Devore J.L., Berk K.N. Modern Mathematical Statistics with Applications

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Error(s) (cont.)

type I, 429

type II, 429

Estimated regression function,

676, 685

Estimated regression line, 625

Estimated standard error, 344,

646, 713

Estimator, 332

Event(s)

complement of, 53

compound, 52, 62

definition of, 52

dependent, 84–88

disjoint, 54

exhaustive, 79

independent, 84–88

indicator function for, 364

intersection of, 53

mutually exclusive, 54

mutually independent, 87

simple, 52

union of, 53

Venn diagrams for, 55

Expected mean squares

in ANOVA, 573, 577, 600, 614

F test and, 589, 593, 600, 604

in mixed effects model, 593, 604

in random effects model, 580,

593–594

in regression, 681

Expected value

conditional, 255

of a continuous random

variable, 171

covariance and, 247

of a discrete random variable, 112

of a function, 115, 245–246

heavy-tailed distribution and,

114–115, 120

of jointly distributed random

variables, 245

Law of Large Numbers and, 303

of a linear combination, 306

of mean squares (see Expected

mean squares)

moment generating function

and, 122, 175

moments and, 121

in order statistics, 272–273, 277

of sample mean, 277, 296

of sample standard deviation,

340, 379

of sample total, 296

of sample variance, 339

Experiment

binomial, 128, 240, 724

definition of, 52

double-blind, 523

observational studies in, 488

paired data, 515

paired vs. independent samples,

520–521

randomized block, 590–593

randomized controlled, 489

repeated measures designs in, 591

with replacement, 69, 141, 287

retrospective, 488

simulation, 291–294

Explanatory variable, 614

Exponential distribution

censored experiments and, 343

chi-squared distribution

and, 317

confidence interval for

parameter, 389

double, 477

estimators for parameter, 343, 351

goodness-of-fit test for, 739

mixed, 229

in pure birth process, 378

shifted, 360, 479

skew in, 277

standard gamma distribution

and, 198

Weibull distribution and, 203

Exponential random variable(s)

