JERROLD H. ZAR Department of Biological Sciences Northern Illinois University

PRENTICE HALL Upper Saddle River. New Jersey 07458

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PREFACE x

INTRODUCTION 1 1.1 Typre of Biological Data 2 19 Frequency Dlsnibutions 6 13Accuracy and Significant Pigums 5 1.4 Cumulative Frequency Distributions 13

POPULATIONS AND SAMPLES 16 2.1 Populations 16 2.3 Random Sampling 17 2.2 Samples fmmPopuIations 17 2.4 Parameters and Smtistiss 18

MEASURES OF CENTRAL TENDENCY 20 3.1 The Arilhmetic Mom 20 3.5 Olhor Measurer of Cenmal Tendency 28 3.2 The Median 23 3.6 The Bffeoot of Coding Data 29 3.3 Olhcr Quanrilss 26 Exercises 31 3.4 The Mode 27

MEASURES OF DISPERSION AND VARIABILITY 32 4.1 The Range 32 4.6 The C ~ f S c i c n tof Variation 40 4.2 Dispersion Mcasuce4 with Qvantilss 34 4.7 Indices of Divosity 40 4 9 Thc Moan Deviation 34 4.8 The Effect of Coding Data 44 4.4 Tha Variance 35 E-i~ss 47 4.5 Tho StandardDeviation 39

PROBABILITIES 48 5.1 Counting Possible Outcomca 49 5.5 Robability of an Event 58 5.2 Pcrmurntions 50 5.6 Adding Robabilitiu 59 5.3 Combinations 54 5.7 Multiplying Probabilitiu 61 5.4 Sow 56 Exercises 63

THE NORMAL DISTRIBUTION 65 6.1 Sy-try and Kurtosis 67 6.4 inmduction to Ststistical Hypomcsis Testing 79 6.2 Roportians of a N d Dishbution 72 6.5 Assessing Depamrea hornNonnaliry 86 6.3 The Distribution ofMeans 76 Excmisu W

ONE-SAMPLE HYPOTHESES 91 7.1 Two-Tsiled HypOlhasw C o o d o g Ula Mean 91 7.9 H y p o U l a ~Concaning tho Median 110 7.2 One-=led HypHhcas Cotmxning the Mean 96 7.10 h f i d e n w Limits for the Population 7.3 Confidence Limits for the Population M- 98 VOI~MCC 110 7.4 RCpoRlag V d i l l t y abovt tho Mean 100 7.11 Hypahcses Concerning the Vmi- 112 7.5 Ssmpls Size and Bslimdon of the Population

Mean 105 7.12 Power and Sample Size in Tests Concsrmng

thevari- 113

7.7 Sampling Pidm Populations 108 7.14 Hypothcscs Concerning Symmctry and 7.8 Confidence Limits for tho Popution Kunosis 115

120Median 110 Excreisss

TWO-SAMPLE HYPOTHESES 122 8.1 Tastlng for Difference between TWOM a s 122 a 8 %tins for DiffWaIc~ hrxecn Two CoefEdents 8.2 ConRdance Li- for Populstion Means 129 of Variation 141

145 146

8 3 Sample Size and Estimation of the Difference 8 9 Nonparmnaic Staristical McUods bctwcur Two Population M a n s 131 8.10 TwoSampls Rpnlr Testing

8.4 Power m d Sample Siz. in Tesu for Diff-cc 8.11 Testing for Di&rance between l b o 155between W o M-n~-~132..- Mcdlans

8.5 Toiling for Difference between Two 8.12 m e EEec~=f Coding 155

bctwccn Two Variances 140 Excrcisu 159

PAIRED-SAMPLE HYPOTHESES 161 1659.1 Testing Mean DS-nss 161 9.5 Paired-Sample Testing by Rmks

of Means 164 Data

9 3 Confidence Llmira for the Powlation Mean Difference 164

9A T&s for lhe Dtffemncs haween Varlanw fmm G o C-lad Populations 164

169 9.3

ContentsIv

7.6 Power and Sample Si* thein %sts C o n d ~ Mcan 105

7.13 Hyplheaaa Concerning the Cmfficient of Variation 114

Data 156 Variances 136

8.6 Confidence Interval for the Population Variance Ratio 139

8.7 Sample Size and Powsr in Tests far Difference

8.13 Wo-Sample Testing of Nominal-Scale

8.14 Testing for Difference between Two Divenity Indiws 156

.'

