1. .
Introductory Statistics Explained
Edition 1.10CC
©2015 Jeremy Balka
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2. Balka ISE 1.10 2

3. Contents
1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Inferential Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Gathering Data 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Populations and Samples, Parameters and Statistics . . . . . . . 9
2.3 Types of Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Simple Random Sampling . . . . . . . . . . . . . . . . . . 11
2.3.2 Other Types of Random Sampling . . . . . . . . . . . . . 13
2.4 Experiments and Observational Studies . . . . . . . . . . . . . . 14
2.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Descriptive Statistics 21
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Plots for Categorical and Quantitative Variables . . . . . . . . . 23
3.2.1 Plots for Categorical Variables . . . . . . . . . . . . . . . 23
3.2.2 Graphs for Quantitative Variables . . . . . . . . . . . . . 25
3.3 Numerical Measures . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.1 Summation Notation . . . . . . . . . . . . . . . . . . . . . 33
3.3.2 Measures of Central Tendency . . . . . . . . . . . . . . . . 34
3.3.3 Measures of Variability . . . . . . . . . . . . . . . . . . . . 37
3.3.4 Measures of Relative Standing . . . . . . . . . . . . . . . 45
3.4 Boxplots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 Probability 59
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Basics of Probability . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2.1 Interpreting Probability . . . . . . . . . . . . . . . . . . . 62
i

4. Balka ISE 1.10 CONTENTS ii
4.2.2 Sample Spaces and Sample Points . . . . . . . . . . . . . 63
4.2.3 Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3 Rules of Probability . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.1 The Intersection of Events . . . . . . . . . . . . . . . . . . 65
4.3.2 Mutually Exclusive Events . . . . . . . . . . . . . . . . . . 65
4.3.3 The Union of Events and the Addition Rule . . . . . . . . 66
4.3.4 Complementary Events . . . . . . . . . . . . . . . . . . . 67
4.3.5 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3.6 Conditional Probability . . . . . . . . . . . . . . . . . . . 69
4.3.7 Independent Events . . . . . . . . . . . . . . . . . . . . . 71
4.3.8 The Multiplication Rule . . . . . . . . . . . . . . . . . . . 73
4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5 Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.5.2 The Law of Total Probability and Bayes’ Theorem . . . . 85
4.6 Counting rules: Permutations and Combinations . . . . . . . . . 88
4.6.1 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.6.2 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.7 Probability and the Long Run . . . . . . . . . . . . . . . . . . . . 91
4.8 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5 Discrete Random Variables and Discrete Probability Distribu-
tions 97
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 Discrete and Continuous Random Variables . . . . . . . . . . . . 99
5.3 Discrete Probability Distributions . . . . . . . . . . . . . . . . . . 101
5.3.1 The Expectation and Variance of Discrete Random Variables102
5.4 The Bernoulli Distribution . . . . . . . . . . . . . . . . . . . . . . 110
5.5 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . 111
5.5.1 Binomial or Not? . . . . . . . . . . . . . . . . . . . . . . . 114
5.5.2 A Binomial Example with Probability Calculations . . . . 115
5.6 The Hypergeometric Distribution . . . . . . . . . . . . . . . . . . 117
5.7 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . 121
5.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.7.2 The Relationship Between the Poisson and Binomial Dis-
tributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.7.3 Poisson or Not? More Discussion on When a Random Vari-
able has a Poisson distribution . . . . . . . . . . . . . . . 125
5.8 The Geometric Distribution . . . . . . . . . . . . . . . . . . . . . 127
5.9 The Negative Binomial Distribution . . . . . . . . . . . . . . . . 131
5.10 The Multinomial Distribution . . . . . . . . . . . . . . . . . . . . 133
5.11 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5. Balka ISE 1.10 CONTENTS iii
6 Continuous Random Variables and Continuous Probability Dis-
tributions 139
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.2 Properties of Continuous Probability Distributions . . . . . . . . 143
6.2.1 An Example Using Integration . . . . . . . . . . . . . . . 144
6.3 The Continuous Uniform Distribution . . . . . . . . . . . . . . . 146
6.4 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . 150
6.4.1 Finding Areas Under the Standard Normal Curve . . . . . 153
6.4.2 Standardizing Normally Distributed Random Variables . . 158
6.5 Normal Quantile-Quantile Plots . . . . . . . . . . . . . . . . . . . 162
6.5.1 Examples of Normal QQ Plots . . . . . . . . . . . . . . . 163
6.6 Other Important Continuous Probability Distributions . . . . . . 166
6.6.1 The χ2 Distribution . . . . . . . . . . . . . . . . . . . . . 166
6.6.2 The t Distribution . . . . . . . . . . . . . . . . . . . . . . 168
6.6.3 The F Distribution . . . . . . . . . . . . . . . . . . . . . . 169
6.7 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 172
7 Sampling Distributions 175
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
7.2 The Sampling Distribution of the Sample Mean . . . . . . . . . . 180
7.3 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . 183
7.3.1 Illustration of the Central Limit Theorem . . . . . . . . . 185
7.4 Some Terminology Regarding Sampling Distributions . . . . . . . 187
7.4.1 Standard Errors . . . . . . . . . . . . . . . . . . . . . . . 187
7.4.2 Unbiased Estimators . . . . . . . . . . . . . . . . . . . . . 187
7.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 188
8 Confidence Intervals 191
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.2 Interval Estimation of µ when σ is Known . . . . . . . . . . . . . 194
8.2.1 Interpretation of the Interval . . . . . . . . . . . . . . . . 198
8.2.2 What Factors Affect the Margin of Error? . . . . . . . . . 200
8.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
8.3 Confidence Intervals for µ When σ is Unknown . . . . . . . . . . 205
8.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 205
8.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
8.3.3 Assumptions of the One-Sample t Procedures . . . . . . . 212
8.4 Determining the Minimum Sample Size n . . . . . . . . . . . . . 218
8.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 220
9 Hypothesis Tests (Tests of Significance) 223
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
9.2 The Logic of Hypothesis Testing . . . . . . . . . . . . . . . . . . 227