Box–Muller transformation

and, 271

cdf of, 199

expected value of, 198

independence of, 242

mean of, 198

in order statistics, 272, 275

pdf of, 198

transformation of, 220, 267, 270

variance of, 198

Exponential regression model, 721

Exponential smoothing, 48

Extreme outliers, 39–41

Extreme value distribution, 217

Factorial notation, 69

Factorization theorem, 363

Factors, 552

Failure rate function, 230

Family of probability distributions,

104, 213

F distribution

chi-squared distribution and, 323

definition of, 323

expected value of, 325

for model utility test, 649, 687, 709

noncentral, 574–575

pdf of, 324

Finite population correction factor, 141

Fisher information, 371

Fisher–Irwin test, 525

Fisher transformation, 669

Fitted values, 588, 629, 674

Fixed effects model, 579, 592, 597

Fourth spread, 37, 41, 285

Frequency, 13

Frequency distribution, 13

Friedman’s test, 785

F test

in ANOVA, 558, 580, 587,

593, 600

Bartlett’s test and, 562

coefficient of determination

and, 687

critical values for, 324, 528, 558

distribution and, 323, 527, 558

for equality of variances, 527, 537

expected mean squares and, 573,

589, 593, 600, 604

Levene test and, 562

power curves and, 574–575

P-value for, 529, 537, 559

in regression, 687, 709

sample sizes for, 574

single-factor, 558, 580

vs. t test, 576

two-factor, 587, 593, 600

type II error in, 574

Full quadratic model, 695

Galton–Watson branching process,

281

Gamma distribution

chi-squared distribution and, 200

definition of, 195

density function for, 195

Erlang distribution and, 201

estimators of parameters, 351,

355, 358

exponential distribution and,

198–200

Poisson distribution and, 783

standard, 195

Weibull distribution and, 203

Gamma function

incomplete, 196, 217

properties of, 195

Gamma random variables, 195

Geometric distribution, 143, 225

Geometric random variables, 143

Goodness-of-fit test

for composite hypotheses,

732, 741

definition of, 723

for homogeneity, 745–747

for independence, 747–749

simple, 724–730

Grand mean, 555, 584

838 Index

Half-normal plot, 220

Histogram

bimodal, 18

class intervals in, 15–17

construction of, 12–20

density, 17–18

multimodal, 19

Pareto diagram, 24

for pmf, 103

symmetric, 19

unimodal, 18

Hodges–Lehmann estimator, 379

Homogeneity, 745–747

Hyperexponential distribution, 229

Hypergeometric distribution,

138–141

and binomial distribution, 141

Hypergeometric random variable,

138–141

Hypothesis

alternative, 426

composite, 732–741, 744

definition of, 426

errors in testing of, 428–434

notation for, 426

null, 426

research, 427

simple, 469

Hypothetical population, 6

Inclusive inequalities, 136

Incomplete beta function, 207

Incomplete gamma function,

196–197, 217

Independence

chi-squared test for, 749

conditional distribution and,

257–258

correlation coefficient

and, 250

covariance and, 250, 252

of events, 84–88

of jointly distributed random

variables, 238–239, 241

in linear combinations, 306–307

mutual, 87

pairwise, 90, 94

in simple random sample, 287

Independent variable, 614

Indicator variables, 696

Inductive reasoning, 6

Inferential statistics, 5–6

Inflection point, 180

Intensity function, 156

Interaction, 597–602, 603–606,

693–698

Intercept, 214, 617, 627

Intersection of events

definition of, 53

multiplication rule for probability

of, 77–79

Invariance principle, 357

Inverse matrix, 712

Jacobian, 267

Jensen’s inequality, 231

Joint cumulative distribution

function, 282

Jointly distributed random variables

bivariate normal distribution

of, 258–260

conditional distribution of,

253–263

correlation coefficients for, 249

covariance between, 248

expected value of function of,

245–246

independence of, 238–239

linear combination of, 306–312

in order statistics, 274–276

pdf of (see Joint probability

density functions)

pmf of (see Joint probability mass

functions)

