Advanced High School Statistics

536 Pages·2019·12.852 MB·English

by  David Diez, Mine Cetinkaya-Rundel, Leah Dorazio, Christopher D. Barr

Advanced High School Statistics
Second Edition,
®
with updates based on AP Statistics Course Framework
David Diez
Data Scientist
OpenIntro
Mine C¸etinkaya-Rundel
Associate Professor of the Practice, Duke University
Professional Educator, RStudio
Leah Dorazio
Statistics and Computer Science Teacher
San Francisco University High School
Christopher D Barr
Investment Analyst
Varadero Capital
Copyright © 2019. Modified Second Edition.
Updated: November 18th, 2019.
This book may be downloaded as a free PDF at openintro.org/ahss. This textbook is also avail-
able under a Creative Commons license, with the source files hosted on Github.
AP®isatrademarkregisteredandownedbytheCollegeBoard,whichwasnotinvolvedintheproductionof,anddoesnot
endorse,thisproduct.
3
Table of Contents
1 Data collection 10
1.1 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.1 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Data basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.1 Observations, variables, and data matrices . . . . . . . . . . . . . . . . . . . . 17
1.2.2 Types of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2.3 Relationships between variables . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3 Overview of data collection principles . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.1 Populations and samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.2 Anecdotal evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.3.3 Explanatory and response variables. . . . . . . . . . . . . . . . . . . . . . . . 30
1.3.4 Observational studies versus experiments . . . . . . . . . . . . . . . . . . . . 31
1.4 Observational studies and sampling strategies . . . . . . . . . . . . . . . . . . . . . . 35
1.4.1 Observational studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.4.2 Sampling from a population . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.4.3 Simple, systematic, stratified, cluster, and multistage sampling . . . . . . . . 40
1.5 Experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
1.5.1 Reducing bias in human experiments . . . . . . . . . . . . . . . . . . . . . . . 48
1.5.2 Principles of experimental design . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.5.3 Completely randomized, blocked, and matched pairs design . . . . . . . . . . 50
1.5.4 Testing more than one variable at a time . . . . . . . . . . . . . . . . . . . . 53
Chapter highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Chapter exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2 Summarizing data 60
2.1 Examining numerical data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.1.1 Scatterplots for paired data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.1.2 Stem-and-leaf plots and dot plots . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.1.3 Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.1.4 Describing Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.1.5 Descriptive versus inferential statistics . . . . . . . . . . . . . . . . . . . . . . 71
2.2 Numerical summaries and box plots . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.2.1 Learning objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.2.2 Measures of center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.2.3 Standard deviation as a measure of spread. . . . . . . . . . . . . . . . . . . . 79
2.2.4 Z-scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.2.5 Box plots and quartiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.2.6 Calculator/Desmos: summarizing 1-variable statistics . . . . . . . . . . . . . 86
2.2.7 Outliers and robust statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.2.8 Linear transformations of data . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.2.9 Comparing numerical data across groups . . . . . . . . . . . . . . . . . . . . 92
2.2.10 Mapping data (special topic) . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
2.3 Considering categorical data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4 TABLE OF CONTENTS
2.3.1 Contingency tables and bar charts . . . . . . . . . . . . . . . . . . . . . . . . 104
2.3.2 Row and column proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2.3.3 Using a bar chart with two variables . . . . . . . . . . . . . . . . . . . . . . . 107
2.3.4 Mosaic plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
2.3.5 The only pie chart you will see in this book . . . . . . . . . . . . . . . . . . . 110
2.4 Case study: malaria vaccine (special topic) . . . . . . . . . . . . . . . . . . . . . . . 113
