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763

Acceptance intervals,240242Addition rule of probabilities, 9293Adjusted coefficient of determina-

tion R2,472Allocation

proportional, 697of sample effort among strata,

703705Alternative hypothesis, 327, 331,

336339,356,386,388. See alsoHypothesis tests/testing

Analysis of variance (ANOVA)comparison of several population

means, 625627introduction to, 625

Kruskal-Wallis test and, 638640one-way, 627636for regression, 412413two-way, more than one

observation per cell, 650656two-way, one observation per cell,

randomized blocks, 641647Analysis of variance tables,

two-way, 646647Approximate mean,6162ARIMA (autoregressive integrated

moving average) models,

693694Arithmetic mean,40Association test, 595598Asymmetric distribution, 4243Autocorrelated errors

Durbin-Watson test and, 564566estimation of regressions with,

566570explanation of, 562564with lagged dependent variable,

570571Autocorrelation, 688689

Autoregressive integrated movingaverage (ARIMA) models,693694

Autoregressive modelsestimation and,689example of,689690explanation of, 688first-order, 688forecasting from, 689692second-order, 688

Bar charts, 810,3233Basic outcomes, 75Bayes, Thomas, 112

Bayes theorem, 112119alternative statement, 115examples, 112117explanation of, 112management decision making, 118solution steps for, 114

Bernoulli distribution,139141Bernoulli random variable,140Beta coefficients, 436438Beta measure, of financial risk,

436438Between-groups mean square

(MSG),662663Between-groups variability, 629Bias

explanation of, 267specification, 551553

Biased estimators, 267Binomial distribution,139145

compared with normaldistribution, 201

compared with Poissondistribution, 152

derived mean and variance of,

140,175176examples of, 142145explanation of, 142normal distribution

approximation for, 199204Poisson approximation to,

151152probability function table, 720723

Binomial probabilities, cumulative,724728

Bivariate probabilities, 102112Blocking variables, 539540,641

Block means, 651Box-and-whisker plots, 4951

Categorical data analysiscontingency tables and, 594598goodness-of-fit tests, population

parameters unknown,589593

goodness-of-fit tests, specifiedprobabilities, 583588

INDEX

nonparametric tests forindependent random samples,608612,616619

nonparametric tests for paired ormatched samples,599606

Spearman rank correlation and,614615

Categorical variables,5graphs to describe, 815

Cell means,651652Central limit theorem, 234240

from linear sum of randomvariables,260

Central tendency, measures of,3948

Chebychevs theorem, 5557Chi-square distribution, 286288

lower critical values table,749population variance,251252upper critical values table, 748variance of normal distribution,

355Chi-square random variable, 585

for contingency tables, 595Chi-square test

examples of, 586588of variance of a normal

distribution, 355356Classical probability,8182,85Cluster bar charts, 10, 3233Cluster sampling

estimators for, 709712explanation of, 709

Cobb-Douglas production function,499

Coefficient estimation,533534

Coefficient estimatorsderivation of, 533534

least squares, 407410, 419variance, 417, 485486

Coefficient of determination R2

adjusted,472explanation of, 413417regression models and, 472sum of squares decomposition

and,469470Coefficient of multiple correlation,

472
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764 Index

Coefficient of multiple regression,461467

Coefficient of standard errors,475Coefficient of variation (CV),55Collectively exhaustive events,78Combinations

formula for determining numberof,8485

number of,8283Complement rule,9192,9899Complements,7880Component bar charts,10Composite hypothesis,331,336339Computer applications. See alsoExcel

for jointly distributed discreterandom variables,160

of regression coefficient,409410Conditional coefficients,466Conditional mean,160Conditional probability,9394Conditional probability

distribution,158Conditional variance,160Confidence interval estimator,271Confidence intervalsbased on normal distribution,273for difference between two normal

population means, dependentsamples,309312

for difference between two normalpopulation means,independent samples,313319

for difference between two popu-lation proportions,320321

examples of,274, 280282,284285,288,290293

explanation of,273finite populations,289293forecast, and prediction intervals,

427428for mean of normal distribution,

population variance known,271276

for mean of normal distribution,population variance un-known,277282

for population mean,271282,289292

for population proportion,283285,292293

for population regression slope,420421

for population total,289292for predictions,427428reducing margin of error of,

275276

for regression coefficients,418425, 475

sample size determination, largepopulations,320321

Students tdistribution,277282of two means, dependent samples,

309of two means, unknown

population variances that areassumed to be equal,315317

of two means, unknown popula-tion variances that are notassumed to be equal,317319

for variance of normaldistribution,286289

Confidence level,272Consistent estimators,306Contingency tables,32. See also

cross tableschi-square random variable for,

595explanation of,594595test of association in,595598

Continuous numerical variables,6Continuous random variables,

127128, 177185covariance of,209(See alsoJointly

distributed continuousrandom variables)

expectations for,183185jointly distributed,208216probability density functions and,