transformation of, 265–270

variance of function of, 252, 307

Joint marginal density function, 245

Joint probability mass function,

233–234

Joint probability table, 233

k-out-of-n system, 153

Kruskal–Wallis test, 784–785

k-tuple, 68–69

lag 1 autocorrelation coefficient, 674

Laplace distribution, 478

Laplace’s rule of succession, 782

Largest extreme value distribution, 228

Law of Large Numbers, 303–304,

322–323, 376

Law of total probability, 79

Least squares estimates, 626, 645,

679, 683–684

Level a test, 433

Level of a factor, 552, 583, 593

Levene test, 562–563

Leverages, 714–715

Likelihood function, 354, 470, 475

Likelihood ratio

chi-squared statistic for, 477

definition of, 470

mle and, 475

model utility test and, 721

in Neyman–Pearson theorem, 470

significance level and, 470, 471

sufficiency and, 380

tests, 475

Limiting relative frequency, 58, 59

Linear combination

distribution of, 309

expected value of, 306

independence in, 306

variance of, 307

Linear probabilistic model, 617, 627

Linear regression

additive model for, 614, 682, 705

ANOVA in, 649, 699, 768

confidence intervals in, 643, 656

correlation coefficient in,

662–671

definition of, 617

degrees of freedom in, 631,

685, 708

least squares estimates in,

625–636, 679

likelihood ratio test in, 721

mles in, 631, 639

model utility test in, 648, 687, 708

parameters in, 617, 624–636, 682

percentage of explained variation

in, 633–634

prediction interval in, 654, 658, 689

residuals in, 629, 674, 685

summary statistics in, 627

sums of squares in, 631–636, 686

t ratio in, 648, 669, 690

Line graph, 102–103

Location parameter, 217, 367

Logistic distribution, 279

Logistic regression model

contingency tables for, 749–751

definition of, 620–622

fit of, 650–651

mles in, 650

in multiple regression analysis, 699

Logit function, 621, 650

Lognormal distribution, 205–205, 233

Lognormal random variables, 205–206

Mann–Whitney test, 766–770

Marginal distribution, 234, 236, 253

Marginal probability density

functions, 236

Marginal probability mass

functions, 234

Matrices in regression analysis,

705–715

Maximum likelihood estimator

for Bernoulli parameter, 377

for binomial parameter, 377

Index 839

Maximum likelihood estimator

(cont.)

