ )
WHERE
5UPPO5E A NEW 5TUPENT ENROLL5, WHO HA5 HEIGHT HOW WELL CAN
WE PREPlOT YNEW WITHOUT MEA5URINO IT?
THE 95% PREPIOTION INTERVAL
FOR A NEW INPIVIPUAL YNCW
WITH OB5ERVEP %NEW 15
У NEW ® л + b^NEW ^.P25 ^(Ynew}
WHERE
_ < | / ^'NCW
^YnEW^® V + + “44----
height
BOTH THE5E 5TANPARP ERROR5 CONTAIN A TERM
THAT 6ROW5 LAR&ER A5 THE %-VALUE, OR
^NEW • 6ET5 FARTHER FROM THE MEAN VALUE %
WHY P0E5 THE ERROR IN6REA5E FARTHER FROM
BE6AU5E, IF YOU WI55LE THE RE5RE55ION
LINE, IT MAKE5 MORE OF A PIFFEREN6E FARTHER
FROM THE MEAN.' ("REMEMBER, THE LINE ALWAY5
PA55E5 THROUGH
2/24
LETS WORK IT OUT FOR THE
RI&6EP PATA: FOR THE Ж4А/
WCI&IT WHEN 76 INCHES»
WE HAVE b = —200 ANP Я = 5.
THEN
У = -200 + 5(76) + (2365X25.15)
= 190 ± (23Ь5)(25Л5) V .3777
= WO ± 36.34 POUNPS
THE ESTI/MATEP /MEAN OF
6'4" STUPENTS 15 190
POUNPS, ANP WE'RE 95%
CONFIPENT THAT WE'RE
WITHIN U POUNP5 OF
THE TRUE /MEAN.
FOR A N£W 5TUPENT WHO‘5 6’4”, WE U5E OUR RI66-EP SAMPLE OF NINE
POINT5 TO PREPI6T THAT
yNCW = -200 +5(76) ± (2365)(25.15) у1+~+(ZiL^?
= 190 ± (2365X2.9.51)
- WO ± 70 POUNPS
WE TELL THE
FOOTBALL
6OA^H THAT
WE'RE PRETTY
SURE THE
NEW 6W
WEIGHS
SOMEWHERE
BETWEEN HO
ANP 290!!!
<105
THE INTERVALS ARE PRETTY TERRIBLE' WHAT'S THE PROBLEM? THERE ARE
TWO PROBLEMS, ACTUALLY:
HEIGHT ALONE IS NOT A VERY 6OOP
PREPICTOR OF WEIGHT.
nine pata points weren't enough.
IN PARTICULAR, THERE WAS ONLY OW
STUPENT WITH HEIGHT 76 INCHES.
THE PENN STATE STUPENTS 6IVE BETTER ESTIMATES.
250 -i
200 -
150 -
ф
5
100 -
50
60
65
70
height
2Ot>
hypothesis testing
THE COMPLETE SKEPTIC MIGHT
SUGGEST THAT THERE IS NO
RELATIONSHIP BETWEEN HEIGHT
ANP WEIGHT. THIS AMOUNTS TO
SAVING THAT p--O.
НД6 HO EFFECT 0^
WE TAKE THIS AS THE NULL
HYPOTHESIS.
W = o
IN THAT CASE, THE TEST STATISTIC
t ' b
&(b)
HAS THE t PISTRIBUTION WITH
Z?-2 PEGREES OF FREEPOM.
AS USUAL, THE SIGNIFICANCE ТБ5Т
PEPENPS ON THE ALTERNATE
НУРОТНБ616.
t > ta FOR Ha .0 > О
t<ta FOR Ия s < О
FOR THE RIGGEP WEIGHT PATA, WE
STRONGLY SUSPECT THE ALTERNATE
HYPOTHESIS SHOULP BE
Ня:^>0
WE TEST
_5_
1.62
= 3.(?0
FOR 7 PEGREES OF FREEPOM,
t „ = 1.095. SINCE tO9i > tD5 , WE
REJECT THE NULL НУРОТНБ616 AT THE
a = .05 SIGNIFICANCE LEVEL ANP
CONCLUPE THAT THERE IS A
SIGNIFICANT, POSITIVE RELATIONSHIP
BETWEEN HEIGHT ANP WEIGHT.
207
Multiple linear
regression
WE CAN USE ТИБ SAME PASlC
/PEAS ТО ANALVZE
RELATIONSHIPS BETWEEN A
PEPENPENT VARIABLE ANP
SEVERAL INPEPENPENT
VARIABLES:
poh’t ycxj
IT'S Ju9r mi HFihE
П-» DIMENSIONAL uypcp-
flane im n-space'
NOTHING 7Z2 IT»
6АЦ!
Pou‘T po
TH AT I
FOR EXAMPLE, WEIGHT IS
PETERMINEP ВУ A NUMBER
OF FACTORS OTHER THAN
HEIGHT: A£E, SEX, PIET, ВОРУ
ТУРБ, ETC.
MATRIX ALGEBRA ANP A COMPUTER COMBINE TO MAKE SUCH PROBLEMS EASy
TO ANALVZE.
Non-linear
regression
SOMETIMES PATA OBVIOUSLy
FIT A NON-LINEAR CURVE.
STATISTICIANS HAVE A BAG OF
TRICKS FOR USIN& LINEAR
REGRESSION TECHNIQUES FOR
NON-LINEAR PROBLEMS. THE
SIMPLEST OF THESE IS TO
WRITE У AS A POLYNOMIAL
У = a + + G.
ANP TREAT ANP Я1 AS
INPEPENPENT VARIABLES IN A
LINEAR MOPEL.
109
Regression diagnostics
FITTING A COMPLEX MOPEL TO PATA 6AM SOMETIMES OBS6URE МАМУ
PIFFI6ULTIES. WE USE REGRESSION PIAGMOSTI6 PROCURES TO UN6OVER АМУ
LURKING NASTY SURPRISES.
HAVE VDD EVER
Р1А6К0Ш> Д
6RAFH ESEFORfc,
PR- PLUPPE6U6^UE’
ТИБ SIMPLEST PRO6EPURE IS TO PLOT THE RG4IPUAL4 e, AGAINST THE
PREDATOR REMEMBER, THE ERROR G. IS ASSUMEP TO ВБ 1МРБРБМРБМТ
OF %.