9.6 Confldcnca Limits for the Population Msdiah

Power and Sample Size in Psired-Sample Testing Difference 169 ',

9.7 Paired-Sample T-ting of Nominal-Scale

Exerdscn 175

MULTISAMPLE HYPOTHESES: THE ANALYSIS OF VARIANCE 1?7

10.1 10.2 Clmfldence Limits for Population Moans I89 10.3 Power and Sample Size in Analysis of 10.8 me E&cr of Coding 206

Variance 189 10.4 Nonp-ohic Analysis of Vsriancrr 195 Data 206

2061 0 5 Testing for Dlfiercncc among S c v w l Exercises Medians 200

MULTIPLE COMPARISONS 208 11.1 11.2 The Newmen-Kculs Toa 214

comparisons 215 11.9 Multiple Comparisons among Variances 228 11.4 Cornoarison of a Contml Mean to Each -5r

Single-Fsotor A~alysie of Variance 178 10.6 Homogeneity of VaMnces 202 10.7 Homogeneity of Cocmcicnts of Vanation 204

10.9 Multisampls Testing for Nominal-Scale

The Tukey Test 210 115 schcff6's Multiplc ContRJts 219 11.6 Nonp-cuio Multiple Comparisoos 223

113Confldence hmrvals PoUowhg Multiplc 11.7 Nonp~~amcrnc Multiplc Contrasts 226 11.8 Multiple Comparisons among Medians 226

Contents

12 TWO-FACTOR ANALYSIS OF VARIANCE 231 12.1 l b o - F e s m Analyaia ofVari-c wilh Equal lZ.8 N o n p Randomized Block ~ ~ or

Replication 232 Repeated-Mcasurea Analysis of Variance 263! 123 lbo-Faswr Analysis ofVariance with Unequal 12.9 Multiple Comparisons fmNonp-~tri~

Rcnlication 245 Randomized Block or Repeated-Mesa-IU -b.~aswr Annbsis of variance without Analysis of Variance 267

R.cplisation 248 12.10 Diohotomous Nominal-Scale Dara in 12.4 The Randomized Block Ekpetimental Randomid Blocks or from Rcosated

Dosign Y O Mcaslsuns 268 123 Repeated-M-urn8 Expctimsnral Designs 255 12.11 Multiple Cornpadsons with Dichowmoua 12.6 Multipb Comparisons and Confidence Intervals R ~ d o m i z e dBlock or Repeated-Measus

in Two-FMor Analysis of Variance 260 Dam 270 12.7 Powar and Sample Size in Two-Factor Analysis 12.12 introduction w Analyeis of Covariance 27b

of Varimc 261 Kxcrsiscs 271

13 DATA TRANSFORMATIONS 273 I3.1 The Loganthrms Traneformauon 275 133 The Square Rmt TVmoformauon 275 153 The Amaim Transformahon 278

14 MULTlWAY FACTORIAL ANALYSIS OF VARIANCE ---282~ ~ ~

14.1 The-Pastor Andyais of Variance 283 14.6 Multiple Comparisons and Confidence Intervals 14.2 The Latin Sqv- Expsrimsntal Design 286 in Multiway Analysis of V a r i ~ c c299 143 Highu-Order Factorial Analysis of Vwimco 287 14.7 Power and Sample Sire in Multiway Analysis of 14.4 Blocked and Repented-Measures Exocrimental WEC300

Designs 288 Exercises 300 145 Paswlial Anplyeis of Variance with Unequal

Replication 298

15 NESTED (HIERARCHICAL) ANALYSIS OF VARIANCE 303 15.1 Nesting within One Main Factor 305 15.4 Power and Sample S l u in Nssrsd Analysis of 15.2 Nesting in Factorial Bxpcrimcnrs 308 Variance 311 l53 Multiple Comparisons and Confidence Exncisc 311

I intervals 310 i

16 MULTIVARIATE ANALYSIS OF VARIANCE 312 1 16.1 The Multivariate Normal Distribution 313 16.3 Further Analysia 322

16.2'1 '!

