Statistical Inference: A Short Course

391 Pages·2012·3.465 MB·English

by  Michael J. Panik

STATISTICAL
INFERENCE
STATISTICAL
INFERENCE
A Short Course
MICHAEL J. PANIK
Universityof Hartford
West Hartford,CT
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LibraryofCongressCataloging-in-PublicationData:
Panik,MichaelJ.
Statisticalinference:ashortcourse/MichaelJ.Panik.
p.cm.
Includesindex.
ISBN978-1-118-22940-8(cloth)
1. Mathematicalstatistics–Testbooks. I.Title.
QA276.12.P362011
519.5–dc23
2011047632
PrintedintheUnitedStatesofAmerica
ISBN:9781118229408
10 9 8 7 6 5 4 3 2 1
To thememory of
Richard S.Martin
CONTENTS
Preface xv
1 The NatureofStatistics 1
1.1 Statistics Defined 1
1.2 ThePopulation andthe Sample 2
1.3 Selecting aSample from aPopulation 3
1.4 Measurement Scales 4
1.5 Let usAdd 6
Exercises 7
2 Analyzing QuantitativeData 9
2.1 Imposing Order 9
2.2 TabularandGraphical Techniques: Ungrouped Data 9
2.3 TabularandGraphical Techniques: Grouped Data 11
Exercises 16
Appendix 2.A Histograms with Classes ofDifferent Lengths 18
3 Descriptive Characteristics ofQuantitative Data 22
3.1 TheSearchfor Summary Characteristics 22
3.2 TheArithmetic Mean 23
3.3 TheMedian 26
vii
viii CONTENTS
3.4 The Mode 27
3.5 The Range 27
3.6 The Standard Deviation 28
3.7 RelativeVariation 33
3.8 Skewness 34
3.9 Quantiles 36
3.10 Kurtosis 38
3.11 Detection of Outliers 39
3.12 So What Do We Dowith All This Stuff? 41
Exercises 47
Appendix 3.A DescriptiveCharacteristics ofGrouped Data 51
3.A.1 The Arithmetic Mean 52
3.A.2 The Median 53
3.A.3 The Mode 55
3.A.4 The Standard Deviation 57
3.A.5 Quantiles(Quartiles, Deciles, and Percentiles) 58
4 Essentials ofProbability 61
4.1 Set Notation 61
4.2 Eventswithin the Sample Space 63
4.3 Basic Probability Calculations 64
4.4 Joint,Marginal, andConditional Probability 68
4.5 Sources ofProbabilities 73
Exercises 75
5 Discrete Probability Distributions and Their Properties 81
5.1 TheDiscrete Probability Distribution 81
5.2 TheMean, Variance, andStandard Deviation of aDiscrete
Random Variable 85
5.3 TheBinomial ProbabilityDistribution 89
5.3.1 CountingIssues 89
5.3.2 The BernoulliProbability Distribution 91
5.3.3 The BinomialProbability Distribution 91
Exercises 96
6 The Normal Distribution 101
6.1 TheContinuous ProbabilityDistribution 101
6.2 TheNormalDistribution 102
6.3 Probabilityas anAreaUnderthe NormalCurve 104
CONTENTS ix
6.4 Percentiles ofthe Standard NormalDistributionand Percentiles
of the Random Variable X 114
Exercises 116
Appendix 6.A The NormalApproximation toBinomialProbabilities 120
7 Simple Random Samplingand the SamplingDistribution
ofthe Mean 122
7.1 Simple Random Sampling 122
7.2 TheSampling Distributionof the Mean 123
7.3 Comments on the SamplingDistributionofthe Mean 127
7.4 ACentral Limit Theorem 130
Exercises 132
Appendix 7.A Using a Table ofRandom Numbers 133
Appendix 7.B AssessingNormality via the NormalProbability Plot 136
Appendix 7.C Randomness, Risk, and Uncertainty 139
7.C.1 Introduction toRandomness 139
7.C.2 Types of Randomness 142
7.C.2.1 Type IRandomness 142
7.C.2.2 Type II Randomness 143
7.C.2.3 Type III Randomness 143
7.C.3 Pseudo-Random Numbers 144
7.C.4 ChaoticBehavior 145
7.C.5 Risk andUncertainty 146
8 Confidence Interval Estimation ofm 152
8.1 TheError BoundonX(cid:2) as anEstimator ofm 152
8.2 AConfidence Interval for the PopulationMean m (s Known) 154
8.3 A Sample Size Requirements Formula 159
8.4 AConfidence Interval for the PopulationMean m (s Unknown) 160
Exercises 165
Appendix 8.A AConfidence Interval for the PopulationMedianMED 167
9 The SamplingDistribution ofa Proportionand itsConfidence
IntervalEstimation 170
9.1 TheSampling Distributionof aProportion 170
9.2 TheError Boundon^p asan Estimator for p 173
9.3 AConfidence Interval for the PopulationProportion
(of Successes)p 174
9.4 A Sample Size Requirements Formula 176
x CONTENTS
Exercises 177
Appendix 9.A Ratio Estimation 179
10 Testing Statistical Hypotheses 184
10.1 What is a Statistical Hypothesis? 184
10.2 Errors inTesting 185
10.3 The Contextual Framework ofHypothesis Testing 186
10.3.1 Types ofErrorsin aLegalContext 188
10.3.2 Types ofErrorsin aMedical Context 188
10.3.3 Types ofErrorsin aProcessing or
ControlContext 189
10.3.4 Types ofErrorsin aSportsContext 189
10.4 Selecting a Test Statistic 190
10.5 The Classical Approach toHypothesis Testing 190
10.6 Types ofHypothesis Tests 191