A RAHDO# S6ATTERPLOT INPI6ATES
THAT THE MOPEL ASSUMPTIONS
ARE PROBABLY OK.
АЫУ PATTGRN IMPI6ATES A
PEFINITE PROBLEM WITH THE
MOPEL ASSUMPTIONS.
A TYPI6AL LURKIM6-
MASTY SURPRISE 6WHI6H
EXISTS IM THE
HEI6HT/WEI6HT PATA)
IS THAT ERRORS ARE
HGTGROUGM^T!^. 1.Б,
THE SPREAP OF e
IM6REASES AS V
INCREASES.
2P9
IN THIS CHAPTER, WE
HAVE SUMMARIZE?
THE BASIC (PEAS ANP
TECHNIQUES OF
REGRESSION
ANALYSIS, THE STUPY
OF STATISTICAL
RELATIONSHIPS
BETWEEN VARIABLES.
THIS CONCLUPES OUR
PETAILEP PISCUSSION
OF BASIC STATISTICAL
METHOPS. IN OUR
FINAL CHAPTER, WE'LL
BRIEFLY REVIEW A
FEW REMAINING
TOPICS ANP ISSUES.
< Y£*. >
IN My PROFESSIONAL
opinion, You've
RE6RES6EP
ч ENOU6H- У
гю
♦ Chapter 12<
CONCLUSION
' THE BASIC PRINCIPLES, TOOLS, ANP
CALCULATIONS COVEREP IN TH 15 BOOK CAN
BE EXTENPEP TO SOLVE MORE COMPLEX
PROBLEMS. HERE'S A BIAttP SAMPLE OF
MORE APVANCEP STATISTICAL METHOPS'
211
f DATA DISPLAY
WE SAW HOW TO PISPLAY ONE VARIABLE WITH A POT PLOT ANP 7WO
VARIABLES USING A SCATTERPLOT—BUT HOW PO WE GRAPHICALLY PISPLAY
MORE TUAN TWO VARIABLES ON A FLAT PAGE? AMONG THE MANY
POSSIBILITIES, A CARTOON GUlPE HAS TO MENTION HERMAN OUERNOFF'S
SIMPLE IPEA: USING THE HUMAN FACE, ASSIGN EACH FEATURE TO A VARIABLE
ANP PRAW THE RESULTING OUERNOFF FACES:
t-EWRCNl SLANT
y^EYE SIZE
г-NOSE LENGTH
t —MOUTH LENGTH
0 - FACE HEIGHT
ETC...
Statistical analysis of
MULTIVARIATE DATA
AN ASSORTMENT OF MULTIVARIATE MOPELS HELP TO ANALYZE ANP PISPLAY
Z7-PIMENSIONAL PATA. SOME MULTIVARIATE TECHNIQUES:
Cluster analysis
SEEKS TO PIVIPE THE
POPULATION INTO
HOMOGENEOUS SUBGROUPS.
FOR EXAMPLE, BY ANALYZING
CONGRESSIONAL VOTING
PATTERNS, WE FINP THAT
REPRESENTATIVES FROM THE
SOUTH ANP WEST FORM TWO
PISTINCT CLUSTERS.
in
Discriminant analysis
15 THE REVERSE PROCESS. FOR EXAMPLE, A COLLEGE ADMISSIONS OFFICE MIGHT
LIKE TO FINP PATA GIVING ADVANCE WARNING WHETHER AN APPLICANT WILL GO
ON TO BE A $UCC&$FUL fRAPUATK (PONATE5 HEAVILY TO THE ALUMNI FUNP?
OR AN UNWta&FVL ONE (GOES OUT TO PO GOOP IN THE WORLP ANP 15
NEVER HEARP FROM AGAIN?.
Factor analysis
SEEKS TO EXPLAIN HIGH-
PIMEN5IONAL PATA WITH A
SMALLER NUMBER OF
VARIABLES. A PSYCHOLOGIST
MAY GIVE A TEST WITH WO
QUESTIONS, WHILE SECRETLY
ASSUMING THAT THE
ANSWERS PEPENP ON ONLY
A FEW FACTORS
EXTROVERSION,
AUTHORITARIANISM, ALTRUISM,
ETC. THE TEST RESULTS
WOULP THEN BE 5UMMARIZEP
USING ONLY A FEW
COMPOSITE 5CORE5 IN
THOSE PIMEN5ION5.
IHTEUO02-
on A SCALE FROM ONE Ю T&L
loU'RE T-b EKlfc)\|ER-na>, 4-9'
altruist^, Дир 2.7 AU-THOflriWW-
TVAt'5 ybO, щ Д NUTSHELL I
21Э
THERE 15 AL*O MORE TO
PROBABILITY:
Random walks be^in with
A £OIN FLIP. *UPPO*£ YOU MOVE AHEAP
ONE *TEP FOR A HEAP ANP BA£K ONE *TEP
FOR A TAIL 6U5IN6- TWO £OIN*, YOU 6AN
PO THI* IN TWO PIMEN*ION*J REPEATEP
FLIP* PROPOSE A *ТОШ*Ш PRO£E**
6ALLEP А Я4А/РО/И WALK. RANPOM WALK
MOPEL* ARE V*EP IN 57О/ГК OPTION
TRAPIN6 ANP PORTFOLIO №ANA6G№NT
Time series analysis peal* with pata *et*. whuh,
LIKE THE RANPOM WALK, A&UMJLATF OVER TIM£: LO£AL ANP GLOBAL
TEMPERATURE*, THE PRI^E OF OIL, ETL. IN 77< fFRIK ANALY^f, RANPOM
MOPEL* ARE U*EP TO FOREST FUTURE VALUE*-
2M
WCVQ ALREAPY SEEN HOW THE COMPUTER HELPS WITH ANALYSIS ANP
ARITHMETIC. THERE ARE ALSO SOME STATISTICAL IPEAS THAT OWE THEIR VERY
EXISTENCE TO THE COMPUTER:
Image analysis
A COMPUTER IMA6E MICHT CONSIST OF WOO BY WOO PIXELS, WITH EACH PATA
POINT REPRESENTS? FROM A RANCE OF 16.7 MILLION COLORS AT ANY PIXEL-
STATISTICAL IMACE ANALYSIS SEEKS TO EXTRACT MEANINC FROM "INFORMATION"
LIKE THIS.
we use Pictures, to
UELP U№RCTAKV 1W.