179181uniform distribution,181

Control charts,241242Control intervals,241242Correlation

applications of,432coefficient of determination R2

and,416coefficient of multiple correlation,

472hypothesis test for,432433of random variables,162,209zero population,433434

Correlation analysis,432434Correlation coefficient analysis,

6768Correlation coefficients,6468

defined,64example using,6568of random variables,162163,209scatter plots and,65Spearman rank,614615statistical independence and,164

Counterfactual argument,331

Covariance (Cov),64,161162computing using Excel,67continuous random variables,209statistical independence,164

Critical value,333Cross-sectional data,15Cross tables,910Cumulative binomial probabilities,

724728Cumulative distribution function,

178179,182of normal distribution,188

Cumulative line graphs,24Cumulative probability function,

130131Cyclical component, of time series,

665

Datacross-sectional,15interval,6measurement levels,67nominal,6ordinal,6presentation errors,3135qualitative,6quantitative,6ratio,7time-series,1519

Data files descriptions,450451,528530

Davies, O. L.,538Decision making

sampling and,23in uncertain environment,25

Decision rules, guidelines forchoosing, 362363

Degrees of freedom, 253,420Dependent samples,309312,

367370Dependent variables,27

lagged, as regressors,547550Descriptive statistics,5Differences, of random variables,

164,210Discrete numerical variables,56Discrete random variables,127

expected value of,132133expected value of functions,135

jointly distributed,156168probability distributions for,

128131joint probability functions of,158properties of,132137standard deviation of,133135variance of,133135,174
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3/12

Index 765

Distribution shape, 4243. See alsospecific distributions

Diversifiable risk, 436438Dummy variables, 502506, 534545

experimental design models,538543

public sector applications,543545

for regression models,502506,538545

Durbin-Watson test, 564566cut-off points, 757758

Efficient estimators, 268Empirical rule, 5657Equality, 383385

of variances between twonormally distributedpopulations, 383385

Errors, 3134, 475, 557561data presentation, 3135nonsampling, 45reducing margin of, 275276sampling, 4, 273, 329standard error, estimate, 470Type I, 330, 331, 387Type II,330,331,349353,387

Error sum of squares, 407408,412413,469,632

Error variance, estimation of, 470Estimated regression model,404Estimates, 265

confidence interval, 271explanation of, 265point, 266standard error,470

Estimation. See alsoConfidenceintervals

of beta coefficients,436438coefficient, 533534of error variance, 470least squares, 449450, 463of model error variance, 417of multiple regression coefficients,

461467of population proportion,293of regressions with autocorrelated

errors, 566570Estimators, 265biased, 267confidence interval, 271consistent, 306efficient, 268examples of, 268explanation of, 265least squares, 449450

least squares coefficient, 407410,419

least squares derivation of,526527

point, 265269of population mean, 705unbiased, 266267, 269

Events,7681collectively exhaustive, 78complements, 7880independent, 105106intersection of,7680,124125mutually exclusive,7677,97union, 7780,124125

Excel, 67. See alsoMinitabconfidence intervals using,

281282,311312covariance and correlation using,

163jointly distributed discrete

random variables, 160regression analysis using, 409shape of a distribution, 42

Expected valueof continuous random variables,

183185of discrete random variables,

132133of functions of random variables,

135,161,164of sample mean, 230

Experimental design models,538543

Exploratory data analysis (EDA), 26Exponential distribution,

205207Exponential model transformations,

498500Exponential smoothing, 677687Extreme points,439,441,444

Failure to reject, 330,331Fdistribution, 383, 751754Financial investment portfolios,

212216Financial risk, beta measure of,

436438Finite population correction factor,

231,289Finite populations, confidence inter-

val estimation for, 289293First-order autoregressive models,

688689First quartile,4445Fisher, R. A., 538Five-number summary, 45

Forecastingfrom autoregressive models,

689692regression models and,

426430seasonal time series, 684687simple exponential smoothing

and,677687trends and, 666

Fprobability distributionhypothesis test for popula-

tion slope coefficient using,423425

Frequency distributions, 8, 20class width,21construction of, 21cumulative,22inclusive and nonoverlapping

classes,2122interval width, 21number of classes for, 21for numerical data,2023relative, 8,22