2.4.1 Variability within data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
2.4.2 Simulating the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
2.4.3 Checking for independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Chapter highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Chapter exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3 Probability 122
3.1 Defining probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.1.1 Introductory examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.1.2 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.1.3 Disjoint or mutually exclusive outcomes . . . . . . . . . . . . . . . . . . . . . 126
3.1.4 Probabilities when events are not disjoint . . . . . . . . . . . . . . . . . . . . 128
3.1.5 Complement of an event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.1.6 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.2 Conditional probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.2.1 Exploring probabilities with a contingency table . . . . . . . . . . . . . . . . 139
3.2.2 Marginal and joint probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3.2.3 Defining conditional probability . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.2.4 Smallpox in Boston, 1721 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
3.2.5 General multiplication rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
3.2.6 Sampling without replacement . . . . . . . . . . . . . . . . . . . . . . . . . . 145
3.2.7 Independence considerations in conditional probability . . . . . . . . . . . . . 147
3.2.8 Checking for independent and mutually exclusive events . . . . . . . . . . . . 147
3.2.9 Tree diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.2.10 Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
3.3 The binomial formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
3.3.1 Introducing the binomial formula . . . . . . . . . . . . . . . . . . . . . . . . . 160
3.3.2 When and how to apply the formula . . . . . . . . . . . . . . . . . . . . . . . 162
3.3.3 Calculator: binomial probabilities . . . . . . . . . . . . . . . . . . . . . . . . 165
3.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
3.4.1 Setting up and carrying out simulations . . . . . . . . . . . . . . . . . . . . . 169
3.5 Random variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
3.5.1 Introduction to expected value . . . . . . . . . . . . . . . . . . . . . . . . . . 175
3.5.2 Probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
3.5.3 Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
3.5.4 Variability in random variables . . . . . . . . . . . . . . . . . . . . . . . . . . 180
3.5.5 Linear transformations of a random variable. . . . . . . . . . . . . . . . . . . 181
3.5.6 Linear combinations of random variables. . . . . . . . . . . . . . . . . . . . . 182
3.5.7 Variability in linear combinations of random variables . . . . . . . . . . . . . 184
3.6 Continuous distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
3.6.1 From histograms to continuous distributions. . . . . . . . . . . . . . . . . . . 189
3.6.2 Probabilities from continuous distributions . . . . . . . . . . . . . . . . . . . 190
Chapter highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Chapter exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
4 Distributions of random variables 197
4.1 Normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
4.1.1 Normal distribution model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
4.1.2 Standardizing with Z-scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
4.1.3 Normal probability table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
4.1.4 Normal probability examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
4.1.5 Calculator: finding normal probabilities . . . . . . . . . . . . . . . . . . . . . 207
TABLE OF CONTENTS 5
4.1.6 68-95-99.7 rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
4.1.7 Evaluating the normal approximation . . . . . . . . . . . . . . . . . . . . . . 211
4.1.8 Normal approximation for sums of random variables . . . . . . . . . . . . . . 215
4.2 Sampling distribution of a sample mean . . . . . . . . . . . . . . . . . . . . . . . . . 220
4.2.1 The mean and standard deviation of x¯ . . . . . . . . . . . . . . . . . . . . . . 220
4.2.2 Examining the Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . 225
4.2.3 Normal approximation for the sampling distribution of x¯ . . . . . . . . . . . 228
4.3 Geometric distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
4.3.1 Bernoulli distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
4.3.2 Geometric distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