PUT Now WE HAUS To
Pictures !
Resampling
SOMETIMES. STANPARP ERRORS ANP CONFIPENCE LIMITS ARE IMPOSSIBLE TO
FINP. ENTER RESAMPLING, A TECHNIQUE THAT TREATS THE SAMPLE AS THOUGH
IT WERE THE POPULATION. THESE TECHNIQUES CO BY SUCH NAMES AS
RANDOMIZATION, JACKKNIFE» ANP BOOTSTRAPPING.
NACe’. SEEMS
IMPOSSIBLE,
В(3т IT WORKS'
215
resampling (contvd)
TO PO RESAMPLING, THE COMPUTER
RESAMPLES THE SAMPLE
COMPUTES THE ESTIMATE
FOR THE RESAMPLE
REPEATS THE FIRST TWO
STEPS МАМУ TIMES, FIN PIN 6-
THE SPREAP OF THE
RESAMPLEP ESTIMATE*
REMEMBER THE CORRELATION COEFFICIENT r OF THE 92 HEIGHT-WEIGHT PAIRS
OF CHAPTER 11? WHAT'S THE S7AA/PARP ERROR Of Г ? THE COMPUTER TAKES
200 BOOTSTRAP SAMPLES FROM THE 92 PATA POINTS, COMPUTES Г EACH TIME,
ANP PLOTS A HISTOGRAM OF THE Г VALUES.
Bootstrapped Correlations
NOTE THAT THE SPREAP OF THE BOOTSTRAP ESTIMATES IS RELATIVELY SMALL
ANP, FINALLY,
HERE ARE SOME
OTHER ISSUES TO
KEEP IN MINP:
216
DATA QUALITY
SEEMlNGLy SMALL ERRORS IN
SAMPLING, MEASUREMENT, ANP PATA
RECORPING CAN PLAY HAVOC WITH ANy
ANALVSIS. ft 4. F&HSR, GENETICIST
ANP FOUNPER OF MOPERN STATISTICS,
NOT ONLy PESI6-NEP ANP ANALVZCP
ANIMAL BREEPING EXPERIMENTS. HE
ALSO ЛБАНБР ТИБ СА6Б* ANP
ТБМРБР ТИБ AHlMALt, BECAUSE HE
KNEW THAT THE LOSS OF AN ANIMAL
WOULP INFLUENCE HIS RESULTS-
/--------------------------------------------------------------------------------\
MOPERN STATISTICIANS, WITH THEIR COMPUTERS, PATABASES, ANP GOVERNMENT
GRANTS, HAVE LOST SOME OF THIS HANPS-ON ATTITUPE.
IF УОи GRAPHEP THE MEAN
MASS OF RAT PROPPINGS
UNPER STATISTICIANS'
FINGERNAILS OVER TIME. IT
WOULP PROMBf-У LOOK
SOMETHING LIKE THIS:
W
Innovation
ТИБ BEST SOLUTIONS ARE NOT ALWAYS IN ТИЕ BOOK/ FOR EXAMPLE, A
COMPANY HIREP TO ESTIMATE THE COMPOSITION OF A 6ARBA6E PUMP WAS
FACEP WITH SOME INTERESTING PROBLEMS NOT FOUNP IN YOUR STANPARP
TEXT...
Communication
BRILLIANT ANALYSIS IS WORTHLESS UNLESS THE RESULTS ARE CLEARLY
COMMUNICATEP IN PLAIN LANGUAGE, INCLUPING THE PEGREE OF STATISTICAL
UNCERTAINTY IN THE CONCLUSIONS. FOR INSTANCE, THE MEPIA NOW MORE
REGULARLY REPORT THE MARGIN OF ERRORS IN THEIR POLLING RESULTS.
Teamwork
IN OUR COMPLEX SOCIETY, THE SOLUTION TO MANY PROBLEMS REQUIRES A
TEAM EFFORT. ENGINEERS, STATISTICIANS, ANP ASSEMBLY LINE WORKERS ARE
COOPERATING TO IMPROVE THE QUALITY OF THEIR PROPUCTS. BIOSTATISTICIANS,
POCTORS, ANP AIPS ACTIVISTS ARE NOW WORKING TOGETHER TO PESlGN
CLINICAL TRIALS TO MORE RAPIPLY EVALUATE THERAPIES.
218
WELL, ТЙАТ'5 ГТ/ ВУ NOW, YOU 5HOULP BE ABLE TO PO
АМУТШ№ WITH STATISTIC, EXCEPT L/£ Oi£AT, 4TGAL,
ANP 6AM8L£
219
'IIO
BIBLIOGRAPHY
FOR THE STUPENT-.
MOORE, PAVIP S., fTATIfTIGb GONGGPTf ANP
GONTROVGRftGf, 1991, NEW YORK, W. H. FREEMAN.
EMPHASIZES IPEAS, rather than mechanics.
FREEPMAN, PAVIP, PISANI, ROBERT, ANP PURVES,
ROGER, 4TATI6TIG5, 1991, NEW YORK, W.W.
NORTON.
MOORE, PAVIP S. ANP M£CABE, GEORGE P-,
INTROPUGTION TO TUG PRAGTIGG OF
5TATI4TIG4, 1909, NEW YORK, W.H. FREEMAN.
SMITH, GARY, 4TAT&TIGAL RGA4ONIN&, 1990,
BOSTON, ALLYN ANP BACON, INC. MORE TECHNICAL,
EMPHASIZING ECONOMICS ANP BUSINESS, BUT HAS
EXAMPLES FROM ALL OVER.
THESE TEXTS ARE CURRENT, CORRECT, LITERATE, ANP WITTY. BESIPES THE ONES WE CITE,
THERE ARE HUNPREPS OF TEXTBOOKS OUT THERE, ANP WE WOULP RATE MOST AS AT
LEAST ACCEPTABLE.