Multivariate Analysis of Variance Hypothesis 16.4 Other Ekperimenral Designs Testing 316 Exercises 323

322

f i;i 17 SIMPLE LINEAR REGRESSION 324 17.1 Regression ve. Comelation 324 17.8 Power and Sample Sizo in Rcgrrsriop 350 17.2 The Simple Linear Regrossion Equation 326 17.9 Regcession chmugh the Origin 351

i '‘ I73 Testing tkSignScance of a Rsgmsioo 333 17.10 Data Transformstions in Rogrcsion 353 4 17.4 Confidanca Intervals in Regression 337 17.11 meHffoft of Coding 357

i 17.5 Inverse Prediction 342 Exercises 358 17.6 Intsrprctationa of Regression Functions 3 4 4

i 4 17.7 Rcgrcssion with Rcplicntion and Testing for

Linearity 345

Contents

18 COMPARING SIMPLE LINEAR REGRESSION EQUATIONS 360 18.1 Comparing 'hw S I w r 360 18.7 Multiple Comparisons among Elevations 373 18.2 Comparing Two Elevations 364 18s Multipls Comparisons of Points a m ~ n ~

374183 Comparing Points on Two Regression Rcgtensioo Lines

183 Comparing more lhnn Two Elevation. 372 Excreiacs 375 18.6 Multiple Comparisons among Slow 372

19 SIMPLE LINEAR CORRELATION' 377 19.1 Thc Cornlation Coefficient 377 198 Multiple Comparisons among Correlation 19.2 H y p o h e r about the Conelalion Coefficients 392

395Coefflci~nt381 19.9 RanL Cordation 19.3 con ti den^ lntmals for tbc Population 19.10 Weighted Rank Cormlation 398

19.5 Comparios I*ro Cmlat ion GxWcicnts 386 19.12 lntraclass Cornlation 404 19.6 Power and Sample S i in Camp-g Two 19.13 Concordance Cornlstion 407

vi

Lines 368 18.4 Comparing mace than -0 Slop- 369

Cornslation Coefficient 383 19.4 Power and S m p l s Size in Cornlation 385

Conelstion Coefficients 388 19.7 Comparing mom than Two Correlation

Coefficients 390

18.9 An O d T a t for Coincidental R ~ - ~ M s 375

19.11 Comlation for Dichotamous Nominal-Scale Data 401

19.14 Zhc Effsct of Coding 41'0 Exasincs 410

20 MULTfPLE REGRESSfON AND CORRELATlON 413 20.1 Intmedjate Computational Saps 414 20.10 Testing DiRmnce between Two P&al 20.2 The Multiple Rsger~ion Equation 419 R c w s i o n Coefficients 436

. 20.3 Analyeis of Variance of Multiple Regression or 437Conelation 422 20.12 Intcniction of Independent Variables

20.4 Hypotheses Concerning Psnial Regression 20.13 Comparing Multlplc Regression 437

440 440

Cocffioicnu 424 Equations 205 Standardized Panid Regression 20.14 Multiple RCgmsnion thmugh the Origin

Cc45ciwlts 426 2O.U Nonlinear Regression 20.6 Panial Conelation 426 20.16 Descriptive vs. &diotive Modcls 442 20.7

20.9 m*g Y Values 433 Exercises 450

21 POLWOMIAL REGRESSION 452 21.1 Polynomial Cvrvc Fcmkg 452 31.3 Quadretie Regeedon 457 21.2 Roundsff mot and Coding Pat. 457 Ercxiscs 459

22 TESTINQ FOR GOODNESS OF FIT 461 22.1 Chi-square Ooodnuu of Fit 462 2 2 8 Ko)mogm~-SmimovGoodness of Pit for

475222 C h i - S q m O%dness of Pit for More than Two D i s ~ t eData Catag0d.x 464 229 lColmogorov-Smimov Goodness of Fit for