Ftestsfor simple regression coefficient,

424425ttests vs., 488489

Functions, of random variables,135137

Geometric mean, 4344Geometric mean rate of return,43Goodness-of-fit tests

explanations of, 583population parameters unknown,

589593specified probabilities, 583588

Gosset, William Sealy, 277, 306Graphical analysis,438444Graphs

for categorical variables, 815data presentation errors, 3134to describe relationships between

variables,2729distribution shape,2426histograms, 24of multiple regression model,

460for numerical variables, 2030ogives, 24scatter plots, 2729stem-and-leaf displays, 2627for time-series data, 1520

Grouped data, measures of,6162

Group means, 651
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4/12

766 Index

Heteroscedasticityexplanation of, 557559graphical techniques for detecting,

558559test for, 559560

Histograms,24misleading, 3133

Holt-Winters exponentialsmoothing forecastingmodel, 680687

example of,681683nonseasonal series,681683seasonal series,684687

Hypergeometric distribution, 153155Hypothesis

alternative, 331, 332, 336339, 356composite, 331, 336339null, 327331, 356one-sided composite alternative,

331simple, 331two-sided composite alternative,

331,340341Hypothesis test decisions, 331Hypothesis tests/testing,326327

assessing power of, 348353comments on, 386388concepts of, 327331confidence intervals,418425control chart, 388for correlation, 432434for difference between two normal

population means, dependentsamples, 367370

for difference between twonormal population means,independent samples,371378

for difference between two popu-lation proportions, 379382

of equality of variances betweentwo normally distributedpopulations, 383385

flow chart for selecting, 393394introduction to, 332333for mean of a normal distribution,

population variance known,332341,349353

for mean of normal distribution,population varianceunknown,342344

for one-way analysis of variance,631633

of population proportion,346347for population slope coefficient

using Fdistribution, 423425power of,331

for regression coefficients,477482,485489

for regression models, 418425for two-way analysis of variance,

646647for variance of a normal distribu-

tion, 355357for zero population correlation,

433434

Income distribution, 43Independent events,97,105106Independent random samples, non-

parametric tests for,608612Independent samples, 313319,

371378Independent variables, 27jointly distributed,158

Indicator variables,502506. See alsoDummy variables

Inferenceabout population regression, 475model interpretation and, 534

Inferential statistics, 5Integral calculus, 222223Interaction, as source of variability,

650Intercept, 399Interquartile range (IQR), 49Intersection of events, 7677, 7980,

131Interval data, 6Intervals

acceptance, 240242control, 241242for frequency distribution, 24

Interval scales,6Investment portfoliosbeta measure of financial risk,

436438portfolio analysis,212216returns on, 214216

Irregular component of time series,665

moving averages to smooth,669671

Jarque-Bera test for normality,591593

Joint cumulative distributionfunction, 208209

Jointly distributed continuousrandom variables, 156168,208216.See alsoContinuousrandom variables; Randomvariables

examples of, 210211financial investment portfolios,

212216linear combinations of,212

Jointly distributed discrete randomvariables,156167. See alsoDiscrete random variables;Random variables

computer applications, 160conditional mean and variance,

160correlation,162163covariance, 162examples of, 156157,159160,163expected value of functions of, 161independence of, 158portfolio analysis,165167

Joint probability, 76,9495,97,103105

Joint probability distribution,157158

Joint probability function, 157properties of, 158

Knowledge, 5Kruskal-Wallis test, 638640Kurtosis, 591, 593

Lagged dependent variable,547550

autocorrelation errors in modelswith, 570571

Law of large numbers,234Least squares algorithm,494495Least squares coefficient estimators,

407410,419Least squares derivation of

estimators,526527Least squares derived coefficient

estimators,408409Least squares estimation, sample

multiple regression and,463Least squares estimators, derivation

of,449450Least squares procedure, 407408,

462467Least squares regression, 399400Least squares regression line, 399,

426Leverage, 439Linear combinations, of random

variables,212Linear functions, of random

variables,160161,185Linear models, 398400Linear regression equation,411417
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5/12