FOR THE STRUGGLING STUPENT:
PYRCZAK, FREP, 4TAT&TIG* WITH A 4GN6G OF
HUMOR, 1909, LOS ANGELES, FREP PYRCZAK
PUBLISHER. AN ELEMENTARY WORKBOOK ANP
GUIPE TO STATISTICAL PROBLEM SOLVING
HOW TO UG, GHGAT, AW GAMPLG YOUR SAINTLY AUTHORS HAVE LITTLE EXPERIENCE IN
THESE FIELPS. HERE IS SOME APVICE FROM THE PROS:
HUFF, PARRELL, HOW TO LIG WITH
f TATIf TIGWITH PICTURES BY IRVING GEIS,
NEW YORK, 1994, W.W. NORTON. CHEAP ANP
STILL IN PRINT!
JAFFE, AJ. ANP SPIRER, HERBERT F, M&UfGP
STATltTlGb STRAIGHT TALK FOR TW&TGP
NUMFGRf, 1907. NEW YORK, MARCEL PECKER.
PART OF A GOOP POPULAR SERIES ON
STATISTICS.
ORKIN, MIKE, GAN POU WIN?, 1991, NEW YORK,
W.H. FREEMAN. APVICE FROM AN EXPERT ON
PROBABILITY ANP GAMBLING.
M^GERVEY, JOHN P., PROFAPILlTIGf IN GVGRP
МУ LIFG, 1909, N.Y., IVY BOOKS. GAMBLING
FROM BLACKJACK TO SMOKING.
221
Г LAW ANP SOCIETY'.
GASTWIRTH, JOSEPH L-, *TATI*TICAL REA*ON!NC /А/
LAW ANP POLICY, VOL. I L 1, 1900, SAN PIEGO,
ACAPEMIC PRESS- THE LEGAL МПТУ GRITTY, INCLUPING
JURY SELECTION CA*E* LIKE THE ONE THAT BEGAN
CHAPTER 9.
STEERING COMMITTEE OF THE PHYSICIANS’ HEALTHY
STUPY RESEARCH GROUP, “FINAL REPORT ON THE
A*PI RIN COMPONENT OF THE ON CO INC PHY*ICIAN*'
HEALTHY *TUPY,” THE NEW ENCLANP JOURNAL OF
MEP/C/NE, VOL. 321, PP. 129-135.
IN CHAPTER 9, THE NONJUPICIAL COMMENT ON POKER FROM THE BENCH WAS FROM AN
ACTUAL CASE, WE ARE ASSURER IN A PERSONAL COMMUNICATION FROM PR. JOHN PE CANI
UNIVERSITY OF PENNSYLVANIA.
GRAPHICAL PISPLAY OF PATA:
TUFTE, EPWARP R., THE VI*UAL P!*PLAY OF
QUANTITATIVE INFORMATION, 1903, CHESHIRE,
CONNECTICUT, GRAPHICS PRESS.
TUFTE, EPWARP R., ENVI*IONINC INFORMATION, 1990,
CHESHIRE, CONNECTICUT, GRAPHICS PRESS, THE
HISTORY, ART ANP SCIENCE OF GRAPHICS. BOTH
BOOKS ARE CLASSICS.
CLEVELAND WILLIAM S., THE ELEMENT* OF CRAPHINC
PATA, 1905, PACIFIC GROVE CA, WAPSWORTH APVANCEP
BOOKS ANP SOFTWARE. RESIGN PRINCIPLES FOR
COMPUTER GRAPHICS.
HISTORY-'
PAVIP, F. N„ CAME*. COP* ANP CAMBLINC, 1962, NEW
YORK, HAFNER, NEW YORK.
STIGLER, STEPHEN M., THE H!*TORY OF *TAT1*T!C*- THE
MEA*UREMENT OF UNCERTAINTY BEFORE 1900, 1905,
CAMBRIPGE, MA, BELKNAP PRESS OF HARVARP UNIVERSITY
PRESS.
BOX, JOAN FISHER, R- A. F!*HER, THE LIFE OF A
*CIENTI*T, 1970, NEW YORK, WILEY. BIOGRAPHY, BY HIS
PAUGHTER, OF THE MOST INFLUENTIAL ANP CONTROVERSIAL
FIGURE OF 2PTH CENTURY STATISTICS.
KRUSKAL, WILLIAM, “THE *ICNIFICANCE OF F!*HER; A
REVIEW OF RA. F!*HER: THE LIFE OF A *C!ENT1*T
1990. JOURNAL OF THE AMERICAN *TATI*TICAL A**OC!AVON
VOL 75, 1030. SETS THE FISHER BIOGRAPHY IN PERSPECTIVE
ANP HAS EXCELLENT BIBLIOGRAPHY.
222.
STATISTICAL SOFTWARE:
IN THIS BOOK WC U6GP THE MINITAB STATISTICAL SOFTWARE SYSTEM (MINITAB INC.,
STATE COLLEGE, PA). THE PENN STATE STUPENT HEIGHT ANP WEIGHT PATA IS FROM THE
PULSG PATA SET ON THIS SYSTEM. COMPUTER GRAPHICS WERE GENERATE? BY S-PLUS
(STATISTICAL SCIENCES INC., SEATTLE WA), ON A 49i PC CLONE S IS SOPHISTICATE? SOFTWARE,
PEVELOPEP BY AT&T BELL LABS FOR APVANCEP ANALYSIS ANP GRAPHICAL PISPLAYS.
RYAN, BARBARA, JOINER, BRIAN, ANP RYAN, THOMAS,
MINITAB HANDBOOK, (PWS-KENT, BOSTON, 1965) ANP
TUG STUPGNT GPITION OF MINITAB (APPlSON
WESLEY) ARE FAST, INEXPENSIVE INTROPUCTIONS TO
STATISTICAL COMPUTING- MINITAB RUNS ON MAIN-
FRAMES, PC COMPATIBLES, ANP MACINTOSH COMPUTERS.
THERE ARE MANY HIGH QUALITY SOFTWARE PACKAGES
AVAILABLE FOR THE PERSONAL COMPUTER, INCLUPINfr
PATAPGSK (PATA PESCRIPTlON, ITHACA. Ny), FOR THE
MACINTOSH
SAS (SAS INSTITUTE INC, CARY, NO, SPSS (SPSS INC,
CHICAGO, ILA ANP BMPP (BMPP STATISTICAL SOFTWARE,
INC., LOS ANGELES, CA) WERE ORIGINALLY PESlGNEP FOR
MAINFRAME SYSTEMS ANP NOW HAVE MIGRATE? TO THE PC,
COMPLETE WITH WINPOWS.