Crame

r–Rao inequality and, 375

data sufficiency for, 369

Fisher information and, 371, 375

for geometric distribution

parameter, 742

in goodness-of-fit testing, 733

in homogeneity test, 745

in independence test, 748

in likelihood ratio tests, 475

in linear regression, 631, 639

in logistic regression, 650

sample size and, 357

score function and, 377

McNemar’s test, 526, 550

Mean

of Cauchy distribution, 322,

342, 761

conditional, 255–257

correction for the, 560

deviations from the, 33, 206, 563,

631, 739

of a function, 115, 245–246

vs. median, 28

moments about, 121

outliers and, 27, 28

population, 26

regression to the, 260, 636

sample, 25

of sample total, 296

See also Average

Mean square

expected, 573, 589, 593, 594,

600, 604

lack of fit, 681

pure error, 681

Mean square error

definition of, 335

of an estimator, 335

MVUE and, 341

sample size and, 337

Measurement error, 337

Median

in boxplot, 37–38

of a distribution, 27, 28

as estimator, 378, 478

vs. mean, 28

outliers and, 26, 28, 29

population, 28

sample, 27, 271

statistic, 378

Mendel’s law of inheritance, 726–728

M-estimator, 359, 381

Midfourth, 46

Midrange, 333

Mild outlier, 39, 393

Minimal sufficient statistic,

366–367, 369

Minimize absolute deviations

principle, 477, 679

Minimum variance unbiased

estimator, 341–343, 358,

369, 375

Mixed effects model, 593–603

Mixed exponential distribution, 229

mle. See Maximum likelihood

estimate

Mode

of a continuous distribution,

228, 229

of a data set, 46

of a discrete distribution, 156

Model utility test, 647–649

Moment generating function

of a Bernoulli rv, 122, 127

of a binomial rv, 135

of a chi-squared rv, 315

CLT and, 329–330

of a continuous rv, 175–177

definition of, 122, 175

of a discrete rv, 122–127

of an exponential rv, 221

of a gamma rv, 195

of a linear combination, 311

and moments, 124, 176

of a negative binomial rv, 143

of a normal rv, 191

of a Poisson rv, 149

of a sample mean, 329–330

uniqueness property of, 123, 176

Moments

definition of, 121

method of, 350–352, 358, 740

and moment generating function,

124, 176

Monotonic, 221, 353

Multimodal histogram, 19

Multinomial distribution, 240, 725

Multinomial experiment, 240, 724

Multiple regression

additive model, 682, 705

categorical variables in, 696–699

coefficient of multiple

determination, 686, 709

confidence intervals in, 712

covariance matrices in, 711–713

degrees of freedom in, 685,

696, 708

diagnostic plots, 691

fitted values in, 685

F ratio in, 687, 709

interaction in models for, 693–698

leverages in, 714–715

logistic regression model, 699

in matrix/vector format, 705–715

model utility test in, 687,

708–709

normal equations in, 683, 685,

705–708

parameters for, 682

and polynomial regression,

691–693

prediction interval in, 689

principle of least squares in,

683–706

residuals in, 685, 691, 688, 691,

708, 713

squared multiple correlation in,

686, 709

sum of squares in, 686, 708–710

t ratios in, 690, 712

Multiplication rule, 77–88

Multiplicative exponential

regression model, 721

Multiplicative power regression

model, 721

Multivariate data, 3, 20

Multivariate hypergeometric

distribution, 244

Mutually exclusive events, 54, 79

MVUE. See Minimum variance

unbiased estimator

Negative binomial distribution,

141–144

definition of, 141

estimation of parameters, 352, 738

Negative binomial random

variable, 141

Newton’s binomial theorem, 143

Neyman factorization theorem, 363

Neyman–Pearson theorem, 470–475

Noncentrality parameter, 423,

574, 582

Noncentral t distribution, 423

Nonhomogeneous Poisson

process, 156

Nonstandard normal distribution,

185–188

Normal distribution

asymptotic, 298, 371, 375, 377

binomial distribution and,

189–190, 302

bivariate, 258–260, 310, 318,

477, 677–671

confidence interval for mean of,

383–388, 392, 398, 403

continuity correction and, 189–190

density curves for, 180

and discrete random variables,

188–190

goodness-of-fit test for, 730, 740

of linear combination, 309

lognormal distribution and,

205, 303

840 Index

nonstandard, 185–188

pdf for, 179

percentiles for, 182–188, 210

probability plot, 210, 740

Ryan–Joiner test for, 747

standard, 181

t distribution and, 320–322, 325,

402

z table, 181–183

Normal equations, 626, 683, 705

Normal probability plot, 210, 740

Normal random variable, 181

Null distribution, 443–444, 760, 780

Null hypothesis, 426

Null set, 54, 57

Null value, 427, 436

Observational study, 488

Odds ratio, 621–622, 750–751

One-sided confidence interval,

398–399

Operating characteristic curve, 137

Ordered categories, 749–751

Ordered pairs, 66–67

Order statistics, 271–278, 338,

365–367, 478

sufficiency and, 365–367

Outliers

in a boxplot, 37–41

definition of, 11

extreme, 39–41

leverage and, 714

mean and, 29, 415–417

median and, 29, 37, 415, 417

mild, 39

in regression analysis, 679, 688

Paired data

in before/after experiments,

511, 526

bootstrap procedure for, 538–540

confidence interval for, 513–515

definition of, 509

vs. independent samples, 515

in McNemar’s test, 550

permutation test for, 540–541

t test for, 511–513

in Wilcoxon signed-rank test,

762–763

Pairwise average, 772, 773, 775

Pairwise independence, 94

Parallel connection, 55, 88, 89, 90,

272, 273

Parameter(s)