Index 767

analysis of variance and, 413coefficient of determination R2,

413414Linear regression model, 401406

assumptions, 402403examples using,405406outcomes, 404population, 403

Linear regression populationequation model, 403

Linear relationships, 398399Linear sum of random variables, 260Line charts,1519Logarithmic transformations,

497498Lower confidence limit, 273Lower tail test, 600

Mann-Whitney Ustatistic, 608609Mann-Whitney Utest, 608610Marginal distributions, 209Marginal probabilities, 103105,

159160Marginal probability distribution,

157158Margin of error, 273, 279, 284

reducing, 275276Matched pairs, 367368Mathematical derivations, 526528,

662663Matrix plots, 466467Mean, 4044

approximate, 6162of Bernoulli random variable, 140of binomial distribution, 142,

175176conditional, 160of continuous random variables,

184geometric,4344of jointly distributed random

variables,176of linear functions of a random

variable, 135137, 174175measures of variability from,

4858of normal distribution, popula-

tion variance known, 295296,332341,349351

of normal distribution, populationvariance unknown,342344

of Poisson probabilitydistribution, 148

of sampling distribution of samplevariances,263

weighted, 6063

Mean square regression (MSR), 485,486

Mean squaresbetween-groups, 631within-groups, 631

Measurement levels, 67Measures of central tendency, 3948

geometric mean,4344mean, median, mode, 4042shape of a distribution, 4243

Median, 4042,43Minimum variance unbiased

estimator,268Minitab, 67. See alsoExcel

autoregressive models, 689692confidence intervals using,317,

318319,321descriptive measures using, 67Durbin-Watson test, 566exponential model estimation, 499hypothesis testing, 357, 369370,

376lagged dependent variable, 549matrix plots, 466467Monte Carlo sampling

simulations, 260263for probability distributions, 134,

144145regression analysis using, 409410

Missing values, 7, 310311Mode,4042Model error variance, estimation of,

417Model specification, 509511,

532533Monte Carlo sampling simulations,

234240, 260263Minitab,260263

Most efficient estimator,267269Moving averages

explanation of, 669671extraction of seasonal component

through, 672677simple centered (2m+ 1)-point, 671

Multicollinearity, 554557corrections for, 556557indicators for, 555

Multiple comparisons, 634635Multiple regression. See also

Regressionanalysis application procedure

and,509517applications of, 455456confidence intervals and hypoth-

esis tests for individual regres-sion coefficients, 473482

estimation of coefficients and,461467

explanatory power of multipleregression equation and,468472

introduction to, 454least squares procedure, 462467objectives, 456prediction and,491493tests on regression coefficients,

485489Multiple regression equation,

468472Multiple regression model, 454

assumptions, 462development of, 457460, 511512dummy variables for, 502506explanation of, 454460model specification,454456objectives, 456457population,459residuals analysis and, 514517test on all coefficients of,477three-dimensional graphing of,

460transformations for nonlinear,

494500Multiplication rule of probabilities,

9496Mutually exclusive events, 7677,

97

Nominal data, 6Nondiversifiable risk, 436Nonlinear regression models

logarithmic transformations,497498

quadratic transformations,495497

transformations for, 494500Nonparametic tests

for independent random samples,608612

Kruskal-Wallis test, 638640Mann-Whitney Utest, 608610normal approximation to the sign

test, 603604for paired or matched samples,

599606for randomness, 616619sign test, 599601, 606Spearman rank correlation,

614615Wilcoxon rank sum test, 611612Wilcoxon signed rank test,

602606
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6/12


7/12

Index 769

tests of mean of normaldistribution with known,313314,332341,349353

tests of mean of normaldistribution with unknown,315319,342344,376378

tests of normal distribution,355357

Portfolio analysis,166168,212216Portfolio market value,165167Power,330331Power function,350351Prediction

multiple regression and,491493regression models and,426430

Prediction intervals,427428Predictor variables, bias from

excluding significant,551553Price-earnings ratios,269Probability,7374

addition rule of,9293Bayes theorem,112118

bivariate,102112classical,8182complement rule,9192,9899conditional,9394examples,8586

joint,9495,97,103105marginal,103105,159160multiplication rule of,9496for normally distributed random

variables,192overinvolvement ratios and,106109permutations and combinations,

8285random experiments and,7475of range using cumulative

distribution function,179relative frequency,86rules,91102statistical independence and,