6. Balka ISE 1.10 CONTENTS iv
9.3 Hypothesis Tests for µ When σ is Known . . . . . . . . . . . . . 229
9.3.1 Constructing Appropriate Hypotheses . . . . . . . . . . . 229
9.3.2 The Test Statistic . . . . . . . . . . . . . . . . . . . . . . 230
9.3.3 The Rejection Region Approach to Hypothesis Testing . . 233
9.3.4 P-values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
9.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
9.5 Interpreting the p-value . . . . . . . . . . . . . . . . . . . . . . . 243
9.5.1 The Distribution of the p-value When H0 is True . . . . . 243
9.5.2 The Distribution of the p-value When H0 is False . . . . . 245
9.6 Type I Errors, Type II Errors, and the Power of a Test . . . . . . 245
9.6.1 Calculating Power and the Probability of a Type II Error 248
9.6.2 What Factors Affect the Power of the Test? . . . . . . . . 253
9.7 One-sided Test or Two-sided Test? . . . . . . . . . . . . . . . . . 254
9.7.1 Choosing Between a One-sided Alternative and a Two-
sided Alternative . . . . . . . . . . . . . . . . . . . . . . . 254
9.7.2 Reaching a Directional Conclusion from a Two-sided Al-
ternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
9.8 Statistical Significance and Practical Significance . . . . . . . . . 257
9.9 The Relationship Between Hypothesis Tests and
Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . 258
9.10 Hypothesis Tests for µ When σ is Unknown . . . . . . . . . . . . 260
9.10.1 Examples of Hypothesis Tests Using the t Statistic . . . . 261
9.11 More on Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 266
9.12 Criticisms of Hypothesis Testing . . . . . . . . . . . . . . . . . . 268
9.13 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 271
10 Inference for Two Means 273
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
10.2 The Sampling Distribution of the Difference in Sample Means . . 276
10.3 Inference for µ1 − µ2 When σ1 and σ2 are Known . . . . . . . . . 277
10.4 Inference for µ1 − µ2 when σ1 and σ2 are unknown . . . . . . . . 279
10.4.1 Pooled Variance Two-Sample t Procedures . . . . . . . . . 280
10.4.2 The Welch Approximate t Procedure . . . . . . . . . . . . 286
10.4.3 Guidelines for Choosing the Appropriate Two-Sample
t Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 290
10.4.4 More Examples of Inferences for the Difference in Means . 292
10.5 Paired-Difference Procedures . . . . . . . . . . . . . . . . . . . . 297
10.5.1 The Paired-Difference t Procedure . . . . . . . . . . . . . 300
10.6 Pooled-Variance t Procedures: Investigating the Normality As-
sumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
10.7 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 306
11 Inference for Proportions 309

