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 statements and conclusions will involve
a measure of the reliability of our estimates. We will make statements like we
can be 95% confident that the true difference in means lies between 4 and 10, or,
the probability of seeing the observed difference, if in reality the new drug has no
effect, is less than 0.001. So probability plays an important role in statistics, and
we will study the basics of probability in Chapter 4. In our study of probability,
keep in mind that we have an end in mind—the use of probability to quantify
our uncertainty when attempting to answer questions of interest.

15. Chapter 2
Gathering Data
7

16. Balka ISE 1.10 8
Supporting Videos For This Chapter

17. Balka ISE 1.10 2.1. INTRODUCTION 9
2.1 Introduction
In this chapter we will investigate a few important concepts in data collection.
Some of the concepts introduced in this chapter will be encountered throughout
the remainder of this text.
2.2 Populations and Samples, Parameters and Statis-
tics
In 1994 the Center for Science in the Public Interest (CSPI) investigated nutri-
tional characteristics of movie theatre popcorn, and found that the popcorn was
often loaded with saturated fats. The study received a lot of media attention,
and even motivated some theatre chains to change how they made popcorn.
In 2009, the CSPI revisited movie theatre popcorn.1 They found that in addi-
tion to the poor stated nutritional characteristics, in reality the popcorn served
at movie theatres was nutritionally worse than the theatre companies claimed.
What was served in theatres contained more calories, fat, and saturated fat than
was stated by the companies.
Suppose that you decide to follow up on this study in at your local theatre.
Suppose that your local movie theatre claims that a large untopped bag of their
popcorn contains 920 calories on average. You spend a little money to get a sam-
ple of 20 large bags of popcorn, have them analyzed, and find that they contain
1210 calories on average. Does this provide strong evidence that the theatre’s
claimed average of 920 calories false? What values are reasonable estimates of
the true mean calorie content? These questions cannot be answered with the
information we have at this point, but with a little more information and a few
essential tools of statistical inference, we will learn how to go about answering
them.
To speak the language of statistical inference, we will need a few definitions:
• Individuals or units or cases are the objects on which a measurement is
taken. In this example, the bags of popcorn are the units.
• The population is the set of all individuals or units of interest to an
investigator. In this example the population is all bags of popcorn of this
type. A population may be finite (all 25 students in a specific kindergarten
1
http://www.cspinet.org/nah/articles/moviepopcorn.html

18. Balka ISE 1.10 2.3. TYPES OF SAMPLING 10
class, for example) or infinite (all bags of popcorn that could be produced
by a certain process, for example).2
• A parameter is a numerical characteristic of a population. Examples:
– The mean calorie content of large bags of popcorn made by a movie
theatre’s production process.
– The mean weight of 40 year-old Canadian males.
– The proportion of adults in the United States that approve of the way
the president is handling his job.
In practice it is usually impractical or impossible to take measurements
on the entire population, and thus we typically do not know the values
of parameters. Much of this text is concerned with methods of parameter
estimation.
• A sample is a subset of individuals or units selected from the population.
The 20 bags of popcorn that were selected is a sample.
• A statistic is a numerical characteristic of a sample. The sample mean of
the 20 bags (1210 calories) is a statistic.
In statistical inference we attempt to make meaningful statements about popu-
lation parameters based on sample statistics. For example, do we have strong
evidence that a food item’s average calorie content is greater than what the
producer claims?
A natural and important question that arises is: How do we go about obtaining a
proper sample? This is often not an easy question to answer, but there are some
important considerations to take into account. Some of these considerations will
be discussed in the following sections.
2.3 Types of Sampling
Suppose a right-wing radio station in the United States asks their listeners to
call in with their opinions on how the president is handling his job. Eighteen
people call in, with sixteen saying they disapprove.
2
Sometimes the term population is used to represent the set of individuals or objects, and
sometimes it is used to represent the set of numerical measurements on these individuals or
objects. For example, if we are interested in the weights of 25 children in a specific kindergarten
class, then the term population might be used to represent the 25 children, or it might be
used to represent the 25 weight measurements on these children. In practice, this difference
is unlikely to cause any confusion.