4.4 Binomial distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
4.4.1 An example of a binomial distribution . . . . . . . . . . . . . . . . . . . . . . 240
4.4.2 The mean and standard deviation of a binomial distribution. . . . . . . . . . 241
4.4.3 Normal approximation to the binomial distribution . . . . . . . . . . . . . . . 242
4.4.4 Normal approximation breaks down on small intervals (special topic) . . . . . 244
4.5 Sampling distribution of a sample proportion . . . . . . . . . . . . . . . . . . . . . . 248
4.5.1 The mean and standard deviation of pˆ . . . . . . . . . . . . . . . . . . . . . . 248
4.5.2 The Central Limit Theorem revisited. . . . . . . . . . . . . . . . . . . . . . . 249
4.5.3 Normal approximation for the distribution of pˆ . . . . . . . . . . . . . . . . . 250
Chapter highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
Chapter exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
5 Foundations for inference 258
5.1 Estimating unknown parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
5.1.1 Point estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
5.1.2 Understanding the variability of a point estimate . . . . . . . . . . . . . . . . 262
5.1.3 Introducing the standard error . . . . . . . . . . . . . . . . . . . . . . . . . . 264
5.1.4 Basic properties of point estimates . . . . . . . . . . . . . . . . . . . . . . . . 265
5.2 Confidence intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
5.2.1 Capturing the population parameter . . . . . . . . . . . . . . . . . . . . . . . 269
5.2.2 Constructing a 95% confidence interval . . . . . . . . . . . . . . . . . . . . . 270
5.2.3 Changing the confidence level . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
5.2.4 Margin of error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
5.2.5 Interpreting confidence intervals . . . . . . . . . . . . . . . . . . . . . . . . . 274
5.2.6 Confidence interval procedures: a five step process . . . . . . . . . . . . . . . 274
5.3 Introducing hypothesis testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
5.3.1 Case study: medical consultant . . . . . . . . . . . . . . . . . . . . . . . . . . 279
5.3.2 Setting up the null and alternate hypothesis . . . . . . . . . . . . . . . . . . . 280
5.3.3 Evaluating the hypotheses with a p-value . . . . . . . . . . . . . . . . . . . . 282
5.3.4 Calculating the p-value by simulation (special topic) . . . . . . . . . . . . . . 285
5.3.5 Hypothesis testing: a five step process . . . . . . . . . . . . . . . . . . . . . . 286
5.3.6 Decision errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
5.3.7 Choosing a significance level. . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
5.3.8 Statistical power of a hypothesis test . . . . . . . . . . . . . . . . . . . . . . . 288
5.4 Does it make sense? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
5.4.1 When to retreat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
5.4.2 Statistical significance versus practical significance . . . . . . . . . . . . . . . 294
5.4.3 Statistical significance versus a real difference . . . . . . . . . . . . . . . . . . 294
Chapter highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
Chapter exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
6 Inference for categorical data 299
6.1 Inference for a single proportion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
6.1.1 Distribution of a sample proportion (review) . . . . . . . . . . . . . . . . . . 302
6.1.2 Checking conditions for inference using a normal distribution . . . . . . . . . 302
6.1.3 Confidence intervals for a proportion . . . . . . . . . . . . . . . . . . . . . . . 303
6.1.4 Calculator: the 1-proportion Z-interval . . . . . . . . . . . . . . . . . . . . . . 307
6.1.5 Choosing a sample size when estimating a proportion . . . . . . . . . . . . . 308
6 TABLE OF CONTENTS
6.1.6 Hypothesis testing for a proportion . . . . . . . . . . . . . . . . . . . . . . . . 310
6.1.7 Calculator: the 1-proportion Z-test . . . . . . . . . . . . . . . . . . . . . . . . 314
6.2 Difference of two proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
6.2.1 Sampling distribution of the difference of two proportions . . . . . . . . . . . 319
6.2.2 Checking conditions for inference using a normal distribution . . . . . . . . . 320
6.2.3 Confidence interval for p −p . . . . . . . . . . . . . . . . . . . . . . . . . . 320
1 2
6.2.4 Calculator: the 2-proportion Z-interval . . . . . . . . . . . . . . . . . . . . . . 323
6.2.5 Hypothesis testing when H : p =p . . . . . . . . . . . . . . . . . . . . . . . 324
0 1 2
6.2.6 Calculator: the 2-proportion Z-test . . . . . . . . . . . . . . . . . . . . . . . . 329