STATGRAPHICS (STATISTICAL GRAPHICS CORP, PRINCETON,
NJ), FOR THE PC.
STATVIGW (ABACUS CONCEPTS, OAKLANP CA) FOR THE
MACINTOSH.
SVST AT (SYSTAT, INC., EVANSTON IL) HAS SYSTEMS THAT
RUN IN ALL ENVIRONMENTS.
THESE PACKAGES PlFFER IN IMPORTANT PETAILS-, YOU NEEP TO BE A SMART SHOPPER.
WE RECOMMENP CHOOSING A SYSTEM THAT YOUR COLLEAGUES HAVE ALREAPY TESTEP.
FEW OF US ARE CUT OUT TO BE STATISTICAL SOFTWARE PIONEERS. WHEN LEARNING A
NEW SYSTEM, EXPERIMENT WITH SMALL, FAMILIAR PATA SETS. REMEMBER, TUG MOST
GXPGNSIVG PART OF ANY SoFTWARG /S YOUR TIMG. THE CARTOON RULE FOR
LEARNING STATISTICAL COMPUTING IS: FAMILIARITY BREEPS RESULTS.
TRYING TO LEARN STATISTICAL THGORY ANP
STATISTICAL COMPUTING AT THE SAME TIME IS
A LITTLE LIKE TRYING TO WAL-K ANP CHGW
GUM AT THE SAME TIME. AFFERENT SKILLS ANP
THOUGHT PROCESSES ARE INVOLVE? IN EACH.
SET ASIPE SEPARATE TIMES TO LEARN THESE
SUBJECTS, THEN BRING THEM TOGETHER. IN
THIS WAY, YOU CAN BECOME A GROWING,
WALKING, COMPUTING, RGNAISSANCG
STATISTICIAN!
2Г5
Acceptance sampling, 150
Addition rule for events, 38-39,42,44
Alternate hypothesis (Hd), 140-141, 147-149,
152-153, 165-166. See also Hypothesis
testing
left-handed, 144-145
relevant, 144-145
right-handed, 144-145
two-handed, 144-145
Analysis of variance. See ANOVA
ANOVA (analysis of variance), 186, 193-195,
table, 194
Approximate probability, 60
Approximation
binomial, 79-81,86-88
continuous, 87-88
normal, 87-88
Archery lessons, confidence intervals and, 116-124
Area under the curve, 64-66
Arrays, 14-15
Aspirin clin cal trials, 160-167. See also Two
populations compared
Astralagi, 28
Average salary comparison, 168-169.
See also Two populations compared
Average squaied distance, 22
Aveiage value, 15-17
standard deviations from, 22,24-25, 168, 171
Bar graphs, 11
Bayes, joe, 46-50
Bayes, Rev. Thomas, 46-50
Bayesian, 35
Bayes Theorem, 46-50
Bernoulli, James, 79
Bernoulli trial, 74-75, 78
sampling size and, 98-100
Best linear unbiased estimators (BLUE), in
regression analysis, 201-202
Beta (probability of type II error), 151-155
Bias
in polls, 126-127
reducing natural, with paired comparison, 178
in simple random sampling, steps to eliminate,
167
Binomial approximation, 79-81,86-88
Binomial coefficient, 76
multiplication rule and, 76
Pascal’s triangle and, 77
Binomial distribution, 77, 81,83,86, 88
asymmetrical, 82
calculating, for large values, 79-80
continuous density function and, 79-80
mean of, 78
standard normals and, 82
variance of, 78
Binomial distribution table, 78
Binomial probability distribution, 77-78
Binomial random variables, 74-76, 139-140
Blocks
complete randomized, 184-185
in experimental design, 183-184
BLUE (best linear unbiased estimators), in
regression analysis, 201-202
Bootstrapping, 215-216
Box and whiskers plot, 21
Brass tacks, 98-103
Categorical statements, 2
Central limit theorem, 106, 128, 169
fuzzy, 83-88
problems with, 107
Central value, 14. See also Spread
mean, 15-16
median, 17-18
Challenger (space shuttle), 3
Chameleon Motors
comparing small sample means, 170-171
confidence intervals, 134-135
hypothesis testing for, 149-150
Chernoff, Herman, 212
Classical probability, 35
Claudius I, 28
Cluster analysis, 212
Cluster sampling design. 95
Coin toss, 32, 54-55, 58, 60-62,68-70
Communication, 218
Comparing failure rates, 160-163
Comparing small sample means, 170-171'
Comparing success wtes, 160-163
Comparing two populations. See Two
populations compared
Comparison of average salaries, 168-169
Comparisons, paired, 174-178
Complete randomized block, 184-185
Computer image analysis, 215
Computer resampling, 215-216
Conditional probability, 40-41
false positive paradox and, 46-50
multiplication rule and, 42-44
Confidence interval levels
decision theory and, 152-153
measuring, 122-123
Confidence intervals, 112-136
computer simulation of, for samples, 120
error levels and, 124-127
estimating, 114-127
increasing levels of, 121-125
margin of error and, 119, 121
in paired comparisons, 176
population means and, 128-130, 169
population proportion and, 128-130
probability calculation and, 117-119
random sampling used for, 114-115, 119
in regression analysis, 203-206
sample means and, 130,171
standard deviation in, 117, 128-130
standard error in, 118, 128-130
Student’s r based, 131-136
for success rates, 164
table for levels, 122-123
Continuity correction, 87-88
Continuous densities, properties of, 66-67
Continuous density function, binomial distribu-
tion and, 79-80
Continuous probabilities 64
Continuous random variables, 63
mean of, 67
probability density of, 65
variance of, 67
Correlation, squared, in regression analysis, 195
Correlation coefficient, in regression analysis,
196
Cumulative probability, 84
Curve, area under the, 64-66
Data
multivariate, statistical analysis of, 212-213
order of, 17
paired and unpaired compared, 177-178
properties of, 59
rigged, in regression analysis, 189,192,
194-195, 205-207
spread of, in regression analysis, 190—195
Data analysis, 4
Data description, 8-26
Data display, 212
Data points, 11-12, 14-15
average, 17
middle, 17
Data quality, 217
Data summary, 12
Death rate, 13
Decision table, two-by-two, 152