Bayesian approach to, 776–782

concentration, 779

confidence interval for, 389, 394

estimator for a, 332–346

Fisher information on, 371–377

goodness-of-fit tests for,

728–729, 732–736

hypothesis testing for, 427, 450

location, 217, 367

maximum likelihood estimate

of, 354–359, 369

moment estimators for, 350–352

MVUE of, 341–343, 358,

369, 375

noncentrality, 574

null value of, 427

of a probability distribution,

103–104

in regression, 617–618, 622,

624–636, 658, 666, 682

scale, 195, 203, 217–218, 365

shape, 217–218, 365

sufficient estimation of, 361–369

Pareto diagram, 24

Pareto distribution, 170, 178, 226

pdf. See Probability density function

Percentiles

for continuous random variables,

166–168

in hypothesis testing, 458, 740

in probability plots, 211–216, 740

sample, 29, 210–211, 216

of standard normal distribution,

182–184, 211–216

Permutation, 68, 69, 535–541

Permutation test, 535–541

PERT analysis, 207

Plot

probability, 210–218, 369, 499,

668, 676, 688, 691, 740

scatter, 615–617, 632–633,

663, 667

pmf. See Probability mass function

Point estimate/estimator

biased, 337–342

bias of, 335–340

bootstrap techniques for,

345–346, 411–418

bound on the error of

estimation of, 388

censoring and, 343–344

consistency, 304, 357, 375–377

for correlation coefficient, 665–666

and Crame

r–Rao inequality,

373–377

definition of, 26, 287, 332

efficiency of, 375

Fisher information on, 371–377

least squares, 626–631

maximum likelihood (mle),

352–359

of a mean, 26, 287, 332–333, 366

mean squared error of, 335

moments method, 350–352, 358

MVUE of, 340–342, 358, 369,

375

notation for, 332, 334

of a standard deviation and,

286, 340

standard error of, 344–346

of a variance, 334, 339

Point prediction, 405, 628, 684

Poisson distribution

Erlang distribution and, 202

expected value, 149, 152

exponential distribution and, 199

gamma distribution and, 783

goodness-of-fit tests for, 736–738

in hypothesis testing, 470–472,

474, 482, 550

mode of, 156

moment generating function

for, 149

nonhomogeneous, 156

parameter of, 149

and Poisson process, 149–151, 199

variance, 149, 152

Poisson process, 149–151, 194

Polynomial regression model,

691–693

Pooled t procedures

and ANOVA, 477, 504–505, 576

vs. Wilcoxon rank-sum

procedures, 769

Posterior probability, 79–81,

777, 781

Power curves, 574–575

Power function of a test, 473–475,

574–575

Power model for regression, 721

Power of a test

Neyman–Pearson theorem and,

473–475

type II error and, 446–447,

472–476, 505, 593, 749

Precision, 315, 344, 371, 382,

387–388, 397, 405, 417, 514,

516, 592, 781

Prediction interval

Bonferroni, 659

vs. confidence interval, 406,

658–659, 690

in linear regression, 654, 658–659

in multiple regression, 690

for normal distribution, 404–406

Prediction level, 405, 659, 689

Predictor variable, 614, 682,

693–696

Principle of least squares, 625–636,

674, 679, 683

Prior probability, 79, 758

Index 841

Probability

conditional, 74–81, 84–85, 200,

253–255, 362, 365–366

continuous random variables

and, 99, 158–225, 235–242,

253–255

counting techniques for, 66–72

definition of, 50

density function (see Probability

density function)

of equally likely outcomes, 62–63

histogram, 103, 159–160,

188–190, 289–290

inferential statistics and, 6, 9, 284

Law of Large Numbers and,

303–304, 322–323

law of total, 79

mass function (see Probability

mass function)

of null event, 57

plots, 210–218, 369, 499, 668,

676, 688, 691, 740

posterior/prior, 79–81, 758,

777, 781

properties of, 56–63

relative frequency and, 58–59,

291–292

sample space and, 51–55, 56–57,

63, 66, 95

and Venn diagrams, 54–55, 62,

75–76

Probability density function (pdf)