9699subjective,8790

Probability density functions,179180,232

areas under,180181for chi-square distribution,252for exponential distribution,206for normal distribution,187properties of,179180for sample means,232for sample proportions,247of standard normal and Students

tdistribution,278Probability distribution function,

129,179

Probability distributionsBernoulli distribution,139141

binomial distribution,139145chi-square distribution,251252for discrete random variables,

128131exponential distribution,205207hypergeometric distribution,

153155Poisson probability distribution,

147152Students tdistribution,306307uniform,181

Probability functionsbinomial distribution table,

720723conditional,158

joint probability function,157,158marginal probability function,157

Probability plots, normal,195197,515

Probability postulatesconsequences of,8889explanation of,8788

Probability value (p-value),340341Problem definition,5Properties

of cumulative probabilitydistributions,131

of joint probability functions,158of probability distribution

functions,130Proportional allocation,703Proportion random variable,

203204Proportions, confidence interval

estimation for,283285Public sector research,543Public sector research and policy

analysis, dummy variableregression in,543545

p-value,334339for chi-square test for variance,

356for sign test,600

Quadratic transformations,495497Qualitative data,6Quantitative data,6Quartiles,4445Queuing problems,149151Quota sampling,714

Random experiments,74outcomes of,7480

Randomized block design,641642

Random samples/sampling,3independent,313319nonparametric tests for

independent, 608612simple,3,225226

Random variables, 127128continuous (SeeContinuous

random variables)correlation of,209differences between,164differences between pairs of,210linear combinations of,212linear functions of,160161,185linear sums and differences of,164mean and variance of linear

functions of,135137proportion,203204statistical independence and,161,

164sums of,209210

Rangeexplanation of,49interquartile,49

Ratio data,7Ratio of mean squares,663Ratios

overinvolvement,106109price-earnings,269

Regression. See alsoLeast squaresregression; Multipleregression; Simple regression

analysis of variance and,412413autocorrelated errors and,562571dummy variables and experimen-

tal design,534545heteroscedasticity,557561lagged valued of dependent

variable,547550least squares regression,399400linear regression model and,

401406mean square,470,486multicollinearity, 554557specification bias,551553

Regression coefficientscomputer computation of,409410confidence intervals for,475477hypothesis tests for,473475subsets of, tests on,486487tests on,485487

Regression models. See alsoMultipleregression model; Nonlinearregression models

coefficient estimation,533534dummy variables,502506,

534538
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8/12

770 Index

Regression models (continued)interpretation and inference, 534linear, 398406, 411417methodology for building, 532534specification, 532533verification, 534

Regression sum of squares, 412,413,470

Reject, 331Relative efficiency, 268Relative frequency distribution,

8,22Relative frequency probability,86Reliability factor, 273Repeated measurements, 309,

311312Residuals, analysis of, 514517Returns, on financial portfolios,

214216Risk, 213

diversifiable,436438nondiversifiable,436

Runs test,616619

Sample covariance,64Sample means

acceptance intervals, 240242central limit theorem, 234240expected value of, 230explanation of, 229sampling distributions of, 229242standard normal distribution for,

231233Sample proportions

examples of, 247248explanation of, 245sampling distributions of, 245248

Sample sizesdetermining,320321determining, for stratified random

sampling, 705706finite populations, 299302large populations, 295299

Sample space,75Samples/sampling. See alsoRandom

samples/samplingcluster,709712defined, 3dependent, 309312, 366370explanation of, 25independent, 313319,366370Monte Carlo sampling

simulations, 260263nonprobabilistic methods,714from population,225229simple random, 3, 225226

stratified, 696706systematic, 3two-phase, 712714

Sample standard deviation, 251Sample variances, 53

chi-square distribution,251252explanation of, 251sampling distributions of,

250255, 263Sampling distributions

explanation of, 226229of least squares coefficient

estimator,419of sample means, 229242of sample proportions,245248of sample variances, 250255, 263

Sampling error,45,273,329Sampling without replacement,

153154Sampling with replacement, 154Sarbanes-Oxley Act (SOX), 597598Scatter plot analysis, 439444Scatter plots, 2729

for residuals analysis, 515517Seasonal component

extraction of, through movingaverages,672677

of time series, 666667Seasonal index method, 684687Seasonal time series, forecasting,

684687Second-order autoregressive

models, 688Second quartile,44Side-by-side bar chart,10Significance level, 329, 331Sign test

explanation of, 599normal approximation to, 603606for paired or matched samples,

599603p-value for, 600for single population median,606

Simple exponential smoothingexplanation of, 678forecasting through,678680Holt-Winters model and, 680683