Z.3 Subdividing Chi-square Analysts 466 22.4

22.7 The Log-Likelihood Ratio 473 BxerCisos 483

20.11 "Dummy"Vadablas 436

Round-off Emor and Coding Data 428 20.8 Selostion of In&pen&nt Variables 429

20.17 Concordance: Rank Cornlation among Severel Variables 443

Continnous Data 478 Chi-Sqwe C-tion for Continuity 468

215 Bias in Chi-Square Calculations 470 22.6 HCfm#wci?j Chi-Squm 471

22.10 Sample Size Required for Kolmogorov- Smimov Cioodness of At for Continuous Data 481

Contents

23 CONTINQENCY TABLES 486 ; '23.1 Chi-Sq- Aoatysis of Contingency 23.7 The Los-Likelihood R.do f a Contingency .

Tabios 488 Tables 505 23.2 Graphing Contingency Table D m 4% 23.8 Three-Dbemi~nal Contingency Tablea 506 233 Tho 2 x 2 Contingency Table 491 23.9 lag-linear Models for Multidi~nnional 23.4 Hetcmgancity Tenling of 2 r 2 Tables 5M) C a h g e n c y Tabis 512 23.5 Subdividing Conllngcncy Tables 502 Excmises 514 23.6 Bias la Chl-Sq- Contingency TBbk

Ar*l ly~s 504

24 MORE ON DICHOTOMOUS VARIABLES 516 24.1 Binmnial Robabilitles 517 249 Confidence Interval for the Population 24.2 The Hypergoomeldo Disldbution 523 Median 542 243 Sampling a Binomial Population 524 24.10 The Fir E w t Tcst 543 24.4 Coofiden- Limitr for Population 24.11 Comparing Two Proportions 555

Proponions 527 24.12 Power and Sample Size in Comparing Two 24.5 Goodness of Pit for the Binomial Roportions 558

Dlslribulion 530 2423 Cornpazing more ban Two Proportions 562 24.6 The Binomial Tcst 533 24.l4 Multiple Comparisons for Proponions 563 24.7 m e Sign Test 538 24.15 Thuds among Roportions 565 24.8 Power of tho Binomial and Sign Tests 539 Exercises 568

25 TESTING FOR RANDOMNESS 571 ' 25.1 Poisson Robabilirics 571 25.7 Serial Randomncas of Mcssurcments: Paramchis

2 9 3 Ctoodnsssof Fir of thc Poisson Distribution 575 25.8 Serial Randomes8 of McaJImmens: 2 5 3 Confidence Limits for the Poisson P-etcr 574 Tesring 586

W.4 The Binomial Test Reviaired 578 Nonparsmcnic Testing 587 25.5 Comparing Two Poisson Countr 582 Exucipcs 590 25.6 Serial Randomness of Nominal-Scale Categories 583

26 CIRCULAR DlSTRiBUl7ONS: DESCRIPTIVE STATISTICS 592 26.1 Data on a Cirou1.r Scale 592 26.8 Diametrically Bimodal Distributions 607 26.2 Oraphical Pnscntatim of Ckcular Data 595 26.9 Second-Ordm Analysis: T b Mcan of Mean 26.3 Sines and Cosine3 of C i u l a r Data 597 Angles M)8 26.4 TheM- Angle 599 16.10 Confidence Limits for the Second-Order Mean 26.5 Angular Dispersion 602 Angle 611 26.6 The Median and Modd Angles 605 Exsrciscs 614 26.7 Conndenco Limitr far the Population Mean and

! M d s n Anglos 605 , .