conditional, 254–255, 777

definition of, 161

joint, 232–278, 310, 354,

363–365, 368, 470, 475

marginal, 236–238, 268–269

vs. pmf, 162

Probability distribution

Bernoulli, 98, 102–104, 113,

122–123, 127, 134, 302, 304,

308, 360, 373, 375, 377, 777

beta, 206–208

binomial, 128–135, 147–149,

189–190, 302, 352–353,

395–396, 428–431

bivariate normal, 258–260,

477, 669

Cauchy, 226, 231, 271, 342

chi-squared, 200, 224, 315–320

conditional, 253–263

continuous, 99, 158–231

discrete, 96–157

exponential, 198–200, 203, 343

extreme value, 217–218

F, 323–325

family, 104, 213, 216–218, 558

gamma, 194–200, 217–218

geometric, 106–107, 114, 143, 225

hyperexponential, 229

hypergeometric, 138–141,

307–308

joint, 232–283, 665–667, 732

Laplace, 315, 477–478

of a linear combination, 259,

306–312

logistic, 279

lognormal, 205–206, 303

multinomial, 240, 724

negative binomial, 141–144

normal, 179–191, 205, 210–216,

258–260, 297–303, 309, 730

parameter of a, 103–104

Pareto, 170, 178, 226

Poisson, 146–151, 199

Rayleigh, 169, 226, 349, 360

of a sample mean, 285–294,

296–304

standard normal, 181–184

of a statistic, 285–304

Studentized range, 565

symmetric, 19, 28, 121, 168,

174, 180

t, 320–323, 325, 401–403, 443,

462, 511

uniform, 161–162, 164

Weibull, 202–205

Probability mass function

conditional, 253–254

definition of, 101–109

joint, 233–236

marginal, 234

Product rules, 66–68

Proportion

population, 30, 395, 450–454,

519–525

sample, 30, 190, 302, 338, 519,

748

trimming, 29, 333, 340, 342–343

P-value

for chi-squared test, 727–728

definition of, 456

for F tests, 529–530

for t tests, 462–465

type I error and, 457–459

for z tests, 459–461

Quadratic regression model, 691–693

Qualitative data, 19

Quartiles, 28–29

Random effects model, 579–580,

593–594, 603–606

Random interval, 384–386

Randomized block experiment,

590–593

Randomized controlled

experiment, 489

Randomized response technique, 349

Random variable

continuous, 158–231

definition of, 97

discrete, 96–157

jointly distributed, 232, 233–283

standardizing of, 185

types of, 99

Range

definition of, 33

in order statistics, 271–274

population, 394

sample, 33, 271–274

Studentized, 565–566

Rank average, 785

Ratio statistic, 478

Rayleigh distribution, 226,

349, 360

Regression

coefficient, 640–651, 682–685,

705–707, 711–712

effect, 260, 636

function, 614, 676, 682, 685,

693, 696

line, 618–620, 624–636,

640–647, 674–677

linear, 617–620, 624–636,

640–649, 654–659

logistic, 620–622, 650–651

matrices for, 705–715

to the mean, 260

multiple, 682–689

multiplicative exponential

model, 721

multiplicative power model

for, 721

plots for, 676–678

polynomial, 691–693

quadratic, 691–693

through the origin, 381–421

Rejection method, 281

Rejection region

cutoff value for, 428–433

definition of, 428

lower-tailed, 431, 437–438

in Neyman–Pearson theorem,

470–474

two-tailed, 438

type I error and, 429

in union-intersection test, 551

upper-tailed, 429, 437–438

Relative frequency, 13–19, 30,

58–59

Repeated measures designs, 591

Replications, 58, 291–293, 386

Research hypothesis, 427

Residual plots, 588, 602, 676–678

842 Index

Residuals

in ANOVA, 588, 602

definition of, 556

leverages and, 714–715

in linear regression, 629, 674–678

in multiple regression, 685, 688

standard error, 674

standardizing of, 675, 691

variance of, 675, 713

Response variable, 8, 614, 620

Retrospective study, 488

Ryan–Joiner test, 741

Sample

convenience, 7

definition of, 2

outliers in, 38–40

simple random, 7, 287

size of (see Sample size)