10.7 Hypothesis Tests for m (s Known) 194
10.8 Hypothesis Tests for m (s Unknown and nSmall) 195
10.9 Reportingthe Results ofStatistical Hypothesis Tests 198
10.10 Hypothesis Tests for the Population Proportion
(of Successes) p 201
Exercises 204
Appendix 10.A Assessingthe Randomness of aSample 208
Appendix 10.B Wilcoxon Signed Rank Test (of a Median) 210
Appendix 10.C Lilliefors Goodness-of-Fit Test for Normality 213
11 Comparing Two Population Means and Two Population
Proportions 217
11.1 ConfidenceIntervals for the Difference of Means when
Samplingfrom Two Independent Normal Populations 217
11.1.1 Sampling fromTwo Independent Normal Populations
with Equal andKnown Variances 217
11.1.2 Sampling fromTwo Independent Normal Populations
with Unequal butKnown Variances 218
11.1.3 Sampling fromTwo Independent Normal Populations
with Equal butUnknown Variances 218
11.1.4 Sampling fromTwo Independent Normal Populations
with Unequal and Unknown Variances 219
11.2 ConfidenceIntervals for the Difference of Means
when Sampling fromTwo Dependent Populations:
Paired Comparisons 224
CONTENTS xi
11.3 ConfidenceIntervals for the Difference ofProportions when
Sampling from Two IndependentBinomialPopulations 227
11.4 Statistical Hypothesis Testsfor the Difference of Means when
Sampling from Two IndependentNormalPopulations 228
11.4.1 PopulationVariances Equal and Known 229
11.4.2 PopulationVariances Unequal butKnown 229
11.4.3 PopulationVariances Equal and Unknown 229
11.4.4 PopulationVariances Unequal and Unknown
(an Approximate Test) 230
11.5 Hypothesis Tests for the Difference ofMeanswhen Sampling
fromTwo Dependent Populations: Paired Comparisons 234
11.6 Hypothesis Tests for the Difference ofProportions when
Sampling from Two IndependentBinomialPopulations 236
Exercises 239
Appendix 11.A RunsTest for Two Independent Samples 243
Appendix 11.B Mann–Whitney (Rank Sum) Test for Two
Independent Populations 245
Appendix 11.C Wilcoxon Signed Rank Test when Sampling from
Two DependentPopulations: PairedComparisons 249
12 BivariateRegression and Correlation 253
12.1 Introducingan AdditionalDimension toourStatistical Analysis 253
12.2 Linear Relationships 254
12.2.1 Exact Linear Relationships 254
12.3 Estimating the Slope andIntercept ofthe Population
Regression Line 257
12.4 Decomposition ofthe Sample VariationinY 262
12.5 Mean, Variance, andSampling Distribution ofthe Least
Squares Estimators b^ and b^ 264
0 1
12.6 ConfidenceIntervals for b and b 266
0 1
12.7 Testing Hypotheses about b and b 267
0 1
12.8 Predicting the Average Value ofY givenX 269
12.9 The Prediction ofa ParticularValue of Y givenX 270
12.10 Correlation Analysis 272
12.10.1 Case A:X and Y Random Variables 272
12.10.1.1 Estimating the Population Correlation
Coefficientr 274
12.10.1.2 Inferences about the Population
Correlation Coefficientr 275
12.10.2 Case B:X ValuesFixed,Y aRandomVariable 277
xii CONTENTS
Exercises 278
Appendix 12.A AssessingNormality (Appendix7.B Continued) 280
Appendix 12.B On Making Causal Inferences 281
12.B.1 Introduction 281
12.B.2 Rudiments ofExperimental Design 282
12.B.3 Truth Sets, Propositions,and Logical
Implications 283
12.B.4 Necessary and SufficientConditions 285
12.B.5 Causality Proper 286
12.B.6 Logical Implications andCausality 287
12.B.7 Correlation and Causality 288
12.B.8 Causality fromCounterfactuals 289
12.B.9 Testing Causality 292
12.B.10 Suggestions for Further Reading 294
13 AnAssortment of AdditionalStatisticalTests 295
13.1 DistributionalHypotheses 295
13.2 The Multinomial Chi-Square Statistic 295
13.3 The Chi-Square Distribution 298
13.4 Testing Goodness ofFit 299
13.5 Testing Independence 304
13.6 Testing kProportions 309
13.7 AMeasure ofStrength of Associationina
Contingency Table 311
13.8 AConfidenceInterval for s2under Random Sampling from
a Normal Population 312
13.9 The F Distribution 314
13.10 Applications of the F Statistic toRegression Analysis 316
13.10.1 Testingthe Significance ofthe Regression
Relationship Between X andY 316
13.10.2 AJointTest of the Regression Intercept andSlope 317
Exercises 318
AppendixA 323
Table A.1 Standard NormalAreas [Z is N(0,1)] 323
Table A.2 Quantiles ofthe t Distribution(T is tv) 325
Table A.3 Quantiles ofthe Chi-Square Distribution(X is w2) 327
v
Table A.4 Quantiles ofthe F Distribution(F is Fv ;v ) 329
1 2
Table A.5 BinomialProbabilitiesP(X;n,p) 334