27 CIRCULAR DISTRIBUTIONS: HYPOTHESIS 7ESTlNG 616! 27.1 Testing Significance of ihe Mean Anglo: 27.6 Nanparamstric no-Sample and Multisample

I UnLnodal Distributioos 616 Testing of Angles 630 i 2 7 3 Tssti~lgsignificance of the Modian Angle: 27.7 Two-Sample and Multi8amplc Testing of

O d b u s Ten 621 Medim Anglos 635 27.3 Tasting Significance of the Median Angle: 27.8 Two-Sample and Multisampls Testing of

Binomial Twt 624 Angular Discaces 635 . . 27.4 Tasting Symmsuy mound the Median 27.9 Two-Sample a d Muitisample Testing of

Anglo 624 Angular Dispersion 637 i 276 Twc-Sample and Muitisample Testing of Mean 27.10 Parametric Om-Sample Second-Order

Anglos 625 Analysis of Angles 638

vlll Contents

27 CIRCULAR DlSTRlBUTiONS: HYP07HESlS TESTING (continued) 27.11 Nonpawnculc Onc-Sample Second-Ordcr

M y s i s of Angles 639 27.12 P-ctric T ~ h S . m p b Ssond-Omer

Analysis of Angisa 641 27.- Nanp-cult TwoSampic Second-Order

Analysis of Anglcs 643 27.14 Paramcmic Paired-Ssunnlo Tostinn with ~~~~ -

Angles 645 27.15 ~ o n p s m e t r i c Paired-Sample Testing with

Anglcr 647

27.16 Parametric Angular Cornlation and Regresioo 649

27.17 Nonparametric Angular Cornlation 653 27.18 Oood- of Pit Testing for Circular

Distdbutions 654 W.19 Serial Randomness of Nominal-Scale

Camaorics oa a Circle 658 Exexi- 660

APPENDIX A ANALYSIS OF VARIANCE HYPOTHESIS TESTING Appl A.l Determination of Appmpriam F's and Dsm &4 Nasted Analysis of Variance Am7

of M o r n Appl A.5 Split-Plot and Mixed WIG-Subjeers Analysis A 3 Ibo-Paouor Analysis of Variance App5 of Variance App8 A.3 Thrse-Factor Analysis of Variance App6

APPENDIX B STATISTICAL TABLES AND ORAPHS Appll Tabk B.1 Critical Valves of Chi-Squm

Distribution ApplZ a b l e B.2 Pmportioos of the Normal Curve

(One-Tailed) Appl7 Table B 3 Crifical Values of the I Distribution Appl9 Table B.4 Critical Values of tho F

Distribution AppZl Table B.5 Criticnl Valuer d the q

Distribution App58 Table 8.6 .-- Ctitial Values of - 0'- far the One-Tailed

Dunm's Teat Am74 Tabla B.7 wtical Values of q' for the Two-Tailed

Dunnsn's Test App76 Table B.8 critical values of 6,for lbc

Kolmopomv-Smimov Ooodncss of Fit Test for-~ i s n r ; or Gmu* Data.App77

a b l e B.9 Critical Values of D for the Kolmogomv-Smirnov Goodness of FI~Test for Continuous Distributions App83

Table 8.10 Critical Values of 4 for the 8 C o m I e d Kolmogomv-Smimov Goodness of Pit Tcst for Continuous Dtstributiona App87

Tabls B.11 Crlticsl Valuar of cha Mann-Whitncy U Distribution App89

Table B.12 Critical Values of the Wllcoxon T Distribution ApplOl

Tablc B.13 Critical Valuer of the Kruskal-Wallis H Dishibution Am104

Tabk B.14 dritisai 'Ynluss of the Friedman xf Dlambunon App106

Table B.15 Cnucal Valum of Q for Nonparamcms Multiple Comparison TcsDng Appl07

Table 8.16 Critical Values of Q' for Nonparamstric Multiple Cornpariron Tesung with a Contml Appl08

Table 8.17 Critics1 Values of the Cornlation Cafflciant, r Appl09

Table B.18 Fisher's r Transformation for Correlation CorWcicnts. r Appll l

Table B.19 Correlation CocWcisnts, r.. Comsponding to Pisher'a z Transformation Appl

Tabls B.ZO Critical Values of the Sosannan Rank correlation ccaffioisnt, r, ~ p p i16

Table B31 Critical Val- of tho Top-Down Comlation Coefficient. rr AppIl8

Table B.7.2 Critical Values of Ute Symmsery M e w , gl Appll9

Table B.23 Critical Values of the Kurtosis Measure, 62 AppIZL

Table B.24 The Arcsine Trmsfo-tion. p' ApplZQ Tabla B.25 Proponions, p. Corresponding to