19. Balka ISE 1.10 2.3. TYPES OF SAMPLING 11
How informative is this type of sample? If we are hoping to say something about
the opinion of Americans in general, it is not very informative. This sample is
hopelessly biased (not representative of the population of interest). We might say
that the sampled population is different from the target population. (Listeners
of a right-wing radio station are not representative of Americans as a whole.)
What if we consider the population of interest to be listeners of this right-wing
radio station? (Which would be the case if we only wish to make statements
about listeners of this station.) There is still a major issue: the obtained sample
is a voluntary response sample (listeners chose whether to respond or not—
they self-selected to be part of the sample). Voluntary response samples tend to
be strongly biased; individuals are more likely to respond if they feel very strongly
about the issue and less likely to respond if they are indifferent. Student course
evaluations are another example of a voluntary response sample, and individuals
that love or hate a professor may be more likely to fill in a course evaluation
than those who think the professor was merely average.
Self-selection often results in a hopelessly biased sample. It is one potential
source of bias in a sample, but there are many other potential sources. For
example, an investigator may consciously or subconsciously select a sample that
is likely to support their hypothesis. This is most definitely something an honest
statistician tries to avoid.
So how do we avoid bias in our sample? We avoid bias by randomly selecting
members of the population for our sample. Using a proper method of random
sampling ensures that the sample does not have any systematic bias. There are
many types of random sampling, each with its pros and cons. One of the simplest
and most important random sampling methods is simple random sampling. In
the statistical inference procedures we encounter later in this text, oftentimes the
procedures will be appropriate only if the sample is a simple random sample
(SRS) from the population of interest. Simple random sampling is discussed in
the next section.
2.3.1 Simple Random Sampling
Simple random sampling is a method used to draw an unbiased sample from
a population. The precise definition of a simple random sample depends on
whether we are sampling from a finite or infinite population. But there is one
constant: no member of the population is more or less likely to be contained in
the sample than any other member.
In a simple random sample of size n from a finite population, each possible sample
of size n has the same chance of being selected. An implication of this is that

20. Balka ISE 1.10 2.3. TYPES OF SAMPLING 12
every member of the population has the same chance of being selected. Simple
random samples are typically carried out without replacement (a member of the
population cannot appear in the sample more than once).
Example 2.1 Let’s look at a simple fictitious example to illustrate simple ran-
dom sampling. Suppose we have a population of 8 goldfish:
Goldfish number 1 2 3 4 5 6 7 8
Goldfish name Mary Hal Eddy Tony Tom Pete Twiddy Jemmy
With such a small population size we may be able to take a measurement on every
member. (We could find the mean weight of the population, say, since it would
not be overly difficult or time consuming to weigh all 8 goldfish.) But most often
the population size is too large to measure all members. And sometimes even for
small populations it is not possible to observe and measure every member. In
this example suppose the desired measurement requires an expensive necropsy,
and we have only enough resources to perform three necropsies. We cannot afford
to measure every member of the population, and to avoid bias we may choose
to draw a simple random sample of 3 goldfish. How do we go about drawing a
simple random sample of 3 goldfish from this population? In the modern era, we
simply get software to draw the simple random sample for us.34
The software may randomly pick Mary, Tony, and Jemmy. Or Pete, Twiddy,
and Mary, or any other combination of 3 names. Since we are simple random
sampling, any combination of 3 names has the same chance of being picked. If
we were to list all possible samples of size 3, we would see that there are 56
possible samples. Any group of 3 names will occur with probability 1
56, and any
individual goldfish will appear in the sample with probability 3
8.5
Many statistical inference techniques assume that the data is the result of a sim-
ple random sample from the population of interest. But at times it is simply
not possible to carry out this sampling method. (For example, in practice we
couldn’t possibly get a simple random sample of adult dolphins in the Atlantic
ocean.) So at times we may not be able to properly randomize, and we may
simply try to minimize obvious biases and hope for the best. But this is danger-
ous, as any time we do not carry out a proper randomization method, bias may
be introduced into the sample and the sample may not be representative of the
3
For example, in the statistical software R, if the names are saved in an object called gold-
fish.names, the command sample(goldfish.names,3) would draw a simple random sample
of 3 names.
4
In the olden days, we often used a random number table to help us pick the sample.
Although random number tab