Simple hypothesis, 327, 331Simple random samples/sampling,

3,225beta measure of financial risk,

436438correlation analysis and, 432434explanatory power of lin-

ear regression equation and,411417

graphical analysis and, 438444least squares coefficient estimators

and,407410prediction and,426430sample sizes, 300302statistical inference and,418425

Simple regression. SeeRegressionSimple regression coefficient, Ftest

for, 424425Skewed distribution, 2526Skewness, 25,7172,591,593Slope, 399

differences in, 505Spearman rank correlation,614615

cutoff points,756Specification bias, 551553SSE, 407408,412413SSR,413415SST, 413415Stacked bar charts, 10Standard deviation, 5253, 54

of continuous random variables, 184of discrete random variable, 133sample,251

Standard error of the estimate,470Standardized normal random

variable, 231Standardized residual, 441444Standard normal distribution, 189

cumulative distribution functiontable, 718

for sample means, 231233Statistical independence, 9699,

161,164covariance, 164

Statistical inference, 418425Statistical significance,387Statistical thinking, 2Statistics,2,40. See also

Nonparametic testsdefined, 4descriptive, 5inferential, 5

Stem-and-leaf displays, 2627Stock market crash of 2008, 74beta coefficients limitations, 437cautions concerning financial

models, 216Stratified random sampling

allocation of sample effort amongstrata and, 703705

analysis of results from, 698700determining sample sizes for,

705706estimation of population mean,

698699
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9/12

Index 771

Time-series dataexplanation of,664665graphs to describe,1520

Time-series plots,1519misleading,3334

Time-series regression model,567570

Total explained variability,527528

Total sum of squares,413,469, 662

Treatment variables,539540Tree diagrams,103104Trend component, of time series,

665666ttests vs. Ftests,488489Two-phase sampling,712714Two-sided composite alternative

hypothesis,327, 331, 340341Two-tail test,600Two-way analysis of variance

examples of,655656hypothesis tests for,646more than one observation per

cell,650656one observation per cell,

641647several observations per cell,

650656sum of squares decomposition

for,645table format,646647tables,646647

Two-way analysis of variancetables,646647

Type I errors,329331, 333, 387Type II errors,329331,

349350, 387determining probability of,

349353

Unbiased estimator,266267Uncertainty, decision making

under,25Uniform distribution,181,184Uniform probability distribution,

178Unions,7780, 131Upper confidence limit,273

Variabilitybetween-groups,629interaction as source of,650total explained,527528within-groups,629

Variability, measures of,4859

Variables. See alsoContinuousrandom variables

bias from excluding significantpredictor,551553

blocking,539, 641categorical,5,814classification of,58correlation analysis and,

432434defined,5dependent,27dummy,502506, 534545effect of dropping statistically

significant,512514independent,27indicator,502506lagged dependent,547550of linear functions of a random

variable,168measures of relationships

between, 6469numerical,56,2029relationships between,398399tables and graphs to describe

relationships between,2729

treatment,539Variance,5154.See alsoAnalysis of

variance (ANOVA)of Bernoulli random variable,

140of binomial distribution,142,

175176conditional,160of continuous random variables,

184of discrete random variables,

133135, 164, 174for grouped data,6162of jointly distributed random

variables,176of linear functions of a random

variable,135137,174175nonuniform,557558of normal distribution, confi-

dence interval estimation for,286289

of normal distribution, tests for,355357

of Poisson probabilitydistribution,148

sampling distributions of sample,250255

between two normally distributedpopulations, tests of equality,383385

estimation of populationproportion,701703

estimation of population total,700701

examples of,699700explanation of,696697

Students tdistribution,306307confidence intervals and,

277282hypothesis tests,342344for two means with unknown

population variances notassumed to be equal,324

upper critical values table,750Subgroup means, multiple

comparisons between,634Subjective probability,8790Sum of squares,413,469,629Sum of squares decomposition

coefficient of determination and,469470

one-way analysis of variance,630631

two-way analysis of variance,645

Sums, of random variables,164,209210

Survey responsesmissing values in,310311sampling errors,4

Symmetric distributions,25Systematic sampling,3

Tablesfor categorical variables,89cross tables,910to describe relationships between

variables,2729frequency distribution,89

Test of association,595598Tests. SeeHypothesis tests/testingThird quartile,4445Time plots, autocorrelation and,