Arcsinc Tmmf-tions. p' App127 Table B.26n Binomial Coefficients. .Cx Ape129 Tablc B36b Roportions of the Binomial

Distribution for p =q E0.5 App132 Table B37 Crilisal Valuw of C for the Sign Test or

for the Binomial Tcst with p =0.5 App133 Table B.28 Critical Values for Flshcr'r Exact

Test Am143 Table B.29 Critical Valuer for Runs Test Am171 a b l e 8.30 Cdtical Valuca of C for the Meadquarc

Successive D i E m c ~ Test Appl8O Table B 3 1 Critical Valucs for the Runs Up and

Down Tcst Appl82

Contents

APPENDlX B STATISTICAL TABLES AND GRAPHS (continued) lbble 8 3 2 Anpular Dtviarion. r. As e Function of

Vetor Lon@, tb App184 lbble 833 C u ~ u l vSwdard Dovialion $0. As

motion of Vector Length, r Apple6 Table 814 Criticd Wvcr of Rqlsigh'r z App188 lbble B35 Critlcd Vpluu of u for tho V Test of

Circular Uniformity Am190 lbble B.36 Critical Vduos of rn forthe Hodgca-Ajnc

Test Appl91

ANSWERS TO EXERCISES An57

LITERATURE CITED L1

INDEX I1

Tabk 8 3 7 Cowt ion Factor. K, for the Wamn and William Test App193

lbble 838 Crtried V a l w of Wncsun's LIZ Appl95xhhh ~ 3 9cri(icd Values of R' for the Moore Ten

TableBdO Common LO&&& of Pactonids App199 a b l e Bd1 Ten Thousand Random Digita AppMl PIplrr 8.1 Power and Sample Sire in Analysis of

Section 24.1 Binomial PmbabilHies 517

24.1 BINOMIALPROBABILITIES

Consides a population consisting of two categories, where p is the proportion of indi- viduals in one of the categories and q r 1 -p is the pmpo~tion in the other. Then the probability of selecting at random fmm this population a member of the f b t caagoIy is p, and the probability of selecting a member of the second category is q.t

For example, let us say we have a population of female and male animals, in proponions of p = 0.4 . a d q = 0.6, respectively, and we take a random sample of two individuals from the population. The probability of the first being a female is p (i.e.. 0.4) and the Ijrobability of the second boing a female is also p. As the probability of two mutually exclusive events both occurring is the product of the probabilities of the two separate events (Section 5.7), the probability of having two females in a sample of two is (p)(p) = p2 10.16: the probability of the sample of two consisting of two males is (q)(q) =q2 E0.36.

What is the probability ofthe sample of two consisting of one male and one female7 This could occur by the fixst individual being a female and the second a male (with a probability of pq) or by the first being amale and the second a female (which would occur with a probability of qp). The probability of either of two mutually exclusive outcomes is the sum of the probabilities of each outcome (Section 5.6). so the probability of one female and one male in the sample is pq +qp = 2pq =2(0.4)(0.6) = 0.48. Note that ! 0.16 +0.36 +0.48 r.1.00.

I

5'18 More on Dichotomous Variables Chapter 24 :L

If we performed the same exercise with n =4, we would find that the probability of four females is p4 = (0.4)4 = 0.0256, the probability of three females (and one male) is 4p3q = 4(0.4)"0.6) = 0.1536, the probability of two females is 6p2qZ = 0.3456, the probability of one female is 4pq3 = 0.3456. and the probability of no females (i.e.. all four are male) is q4 = 0.1296. (The sum of these five tenns is 1.0000, a good arithmetic check.)