stratified, 7

Sample coefficient of variation, 45

Sample correlation coefficient

in linear regression, 662–664,

669, 719

vs. population correlation

coefficient, 666, 669–671

properties of, 664–665

strength of relationship, 665

Sample mean

definition of, 25

population mean and, 296–304

sampling distribution of, 296–304

Sample median

definition of, 27

in order statistics, 271–272

vs. population median, 417

Sample moments, 350–351

Sample percentiles, 210–211

Sample proportion, 30, 335–336,

338, 391–400, 450–455,

519–526

Sample size

in ANOVA, 574–576

asymptotic relative efficiency

and, 764, 769

bound on the error of estimation

and, 388

Central Limit Theorem and, 302

confidence intervals and,

387–388, 394, 396, 403, 495

definition of, 9

in finite population correction

factor, 140

for F test, 574–576

for Levene test, 562–563

mle and, 357–358, 375

noncentrality parameter and,

574–576, 582

Poisson distribution and, 147

for population proportion,

396–398

power and, 433, 440–441, 445,

452–454, 489, 505, 523

probability plots and, 216

in simple random sample, 287

t distribution and, 445, 505

type I error and, 433, 440-441,

445, 489, 523

type II error and, 433, 440–441,

445, 452–454, 489, 505, 523

variance and, 303

z test and, 440–441, 452–453

Sample space

definition of, 51

probability of, 56–63

Venn diagrams for, 54–55

Sample standard deviation

in bootstrap procedure, 413, 537

confidence bounds and, 398

confidence intervals and, 392,

403

definition of, 33

as estimator, 340, 379

expected value of, 340, 379

independence of, 318–319

mle and, 357

population standard deviation

and, 286, 340, 379

sample mean and, 34, 318–319

sampling distribution of,

288–289, 320, 340, 379, 482

variance of, 482

Sample total, 296, 306, 560

Sample variance

in ANOVA, 555–556

calculation of, 35

definition of, 33

distribution of, 287–289, 320

expected value of, 339

population variance and, 35, 317,

322–323, 339

Sampling distribution

bootstrap procedure and, 413,

532, 758

definition of, 284, 287

derivation of, 288–291

of intercept coefficient, 719

of mean, 288–290, 297–299

permutation tests and, 758

simulation experiments for,

291–294

of slope coefficient, 640–649

Scale parameter, 195, 203–204,

217–218, 365

Scatter plot, 615–617

Scheffe

method, 610

Score function, 373–377

Series connection, 272–273

Set theory, 53–55

Shape parameters, 217–218, 366

Siegel–Tukey test, 786

Significance

practical, 468–469, 727

statistical, 469, 489, 727

Significance level

definition of, 433

joint distribution and, 479

likelihood ratio and, 475

observed, 458

Sign interval, 784

Sign test, 784

Simple events, 52, 62, 66

Simple hypothesis, 469, 732

Simple random sample

definition of, 7, 287

independence in, 287

sample size in, 287

Simulation experiment, 288,

291–294, 417, 463

Skewed data

coefficient of skewness, 121, 178

definition of, 19

in histograms, 19, 413

mean vs. median in, 28

measure of, 121

probability plot of, 216, 411–413

Slope, 617–618, 622, 626, 642, 644

Slope coefficient

confidence interval for, 644

definition of, 617–618

hypothesis tests for, 648

least squares estimate of, 626

in logistic regression model, 622

Standard deviation

normal distribution and, 179

of point estimator, 344–346

population, 117, 173

of a random variable, 117, 173

sample, 33

z table and, 186

Standard error, 344–346

Standardized variable, 185

Standard normal distribution

Cauchy distribution and, 271

chi-squared distribution and,

316, 325

critical values of, 184

definition of, 181

density curve properties for,

181–184

F distribution and, 323, 325

percentiles of, 182–184

t distribution and, 320, 325

Standard normal random variable,

181, 325

Statistic, 286

Index 843

Statistical hypothesis, 426

Stem-and-leaf display, 10–12

Step function, 106

Stratified samples, 7

Studentized range distribution, 565

Student t distribution, 320–323

Summary statistics, 627, 630,

645, 671

Sum of squares

error, 557, 631, 708

interaction, 599