Decision theory, hypothesis testing, 151-155
Deductive reasoning, 113
Degrees of freedom, 131-135
in comparing small sample means, 171
hypothesis testing and, 149-150
de Mere, Chevalier, 28-29,75, 78
de Moivre, Abraham, 79-83, 86-88, 101
Dependent random variable, in regression
analysis, 199-209
Dependent variable, in regression analysis, 189
Dice, 28-45
loaded, 33
Discrete probabilities, 64, 66
Discrete random variables, 63
Discriminate analysis, 213
Dot plots, 9
two-dimensional, 188
Election polls, 114-127
hypothesis testing in, 143-145
Elementary outcomes, 30, 32-38
Error levels, confidence intervals and, 124-127
Errors
heteroscedastic, 209
margin of, confidence intervals and, 119,121
measurement, experimental design and, 183
random error fluctuations, 199-209
standard. See Standard error (SE)
sum of squared (SSE), in regression analysis,
190-195
type I. 151-154
type II, 151-154
Estimates, 102-103, 107
Estimating confidence intervals, 114-127
Estimators, 102-103
best linear unbiased (BLUE), in regression
analysis, 201-202
in comparing the means of two populations,
168-169
Events
addition rule for, 38-39,42,44
mutually exclusive, 39, 42, 44
probability of, 35-37
repeatable, 35
rules for outcomes of, 38-39
subtraction rule for, 39,44
Expected value, 61
Experiment
random, 30, 32, 34, 36
sampling and, 98-100, 104-105
Experiment (continued)
weight, 9-12, 16, 18-26
regression, 188-209. See also Regression
Experimental design
basic principles, 183
blocks in, 183-184
elements of, 182-183
four-by-four table in, 184-185
Latin square in, 184-185
local control in, 183
measurement error and, 183
natural variability and, 183-185
randomization in, 183, 185
replication in, 183, 185
total variability and, 186
Experimental treatments, 182-183
Experimental units, 182-183
Factor analysis, 213
Failure rates, comparing, for two populations,
160-163
False positive paradox, 46-50
Fermat, Pierre de, 28—45
Fisher, R. A., 217
Fitting process, in regression analysis, 189-196
Fixed significance level, in hypothesis testing,
141-142, 145
Four-by-four table, in experimental design,
184-185
Frequency, relative, 10-11,35,57-58, 60
Frequency histograms, 11. 57-58
Frequency tables, intervals in, 10-11
Gallup Poll, 127
Gambling, 27-45
Gasoline comparisons, 172-173
experiment design and, 182-186
paired comparisons of, 174-178
Gosset, William. 108-109, 131-132
Graphic display, 13
Graphs
bar, 11
histograms. See Histograms
probability distribution, 56-58
(Ид). See Alternate hypothesis
Heteroscedastic errors. 209
Histograms, 13
frequency, 57
probability, 56-58
relative frequency, 11,57-58
spread measured in, 19
symmetrical, 24-25, 77
Hite, Shere, 97
Holmes, Sherlock, 113-130
Ho(null hypotheses), 140-141, 144-145,
147-150, 152-153, 165-166. See also
Hypothesis testing
1 lypotheses. See also Hypothesis testing
alternate (Ha), 140-141, 147-149, 152-153,
165-166
left-handed, 144-145
relevant, 144-145
right-handed, 144-145
two-handed, 144-145
null (Ho), 140-141, 144-145, 147-150,
152-153, 165-166
Hypothesis testing. 138-139
decision theory, 151-155
degrees of freedom and, 149-150
fixed significance level in, 141-142, 145
large sample
for population mean. 146-148
significance test for proportions, 143-145
in paired comparisons, 176
population mean and, 146-148, 169
probability statement in, 141-142
in regression analysis, 207
statistical, 140-142
Iguana autos, 170-171
Increments, 9
Independence, 71,74
simple random sampling and, 92-94,96
special multiplication rule and, 43-44
Independent mechanisms, 71
Independent variable, in regression analysis,
189, 199-209
Inductive reasoning, 113
Innovation, 218
Inspection sampling, significance test used in,
146-148
Integral, 66-67
Interquartile range (IQR), spread measured in,
20-21
Intervals
confidence. See Confidence intervals in a
frequency table, 10-11
IQR (interquartile range), spread measured in,
20-21
Jackknife, 215-216
Jury selection, racial bias in, 138-141
Large sample hypothesis testing
for population mean, 146-148
significance test for proportions, 143-145
Large values, calculating binomial distribution
for, 79-80
Latin square, in experimental design, 184-185
Least squares line, 189-190, 208
Left-handed alternate hypothesis, 144-145
Linear regression, in regression analysis,
189-190, 208
226
Local control, in experimental design, 183
Logical operations, 37
Margin of error, confidence intervals and, 119,121
Mean, 15-16, 18
of binomial distribution, 78
central, 15-16
comparing small sample, 170-171
confidence intervals and, 128-130, 169, 171
large sample test for, 146-148
in paired comparisons, 175-176
population, 59,62, 80
confidence intervals and, 128-130, 169
hypothesis testing and, 146-148
of probability distribution, 60-61
of random variables, 61, 67-69
sample
comparing small, 170-171
confidence intervals and, 130,171
distribution of, 104-106, 171
hypothesis testing for, 146-148
standard deviation from, 22, 24-25, 62, 168,
171
Mean response, predicting, in regression
analysis, 204-206
Measurement енот, experimental design and, 183
Measures of spread, 19-25
Median, 17-18, 20-21
Midpoints, 10-11