If a random sample of size n is taken from a binomial population, then the prob- ability of X individuals being in one category (and, therefore. n -X individuals in the second category) is

P ( X ) = (;)pxqn-x. (24.1)

In this equation. pxq"-x refers to the probability of sample consisting of X items. each having a probability of p, and n - X items, each with probability q. The binomial coeficient,

is the number of ways X items of one kind can be arranged with n - X items of a second kind, or, in other words, the number of possible combinations of n items divided into one group of X items and a second group of n - X items. (See Section 5.3 for a discussion of combinations; Equation 5.3 explained the factorial notation. "!".) Therefore, Equation 24.1 can be written as

n! P ( X ) = PXq"-x.

Xl(n - x)!

Thus. (;)pXqn-X is the Xth term in the expansion of (p + q)", and Table 24.1 shows this expansion for powers up through 6. Note that for any power. n, the sumof the two exponents in any term is n. Furthermore, the first term will always be p". the second will always contain pn-'q, the third will always contain pn-'q2, etc.. with the last term always being qn. The sum of all the terns in a binomial expansion will always be 1.0. for p + q = 1, and (p + q)" = 1" = 1.

As for the coefficients of these terms in the binomial expansion, the Xth term of the nth power expansion can be calculated by Equation 24.3. Furthermore, the examination of these coefficients as shown in Table 24.2 has been deemed interesting for centuries.

TABLE 24.1 Expansion of the Binomial, (JJ + q)"

n (P +4) -

1 P + ' l 2 p2 + 2 p q + q2 3 p3 + 3 p l q +3pq2 +q 3 4 p4 + 4 p 3 q + 6 p 1 q 2 + 4 p q 3 + q 4 5 p5 + 5 p 4 q + 10p3q1 + 10plq3 + 5pq4 +4' 6 p6 + 6 p 5 q + 15p4q2 + ZOdq3 + 15p244 +6 p q 5 -q 6

Section 24.1 Binomial Probabilities 619

TABLe 2 4 3 Binomial Coefficient, "Cx

n XI 0 1 2 3 4 5 6 7 8 9 10 Sum of coefficients

This mangement is known as Pascal's triangle: We can see from this triangular array that any binomial coefficient is the sum of two coefficients on the line above it, namely.

This can be more readily observed if we display rhe tdangular array as follows: 1

1 1 I 2 1

1 3 3 1 1 4 6 4 1

1 5 10 10 5 1

Also note that the sum of all coefficients for the nth power binomial expansion is 2". Appendix Table B.26a presents binomial coefficients for much large1 n's and X's, and they will be found useful later in this chapter.

Thus, we can calculate probabilities of category frequencies occurring in random samples from binomial population. If, for example.-a sample of five (i.e.. n = 5) is taken from a population composcd of 50% males and 50% females (i.e., p =0.5 and q =0.5) then Example 24.1 shows how Equation 24.3 is used to determine tho probability of the sample containing 0 males, 1 male, 2 males, 3 males. 4 males, and 5 males. These

.Blcuao Peacal (1623-1662). French mathematidan and physicist and one of Ula founders of probability lhcory (in 1654. immediately before abandoning msthematia to become a mligioua mcluse). H e had his hiangular binomial cooflisicnt &"vation pubii8hed in 1665. allhovgh knowledge of the triangular pmp~rtics appears in Chinos* writings aa aarly as 1303 (Cajori, 1954: David. 1962: SrmiL. 1967: 79). P a s 4 also invented (at age 19) a mechanical adding and sublvaoting machine which, rhough patented in 1649,pmved t m expensive to bs practical to eonsrmct (Asimov. 1982: 130-131). His aigni6cant contributions to Ihe smdy of fluid prua-r have b n honorcd by naming the iotanational unit of pressure the pascal. which is a of one newton ~ c rso- mler (where a ncwtonaamcd for Sir Isaac Newton-is Ulc unit of form reorssmrin. . . -a. one-kilogram mass accalerating at lhe rate of one meter wr second pr second). Pascal is dso ths name. given to a modom computer Imguagc. The relationship of Pascal's mangle to. Cx w.s first published in 1685 by lhc Engtish mathLmatioian. John Wallis (1616-1703) (David. 1962: 123-124).

,520 More on Dichotomous Variables Chapter 24

becomes

EXAMPLE 24.1 Computing binomid pmbabilltleies, P ( X ) ,"hue n = 5,p = 0.5, and p = 0.5 (following Equation 24.3).