7. Balka ISE 1.10 CONTENTS v
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
11.2 The Sampling Distribution of the Sample Proportion . . . . . . . 312
11.2.1 The Mean and Variance of the Sampling Distribution of p̂ 312
11.2.2 The Normal Approximation . . . . . . . . . . . . . . . . . 313
11.3 Confidence Intervals and Hypothesis Tests for the Population Pro-
portion p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
11.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
11.4 Determining the Minimum Sample Size n . . . . . . . . . . . . . 320
11.5 Inference Procedures for Two Population Proportions . . . . . . . 321
11.5.1 The Sampling Distribution of p̂1 − p̂2 . . . . . . . . . . . . 322
11.5.2 Confidence Intervals and Hypothesis Tests for p1 − p2 . . 323
11.6 More on Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 327
11.7 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 329
12 Inference for Variances 331
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
12.2 The Sampling Distribution of the Sample Variance . . . . . . . . 334
12.2.1 The Sampling Distribution of the Sample Variance When
Sampling from a Normal Population . . . . . . . . . . . . 334
12.2.2 The Sampling Distribution of the Sample Variance When
Sampling from Non-Normal Populations . . . . . . . . . . 336
12.3 Inference Procedures for a Single Variance . . . . . . . . . . . . . 337
12.4 Comparing Two Variances . . . . . . . . . . . . . . . . . . . . . . 343
12.4.1 The Sampling Distribution of the Ratio of Sample Variances343
12.4.2 Inference Procedures for the Ratio of Population Variances 345
12.5 Investigating the Effect of Violations of the Normality Assumption 352
12.5.1 Inference Procedures for One Variance: How Robust are
these Procedures? . . . . . . . . . . . . . . . . . . . . . . 352
12.5.2 Inference Procedures for the Ratio of Variances: How Ro-
bust are these Procedures? . . . . . . . . . . . . . . . . . 355
12.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 357
13 χ2 Tests for Count Data 359
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
13.2 χ2 Tests for One-Way Tables . . . . . . . . . . . . . . . . . . . . 361
13.2.1 The χ2 Test Statistic . . . . . . . . . . . . . . . . . . . . . 362
13.2.2 Testing Goodness-of-Fit for Specific Parametric Distributions366
13.3 χ2 Tests for Two-Way Tables . . . . . . . . . . . . . . . . . . . . 368
13.3.1 The χ2 Test Statistic for Two-Way Tables . . . . . . . . . 370
13.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
13.4 A Few More Points . . . . . . . . . . . . . . . . . . . . . . . . . . 375
13.4.1 Relationship Between the Z Test and χ2 Test for 2×2 Tables375
13.4.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 376

8. Balka ISE 1.10 CONTENTS vi
13.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 379
14 One-Way Analysis of Variance (ANOVA) 381
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
14.2 One-Way ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . 384
14.3 Carrying Out the One-Way Analysis of Variance . . . . . . . . . 386
14.3.1 The Formulas . . . . . . . . . . . . . . . . . . . . . . . . . 386
14.3.2 An Example with Full Calculations . . . . . . . . . . . . . 388
14.4 What Should be Done After One-Way ANOVA? . . . . . . . . . 391
14.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 391
14.4.2 Fisher’s LSD Method . . . . . . . . . . . . . . . . . . . . . 393
14.4.3 The Bonferroni Correction . . . . . . . . . . . . . . . . . . 395
14.4.4 Tukey’s Honest Significant Difference Method . . . . . . . 398
14.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
14.6 A Few More Points . . . . . . . . . . . . . . . . . . . . . . . . . . 406
14.6.1 Different Types of Experimental Design . . . . . . . . . . 406
14.6.2 One-Way ANOVA and the Pooled-Variance t Test . . . . 407
14.6.3 ANOVA Assumptions . . . . . . . . . . . . . . . . . . . . 407
14.7 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 408
15 Introduction to Simple Linear Regression 411
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
15.2 The Linear Regression Model . . . . . . . . . . . . . . . . . . . . 413
15.3 The Least Squares Regression Line . . . . . . . . . . . . . . . . . 417
15.4 Statistical Inference in Simple Linear Regression . . . . . . . . . 420
15.4.1 Model Assumptions . . . . . . . . . . . . . . . . . . . . . 421
15.4.2 Statistical Inference for the Parameter β1 . . . . . . . . . 422
15.5 Checking Model Assumptions with Residual Plots . . . . . . . . . 424
15.6 Measures of the Strength of the Linear Relationship . . . . . . . 426
15.6.1 The Pearson Correlation Coefficient . . . . . . . . . . . . 426
15.6.2 The Coefficient of Determination . . . . . . . . . . . . . . 429
15.7 Estimation and Prediction Using the Fitted Line . . . . . . . . . 431
15.8 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
15.9 A Complete Example . . . . . . . . . . . . . . . . . . . . . . . . . 435
15.10Outliers, Leverage, and Influential Points . . . . . . . . . . . . . . 437
15.11Some Cautions about Regression and Correlation . . . . . . . . . 439
15.11.1Always Plot Your Data . . . . . . . . . . . . . . . . . . . 439
15.11.2Avoid Extrapolating . . . . . . . . . . . . . . . . . . . . . 440
15.11.3Correlation Does Not Imply Causation . . . . . . . . . . . 441
15.12A Brief Multiple Regression Example . . . . . . . . . . . . . . . . 442
15.13Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 446