lack of fit, 681

pure error, 681

regression, 636, 699, 708

total, 559–560, 587, 591, 645,

686

treatment, 557–560

Symmetric distribution, 19,

121, 168

Taylor series, 174, 579

t confidence interval

heavy tails and, 764, 769, 774

in linear regression, 643, 656

in multiple regression, 689, 712

one-sample, 403–404

paired, 513–515

pooled, 505

two-sample, 500, 515

t distribution

central, 423

chi-squared distribution and,

320, 325, 500, 504

critical values of, 322, 402,

444, 461

definition of, 320

degrees of freedom in,

320–321, 401–402

density curve properties for,

322, 402

F distribution and, 325, 576

noncentral, 423

standard normal distribution

and, 320, 322, 403

Student, 320–323

Test statistic, 428

Time series, 48, 674

Tolerance interval, 406

Treatment, 553, 555–556, 583

Tree diagram, 67–68, 78, 81, 87

Trial, 128–131

Trimmed mean

definition of, 28–29

in order statistics, 271–272

outliers and, 29

as point estimator, 333,

340, 343

population mean and, 340, 343

Trimming proportion, 29, 343

True regression function, 615

True regression line, 618–620, 625,

640–641

t test

vs. F test, 576

heavy tails and, 764, 769, 774

likelihood ratio and, 475, 476

in linear regression, 648

in multiple regression,

688–690, 712

one-sample, 443–445, 461,

474–476, 511, 769

paired, 511

pooled, 504–505, 576

P-value for, 461–462

two-sample, 499–504, 576, 515

type I error and, 443–445, 501

type II error and, 445–447, 505

vs. Wilcoxon rank-sum

test, 769

vs. Wilcoxon signed-rank test,

763–764

Tukey’s procedure, 565–570, 578,

589–590, 603

Two one-sided tests, 551

Type I error

definition of, 429

Neyman–Pearson theorem

and, 470

power function of the test

and, 473

P-value and, 457–458

sample size and, 441

significance level and, 433

vs. type II error, 433

Type II error

definition of, 429

vs. type I error, 433

Type II error probability

in ANOVA, 574–576, 596

degrees of freedom and, 516

for F test, 574–576, 596

in linear regression, 653

Neyman–Pearson theorem

and, 469–472

power of the test and, 446, 473

sample size and, 440, 505,

477–478, 468, 495

in tests concerning means,

440, 445, 468, 489, 505

in tests concerning proportions,

452–453, 522–524

test and, 445, 505

vs. type I error probability, 433

in Wilcoxon rank-sum

test, 769

in Wilcoxon signed-rank test,

763–764

Unbiased estimator, 337–344

minimum variance, 340–343

Uncorrelated random variables,

251, 307

Uniform distribution

beta distribution and, 778

Box–Muller transformation

and, 271

definition of, 161

discrete, 120

transformation and, 223–224

Uniformly most powerful test,

473–474

Unimodal histogram, 18–19

Union-intersection test, 551

Union of events, 53

Univariate data, 3

Variable(s)

covariate, 699

in a data set, 10

definition of, 3

dependent, 614

dummy, 696–699

explanatory, 614

independent, 614

indicator, 696–699

predictor, 614

random, 96–231

response, 614

Variance

conditional, 255–257

of a function, 118–119,

174–175, 328

of a linear function, 118–120, 307

population, 34–35, 117, 173

precision and, 781

of a random variable, 117, 173

sample, 33–37

Venn diagram, 54–55, 62, 75, 76

Weibull distribution

basics of, 202–205

chi-squared distribution and, 231

estimation of parameters, 356,

359–360

extreme value distribution

and, 217

probability plot, 217–218

Weighted average, 112, 171, 261,

504, 779, 781

Weighted least squares

estimates, 679

Wilcoxon rank-sum test, 766–769

Wilcoxon signed-rank test,

759–764

844 Index

z confidence interval

for a correlation coefficient, 671

for a difference between

means, 493

for a difference between

proportions, 524

for a mean, 387, 392

for a proportion, 395

z curve

area under, maximizing

of, 479

rejection region and, 438

t curve and, 322, 402

z test

chi-squared test and, 752

for a correlation

coefficient, 669

for a difference between

means, 485–493

for a difference between

proportions, 521

for a mean, 438, 442

for a Poisson parameter,

400, 482

for a proportion, 451

P-value for, 459–461

Index 845