562563Time series

autoregressive integrated movingaverage models,693694

autoregressive models,688692components of,665669explanation of,664665exponential smoothing and,

677687forecasting seasonal,684687moving averages,669677

Time-series component analysis,668
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10/12

772 Index

Waiting line problems, 149151Weighted mean, 6063Width, 273Wilcoxon rank sum statistic T,611Wilcoxon rank sum test, 611612

cutoff points for statistic, 755Wilcoxon signed rank test,

602603normal approximation to, 604606

Within-groups mean square (MSW),662

Within-groups variability, 629

y-intercept, 399

Zero population correlation,433434

z-score, 5758

Variation, coefficient of, 55Venn diagrams

for addition rule,92for complement of event, 78for intersection of events, 77,80,

124125for union of events, 7678,

124125Verifications,174176
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Standard Normal

Distribution Table

z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359

0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753

0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141

0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517

0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879

0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.22240.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549

0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852

0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133

0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389

1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621

1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830

1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015

1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177

1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319

1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441

1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545

1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633

1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706

1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767

2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817

2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857

2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890

2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916

2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936

2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952

2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964

2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974

2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981

2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986

3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990

3.1 0.4990 0.4991 0.4991 0.4991 0.4992 0.4992 0.4992 0.4992 0.4993 0.4993

3.2 0.4993 0.4993 0.4994 0.4994 0.4994 0.4994 0.4994 0.4995 0.4995 0.4995

3.3 0.4995 0.4995 0.4995 0.4996 0.4996 0.4996 0.4996 0.4996 0.4996 0.4997

Dr. William L. Carlson, prepared using Minitab 16.

0 z= 1.25

z

0.3944


12/12

Upper critical values of StudentstDistribution withNDegrees of Freedom

t0 ,

For selected probabilities, a, the table shows the values tn,asuch that P1tn 7 tn,a2 = a,where tnis a Students trandom variable with ndegrees

of freedom. For example, the probability is .10 that a Students trandom variable with 10 degrees of freedom exceeds 1.372.

P E C V

n 0.10 0.05 0.025 0.01 0.005 0.001

1 3.078 6.314 12.706 31.821 63.657 318.313

2 1.886 2.920 4.303 6.965 9.925 22.327

3 1.638 2.353 3.182 4.541 5.841 10.215

4 1.533 2.132 2.776 3.747 4.604 7.173 5 1.476 2.015 2.571 3.365 4.032 5.893

6 1.440 1.943 2.447 3.143 3.707 5.208

7 1.415 1.895 2.365 2.998 3.499 4.782

8 1.397 1.860 2.306 2.896 3.355 4.499

9 1.383 1.833 2.262 2.821 3.250 4.296

10 1.372 1.812 2.228 2.764 3.169 4.143

11 1.363 1.796 2.201 2.718 3.106 4.024

12 1.356 1.782 2.179 2.681 3.055 3.929

13 1.350 1.771 2.160 2.650 3.012 3.852

14 1.345 1.761 2.145 2.624 2.977 3.787

15 1.341 1.753 2.131 2.602 2.947 3.733

16 1.337 1.746 2.120 2.583 2.921 3.686

17 1.333 1.740 2.110 2.567 2.898 3.646

18 1.330 1.734 2.101 2.552 2.878 3.610

19 1.328 1.729 2.093 2.539 2.861 3.579

20 1.325 1.725 2.086 2.528 2.845 3.552

21 1.323 1.721 2.080 2.518 2.831 3.527

22 1.321 1.717 2.074 2.508 2.819 3.505

23 1.319 1.714 2.069 2.500 2.807 3.485

24 1.318 1.711 2.064 2.492 2.797 3.467

25 1.316 1.708 2.060 2.485 2.787 3.450

26 1.315 1.706 2.056 2.479 2.779 3.435

27 1.314 1.703 2.052 2.473 2.771 3.421

28 1.313 1.701 2.048 2.467 2.763 3.408

29 1.311 1.699 2.045 2.462 2.756 3.396

30 1.310 1.697 2.042 2.457 2.750 3.385

40 1.303 1.684 2.021 2.423 2.704 3.307

60 1.296 1.671 2.000 2.390 2.660 3.232

100 1.290 1.660 1.984 2.364 2.626 3.174

` 1.282 1.645 1.960 2.326 2.576 3.090

NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/, September 2011.

index to newton and tomahawk

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