9. Chapter 1
Introduction
“I have been impressed with the urgency of doing. Knowing is not enough; we
must apply. Being willing is not enough; we must do.”
-Leonardo da Vinci
1

10. Balka ISE 1.10 1.1. INTRODUCTION 2
1.1 Introduction
When first encountering the study of statistics, students often have a preconceived—
and incorrect—notion of what the field of statistics is all about. Some people
think that statisticians are able to quote all sorts of unusual statistics, such as
32% of undergraduate students report patterns of harmful drinking behaviour,
or that 55% of undergraduates do not understand what the field of statistics is
all about. But the field of statistics has little to do with quoting obscure percent-
ages or other numerical summaries. In statistics, we often use data to answer
questions like:
• Is a newly developed drug more effective than one currently in use?
• Is there a still a sex effect in salaries? After accounting for other relevant
variables, is there a difference in salaries between men and women? Can
we estimate the size of this effect for different occupations?
• Can post-menopausal women lower their heart attack risk by undergoing
hormone replacement therapy?
To answer these types of questions, we will first need to find or collect appropriate
data. We must be careful in the planning and data collection process, as unfortu-
nately sometimes the data a researcher collects is not appropriate for answering
the questions of interest. Once appropriate data has been collected, we summa-
rize and illustrate it with plots and numerical summaries. Then—ideally—we use
the data in the most effective way possible to answer our question or questions
of interest.
Before we move on to answering these types of questions using statistical in-
ference techniques, we will first explore the basics of descriptive statistics.
1.2 Descriptive Statistics
In descriptive statistics, plots and numerical summaries are used to describe a
data set.
Example 1.1 Consider a data set representing final grades in a large introduc-
tory statistics course. We may wish to illustrate the grades using a histogram or
a boxplot,1 as in Figure 1.1.
1
You may not have encountered boxplots before. Boxplots will be discussed in detail in Sec-
tion 3.4. In their simplest form, they are plots of the five-number summary: minimum, 25th
percentile, median, 75th percentile, and the maximum. Extreme values are plotted in indi-
vidually. They are most useful for comparing two or more distributions.

11. Balka ISE 1.10 1.3. INFERENTIAL STATISTICS 3
20 40 60 80 100
0
40
80
120
Final Grade
Frequency
(a) Histogram of final grades.
25
50
75
100
Final
Grade
(b) Boxplot of final grades.
Figure 1.1: Boxplot and histogram of final grades in an introductory statistics
course.
Plots like these can give an effective visual summary of the data. But we are
also interested in numerical summary statistics, such as the mean, median, and
variance. We will investigate descriptive statistics in greater detail in Chapter 3.
But the main purpose of this text is to introduce statistical inference concepts
and techniques.
1.3 Inferential Statistics
The most interesting statistical techniques involve investigating the relationship
between variables. Let’s look at a few examples of the types of problems we will
be investigating.
Example 1.2 Do traffic police officers in Cairo have higher levels of lead in their
blood than that of other police officers? A study2 investigated this question by
drawing random samples of 126 Cairo traffic officers and 50 officers from the
suburbs. Lead levels in the blood (µg/dL) were measured.
The boxplots in Figure 1.2 illustrate the data. Boxplots are very useful for
comparing the distributions of two or more groups.
The boxplots show a difference in the distributions—it appears as though the
distribution of lead in the blood of Cairo traffic officers is shifted higher than that
of officers in the suburbs. In other words, it appears as though the traffic officers
have a higher mean blood lead level. The summary statistics are illustrated in
Table 1.1.3
2
Kamal, A., Eldamaty, S., and Faris, R. (1991). Blood level of Cairo traffic policemen. Science
of the Total Environment, 105:165–170. The data used in this text is simulated data based on
the summary statistics from that study.
3
The standard deviation is a measure of the variability of the data. We will discuss it in detail
in Section 3.3.3.