Model properties, 59
Models
regression, 199-202
stochastic random, 116-118
for two populations, 162
Monitoring programs
power analysis in, 154-155
probability of type 11 errors in, 151-155
Mortality statistics, 13
Multiple linear regression, in regression
analysis, 208
Multiplication rule, 45
binomial coefficient and, 76
conditional probability and, 42-44
Multivariate data, statistical analysis of
cluster, 212
discriminate, 213
factor, 213
mu. See Population mean
Mutually exclusive events, 39,42, 44
Natural bias, reducing, with paired comparison,
178
Natural variability
experimental design and, 183-185
reducing, with paired comparison, 178
Nightingale, Florence, 13
Non-linear regression, in regression analysis, 208
Normal approximation, 87-88
Normal distribution, standard, 79-85
rule for computing, 85
table to find, 84-85
Null hypothesis (Ho), 140-141,144-145,
147-150, 152-153, 165-166. See also
Hypothesis testing
Numerical outcome, sampling and, 98-100,
104-105
Numerical weight, 32
ObjectiviM, 35
Observed value of r, 149-150
Observed value of z, hypothesis testing and,
144-145, 165-166, 169
Opportunity sampling. 97
Opportunity sampling design, 97
Order of data, 17
Outcomes
elementary, 30, 32-38, 41
of events, rules for, 38-39
numerical, sampling and, 98-100, 104-105
Outliers, 18,21-23
Paired comparisons
of gasolines, 174-178
means in, 175-176
paired and unpaired data compared, 177 178
small-sample i test statistic for, 176
standard deviation in, 175-176
Pascal, Blaise, 29
Pascal’s triangle, 77
Personal probability, 35
Polls
bias in, 126-127
election, 114-127
error levels in, 124-127
Gallup, 127
hypothesis testing in, 143-145
as opposed to actual elections, 126-127
Pollution monitoring, probability of type II
enors in, 151-155
Pool the sum of squares
in comparing small sample means, 171
Population. See also Two populations compared
properties, 59
proportion, 128-130
standard deviation, 59, 62, 80
Population mean, 59, 62, 80. See also Two
populations compared
confidence intervals and, 128-130, 169
hypothesis testing and, 146-148
Power analysis in monitoring programs, 154-155
Prediction line, 189
Predictor variable, in regression analysis, 189
Probabilities, 4, 27-51
approximate, 60
227
Probabilities (continued)
characteristic properties of, 34
classical, 35
conditional
false positive paradox and, 46-50
multiplication rule and, 42-44
continuous, 64
cumulative, 84
discrete, 64, 66
formulas for manipulating, 37-39
non-negative, 34
normal, 83-85
personal, 35
repeatable events and, 35
sample, 100
spread of, 67
Probability calculation, confidence intervals
and. 117-119
Probability density, 66
of continuous random variable, 65
Probability distribution
binomial, 77-78
graphs, 56-58
mean of, 60-61
properties of, 59
random variable, 55-58
table to find normal, 84-85
Probability graphs, 56-58
Probability of type II errors, 151 155
Probability statement, in hypothesis testing,
141-142
Probability zero, 63-64
Propot tion of successes. See Success rates
Pseudo-random numbers, 65
P-value, in hypothesis testing, 141-142. 148
Random error fluctuations, in regression
analysis. 199-209
Random experiment, 30, 32, 34, 36
sampling and, 98-100, 104-105
Randomization. 215-216
in experimental design, 183, 185
Random models, stochastic, 116-118
Random number generator, 65, 94
Random sampling
independence and, 92-94,96
simple, 92-96, 167
steps to eliminate bias in, 167
used for confidence intervals, 114-115, 119
Random sampling design, 92-94
Random selection of jurors, 138-141
Random variables, 53-72
adding, 68-71
binomial. 74-76, 139-140
discrete, 63
mean of, 61, 67-69
piobability distribution, 55-58
sampling and, 98-100, 104-105
r, 107-109
variance of, 62, 67-71
Random variable t, 107-109
Random walk, 214
Regression, 187-209
Regression analysis
best linear unbiased estimators (BLUE) in,
201-202
confidence intervals in, 203-206
correlation coefficient in, 196
dependent random variable in, 199-209
dependent variable in, 189
fitting process in, 189-196
hypothesis testing in, 207
independent variable in, 189,199-209
linear regression in, 189-190, 208
predicting mean response in, 204-206
predictor variable in, 189
random error fluctuations in, 199-209
regression diagnostics in, 209
response variable in, 189
rigged data in, 189, 192, 194-195, 205-207
spread of data in, 190-195
squared correlation in, 195
standard error (SE) without derivation in, 203
statistical inference in, 199-209
student weight experiment and, 188-209
sum of squared errors (SSE) in, 190-195
sum of squared regression (SSR) in, 194-196
Regression coefficient sample, 191-192
Regression line, 189-190, 208
Regression model, 199-202
Relative frequency, 10, 35, 60
Relative frequency histogiams, 11,57-58
Repeatable events, probability and, 35
Replication in experimental design, 183, 185
Resampling, 215-216
Response variable in regression analysis, 189
Right-handed alternate hypothesis, 144-145
Rounding off, 9
Round numbers, 10
Salk polio vaccine, 3
Sample means
comparing small, 170-171
confidence intervals and, 130, 171
distribution of, 104-106
hypothesis testing for the population mean.