X P(X)

510 -(0.5°)(0.5') = (1)(1.0)(0.03125) = 0.031250151 51

1 -(0.5')(0.5') = (5)(0.5)(0.0625) = 0.156251141 51

2 -(0.5')(0.5~) = (10)(0.25)(0.125)= 0.312502131 51

3 3121(0.53)(0.52)= (10)(0.125)(0.25) =0.31250

514 -(0.5')(0.5') = (5)(0.0625)(0.5) = 0.15625

4111 515 ~ ( 0 . 5 ' ) ( 0 . S 0 )= (1)(0.03125)(1.0) = 0.03125

521 S d o n 24.1 Blnornlal Probabilities

Fl- m.1 I h c binodd dirtribution. for n = 5. (a) p = q = 0.5. (b)p = 0.3. q = 0.7. (c) p = 0.1, q =0.9. lheae graphs wtrr drawn utilizing the p m m o n s given by Equation 24.1.

especially in the tails of the distribution (i.e.. for low X and for high X),as shown in Example 24.3. If p is very small,then the use of the Poisson distribution (Section 25.1), should be considered..

The mean of a binomial distribution of counts X, is

/AX =np. the variancet is

0: ="P4.

and the standard deviation of X is

ex = ,biz.

.RafS (1956) and Molsnur (1969~. 1969b) discuss several lppmximations to the binomial dismbution. including the normal pnd Poiason distributions.

*A me-o ofsymmetry (see Secdoa 6.1) far a b i i distribution is ~

y , -4-P (24.7)m. so it can bc seen that yl = 0 only when p - q = 0.05. yl > 0 implies a diarribution skewed to the righl (ar in Pigs. 24.lb md 24.1~)and n < 0 indicates a distribution skewed to the left.

622 More on Dichotomous Variables Chapter 24

-EXAMPLE 14.2 Computing blnomfnl probsbllltles. P(X), where n = 5, p = 0.4. q = 0.7 (lollow- lng Equation 24.3). !

X P(X)

510 E(~.3°)(0.75) = (1)(1.0)(0.16807) = 0.16807

1 ( 0 . 3 ~ ) ( 0 . 7 ' ) = (5)(0.3)(0.2401) = 0.360151141

2 2(0.3')(0.7~) = (10)(0.09)(0.343) = 0.308702131

S!3 -(0.3~)(0.7') = (10)(0.027)(0.49) - 0.13230

3121 51

4 ~ (0 .3 ' ) (0 .7 ' ) = (5)(0.@381)(0.7)= 0.02835

515 -(0.3~)(0~7~)= (l)(O.OM43)(1.0) = 0.002435101

EXAMPLE 24.3 Computing blnomial pmbabiUtlen, P(X). Kith n = 400.11 = 0.02, and q = 0.98.

(Many calculators can operate with Large powers of numbers; otherwise. logarithms may be used.)

X P ( x )

n l' O!(n -0)lP "-'= q" = 0.9- = 0.W031

"1 1 "-1 = npq"-' = (400)(0.02)(0.98~~)= 0.00253I I ( ~- 111' '

"1 1 " - 2 -Z ! ( ~ - Z ) I ~ ' "(n2; 1)p'qi-2 = 2 = 0.01028(400)(399) (0.02~)(0.9839~)

nl , "-3 - 4"- I)(" - 2) dpa-3 = (400)(399)(398) (0,021)(~,~~397) 3 1 ( n - 3 ) ! ~ ' - 31 (3)(2)

= 0.02784

and so on.

Thus, if we have a binomially distributed population where p (e.g., the proponiod of males) = 0.5 and q (e.g.. the proportion of females) = 0.5 and we take ten samples fSom that population, the mean of the ten X's (is., the mean number of males per sample) would be expected to be np = (10)(0.05) = 5 and the standard deviation of the ten X ' S

would be expected to be &Zf = J(10)(0.5)(0.5) = 1.58. Our concern typically is with the distribution of the expected probabiLities rather than the expected X's, as will be explained in Section 24.3.