12. Balka ISE 1.10 1.3. INFERENTIAL STATISTICS 4
10
20
30
40
Lead
Concentration
Cairo Traffic Suburbs
Figure 1.2: Lead levels in the blood of Egyptian police officers.
Cairo Suburbs
Number of observations 126 50
Mean 29.2 18.2
Standard deviation 7.5 5.8
Table 1.1: Summary statistics for the blood lead level data.
In this scenario there are two main points of interest:
1. Estimating the difference in mean blood lead levels between the two groups.
2. Testing if the observed difference in blood lead levels is statistically signifi-
cant. (A statistically significant difference means it would be very unlikely
to observe a difference of that size, if in reality the groups had the same
true mean, thus giving strong evidence the observed effect is a real one.
We will discuss this notion in much greater detail later.)
Later in this text we will learn about confidence intervals and hypothesis
tests—statistical inference techniques that will allow us to properly address these
points of interest.
Example 1.3 Can self-control be restored during intoxication? Researchers in-
vestigated this in an experiment with 44 male undergraduate student volunteers.4
The males were randomly assigned to one of 4 treatment groups (11 to each
group):
1. An alcohol group, receiving drinks containing a total of 0.62 mg/kg alcohol.
(Group A)
2. A group receiving drinks with the same alcohol content as Group A, but
also containing 4.4 mg/kg of caffeine. (Group AC)
4
Grattan-Miscio, K. and Vogel-Sprott, M. (2005). Alcohol, intentional control, and inappropri-
ate behavior: Regulation by caffeine or an incentive. Experimental and Clinical Psychophar-
macology, 13:48–55.

13. Balka ISE 1.10 1.3. INFERENTIAL STATISTICS 5
3. A group receiving drinks with the same alcohol content as Group A, but
also receiving a monetary reward for success on the task. (Group AR)
4. A group told they would receive alcoholic drinks, but instead given a
placebo (a drink containing a few drops of alcohol on the surface, and
misted to give a strong alcoholic scent). (Group P)
After consuming the drinks and resting for a few minutes, the participants car-
ried out a word stem completion task involving “controlled (effortful) memory
processes”. Figure 1.3 shows the boxplots for the four treatment groups. Higher
scores are indicative of greater self-control.
A AC AR P
-0.4
-0.2
0.0
0.2
0.4
0.6
Score
on
Task
Figure 1.3: Boxplots of word stem completion task scores.
The plot seems to show a systematic difference between the groups in their task
scores. Are these differences statistically significant? How unlikely is it to see
differences of this size, if in reality there isn’t a real effect of the different treat-
ments? Does this data give evidence that self-control can in fact be restored by
the use of caffeine or a monetary reward? In Chapter 14 we will use a statistical
inference technique called ANOVA to help answer these questions.
Example 1.4 A study5 investigated a possible relationship between eggshell
thickness and environmental contaminants in brown pelican eggs. It was sus-
pected that higher levels of contaminants would result in thinner eggshells. This
study looked at the relationship of several environmental contaminants on the
thickness of shells. One contaminant was DDT, measured in parts per million
of the yolk lipid. Figure 1.4 shows a scatterplot of shell thickness vs. DDT in a
sample of 65 brown pelican eggs from Anacapa Island, California.
There appears to be a decreasing relationship. Does this data give strong ev-
5
Risebrough, R. (1972). Effects of environmental pollutants upon animals other than man. In
Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability.

14. Balka ISE 1.10 1.3. INFERENTIAL STATISTICS 6
0 500 1000 1500 2000 2500 3000
0.1
0.2
0.3
0.4
0.5
Shell
Thickness
(mm)
DDT (ppm)
Figure 1.4: Shell thickness (mm) vs. DDT (ppm) for 65 brown pelican eggs.
idence of a relationship between DDT contamination and eggshell thickness?
Can we use the relationship to help predict eggshell thickness for a given level
of DDT contamination? In Chapter 15 we will use a statistical method called
linear regression analysis to help answer these questions.
In the world of statistics—and our world at large for that matter—we rarely if
ever know anything with certainty. Our state