146-148
Sample probability, 100
Sample properties, 59
Sample regression coefficient, 191-192
Sample size, 91
comparing small, 170-171
confidence levels and, 124-125
increasing, 124-125
ZZS
sample size (conunued)
standard error and, 98-103
testing large, 143-148
Sample space, 30-31, 33,41
Sample variance, 22
Sampling, 89-109
acceptance, 150
random, 95
independence and, 92-94, 96
steps to eliminate bias in, 167
used for confidence intervals, 114-115, 119
random experiment and, 98-100, 104-105
random variables and, 98-100, 104-105
standard deviation and, 101-103
Sampling design
cluster, 95
opportunity, 97
random, 92-94
simple random, 92-94
stratified, 95
systematic, 96-97
Sampling distribution
of the mean, 104-106
for proportion of successes, 163
Scatterplots, 188-189
random, 209
SD. See Standard deviation
SE. See Standard error
Senator Astute. See Election polls
Sigma, 16. See also Summary statistics
Significance level
fixed, 141-142, 145
in hypothesis testing, 141-142, 145, 147-148
in scientific work, 141-142
Significance test
for proportions, 143-145
used in inspection sampling, 146-148
Simple random sampling, 92-96, 167. See also
Random sampling
Smoke-detectors, as a decision theory example,
151-154
Special addition rule, for mutually exclusive
events, 39, 42, 44
Special multiplication rule
conditional probability and, 42-44
independence and, 43-44
Spinning pointer, 63-64
Spread, 14
of data in regression analysis, 190-192
sunt of squared errors relative to, 193-195
measures of, 19-25
of probabilities, 67
variance in, 22-23
Spread distance, squares of, 22
Squared correlation, in regression analysis, 195
Squared distance, 22, 61-62
Squared errors, sum of (SSE) in regression
analysis. 190-195
Squared regression, sum of (SSR), in regression
analysis, 194-196
Square root, standard deviation defined by, 23
Squares, pool the sum of, 171
SSE (sum of squared errors)
in regression analysis, 190-195
relative to spread of data, 193-195
SSR (sum of squared regression), in regression
analysis, 194-196
Standard deviation (SD)
in comparing small sample means, 171
in comparing the means of two populations,
168
in confidence intervals, 117, 128-130
defined by square root, 23
from mean values, 22, 24-25, 168, 171
in paired comparisons, 175-176
population, 59, 62, 80
sampling and, 101-103, 107
spread measures and, 22
z-scores and, 24-25
Standard error (SE)
in comparing small sample means, 171
in comparing the means of two populations,
168
Standard error (SE)
in confidence intervals, 118, 128-130
sample size and, 98-103
without derivation, in regression analysis, 203
Standard normal distribution, 79-82
table for, 84-85
Statistical analysis of multivariate data,
212-213
Statistical hypothesis testing, 140-142, 144-145,
147-148, 165-166, 169
Statistical inference, 4
in regression analysis, 199-209
Statistical situations, 158-159
Statistics
mortality, 13
summary, 14-26, 148
Stem-and-leaf diagram, 12, 18
Stochastic random models, 116-118
Stratified sampling design, 95
Student’s i. See t-distribution
Subjectivist, 35
Subtraction rule for events, 39,44
Successes, number of, 75
Success rates, 99
comparing, for two populations, 160-163
confidence intervals for, 164
in hypothesis testing, 143-145
sampling distribution for, 163
Summary statistics, 14-26
in hypothesis testing, 148
Summation, 16. See also Summary statistics
in regression analysis, 190-195
relative to spread of data, 193-195
Sum of squared regression (SSR), in regression
analysis, 194-196
Sum of squares, pool the, 171
Systematic sampling design, 96-97
rdistribution, 107-109
in comparing small sample means, 171
confidence intervals based on, 131-136
critical values for, 132-136, 150
hypothesis testing and, 149-150
Teamwork, 218
Test statistic
in hypothesis testing, 140-141, 144-145,
147-148, 165-166, 169
small-sample t, for paired comparisons, 176
Time series analysis, 214-215
r-observed value, 149-150
Total variability
due to the regression, 194-195
experimental design and, 186
Tukey, John, 12, 21
r-values. See r-distribution
Two-by-two decision table, 152
Two-handed alternate hypothesis, 144-145
Two populations compared, 158-179. See also
Population
confidence intervals for, 164, 169
hypothesis testing, 160-163, 169
mean of, 168-169
model for, 162
sampling distribution for proportion of
successes, 163
success rates, 160-164
Type I errors, 151-154
Type II errors, 151-155
Typical value, 14-18. See also Spread
Variability
natural
experimental design and, 183-185
reducing, with paired comparison, 178
total
due to the regression, 194-195
experimental design and. 186
Variables
binomial random, 74-76, 139-140
continuous random, 63, 65,67
dependent, 189
dependent random, 199-209
discrete random, 63
random. See Random variables
in regression analysis, 189, 199-209
Variance
analysis of. See ANOVA
of binomial distribution, 78
of continuous random variables, 67
of random variables, 62, 67-71
sample, 22
in spread, 22-23
Vertical scale, 11
Weight experiment, Penn State student, 9-12,
16, 18-26, 188-209. See also Regres-
sion; Regression analysis
x-axis, 80
y-axis, 80
z-observed value, hypothesis testing and,
144-145, 165-166, 169
z-scores, standard deviation and, 24-25
Z transformation, 84-88, 117-118
250
ABOUT THE AUTHORS
WOOLUOTT ЫМТН IS
PROFESSOR OF STATISTICS
AT TEMPLE UNIVERSITY. WITH
A B.S. ANP MA. FROM
MICHIGAN STATE UNIVERSITY,
ANP A PH.P. FROM TOWNS
HOPKINS, HE IS THE AUTHOR
OR COAUTHOR OF MORE
THAN FORTY PUBLICATIONS
IN SUCH PIVERSE AREAS AS
OIL SPILL SURVEYS,
STATISTICAL THEORY, ANP
ENVIRONMENTAL STATISTICS.
HE HAS APVISEP A NUMBER
OF NATIONAL SCIENTIFIC
PROGRAMS. HE PEBATES ANP
KAYAKS WITH HIS WIFE, LEAH,
ANP HIS TWO GROWN
CHILPREN, KESTON ANP
AMELIA.
LARRY &ОЫЮС IS THE AUTHOR OR
COAUTHOR OF A NUMBER OF BOOKS
OF NON-FICTION CARTOONING, AS
WELL AS THE FEATURE MEME
CWSS/CS, APPEARING BIMONTHLY IN
PLOVER MAGAZINE. A PROP-OUT
OF HARVARP GRAPUATE SCHOOL IN
MATHEMATICS, HE SEEMS TO HAVE
COME FULL CIRCLE. HE LIVES IN SAN
FRANCISCO WITH HIS WIFE, LISA, ANP
TWO PAUGUTERS, SOPHIE ANP ANNA,
ANP HOPES TO UNPERSTANP LIFE
WHILE REMAINING CHAINEP TO HIS
PRAWING BOARP.
I. Ч1ДСС
If you have ever looked for P-values by shopping at P mart, tried to watch
the Bernoulli Trials on “People's Court," or think that the standard deviation
Is a criminal offense in six states, then you need The Cartoon Guido to
Statistics to put you on the road to statistical literacy.
The Cartoon Guide to Statistics covers all the central ideas of modem
statistics: the summary and display of data, probability in gambling and
medicine, random variables, Bernoulli Trials, the Central Limit Theorem,
hypothesis testing, confidence interval estimation, and much more—all
explained in simple, clear, and, yes, funny illustrations. Never again
will you order the Poisson Distribution in a French restaurant!
lAiuy Gonick is the author or coauthor of four previous cartoon guides by
HarperCollins. Woollcott Smith is a statistician active in research
and consulting, and a professor at Temple University.
“Gonick is one of a kind." —Discover
Im HarperPerennial
A Division of HarperCollins Publishers
Cover design and illustration ©1993 by Larry Gonick
USA $15.00
CANADA $21.50
07931