6.3 Testing for goodness of fit using chi-square. . . . . . . . . . . . . . . . . . . . . . . . 335
6.3.1 Creating a test statistic for one-way tables. . . . . . . . . . . . . . . . . . . . 335
6.3.2 The chi-square test statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
6.3.3 The chi-square distribution and finding areas . . . . . . . . . . . . . . . . . . 337
6.3.4 Finding a p-value for a chi-square distribution . . . . . . . . . . . . . . . . . 341
6.3.5 Evaluating goodness of fit for a distribution . . . . . . . . . . . . . . . . . . . 343
6.3.6 Calculator: chi-square goodness of fit test . . . . . . . . . . . . . . . . . . . . 346
6.4 Homogeneity and independence in two-way tables . . . . . . . . . . . . . . . . . . . . 350
6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
6.4.2 Expected counts in two-way tables . . . . . . . . . . . . . . . . . . . . . . . . 352
6.4.3 The chi-square test of homogeneity for two-way tables . . . . . . . . . . . . . 353
6.4.4 The chi-square test of independence for two-way tables. . . . . . . . . . . . . 357
6.4.5 Calculator: chi-square test for two-way tables . . . . . . . . . . . . . . . . . . 361
Chapter highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
Chapter exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
7 Inference for numerical data 369
7.1 Inference for a mean with the t-distribution . . . . . . . . . . . . . . . . . . . . . . . 371
7.1.1 Using a normal distribution for inference when σ is known . . . . . . . . . . . 371
7.1.2 Introducing the t-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
7.1.3 Calculator: finding area under the t-distribution . . . . . . . . . . . . . . . . 375
7.1.4 Checking conditions for inference on a mean using the t-distribution . . . . . 376
7.1.5 One sample t-interval for a mean . . . . . . . . . . . . . . . . . . . . . . . . . 376
7.1.6 Calculator: the 1-sample t-interval . . . . . . . . . . . . . . . . . . . . . . . . 381
7.1.7 Choosing a sample size when estimating a mean . . . . . . . . . . . . . . . . 382
7.1.8 Hypothesis testing for a mean . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
7.1.9 Calculator: 1-sample t-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
7.2 Inference for paired data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
7.2.1 Paired observations and samples . . . . . . . . . . . . . . . . . . . . . . . . . 392
7.2.2 Hypothesis tests for paired data . . . . . . . . . . . . . . . . . . . . . . . . . 393
7.2.3 Calculator: the matched pairs t-test . . . . . . . . . . . . . . . . . . . . . . . 397
7.2.4 Confidence intervals for the mean of a difference . . . . . . . . . . . . . . . . 397
7.2.5 Calculator: the matched pairs t-interval . . . . . . . . . . . . . . . . . . . . . 400
7.3 Inference for the difference of two means . . . . . . . . . . . . . . . . . . . . . . . . . 404
7.3.1 Sampling distribution for the difference of two means. . . . . . . . . . . . . . 405
7.3.2 Checking conditions for inference on a difference of means . . . . . . . . . . . 405
7.3.3 Confidence intervals for a difference of means . . . . . . . . . . . . . . . . . . 406
7.3.4 Calculator: the 2-sample t-interval . . . . . . . . . . . . . . . . . . . . . . . . 410
7.3.5 Hypothesis testing for the difference of two means . . . . . . . . . . . . . . . 411
7.3.6 Calculator: the 2-sample t-test . . . . . . . . . . . . . . . . . . . . . . . . . . 416
Chapter highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
Chapter exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
8 Introduction to linear regression 427
8.1 Line fitting, residuals, and correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 429
8.1.1 Fitting a line to data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
8.1.2 Using linear regression to predict possum head lengths . . . . . . . . . . . . . 431
8.1.3 Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
8.1.4 Describing linear relationships with correlation . . . . . . . . . . . . . . . . . 437
TABLE OF CONTENTS 7
8.2 Fitting a line by least squares regression . . . . . . . . . . . . . . . . . . . . . . . . . 446
8.2.1 An objective measure for finding the best line . . . . . . . . . . . . . . . . . . 446
8.2.2 Finding the least squares line . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
8.2.3 Interpreting the coefficients of a regression line . . . . . . . . . . . . . . . . . 450
8.2.4 Extrapolation is treacherous. . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
8.2.5 Using R2 to describe the strength of a fit . . . . . . . . . . . . . . . . . . . . 452
8.2.6 Calculator/Desmos: linear correlation and regression . . . . . . . . . . . . . . 454
8.2.7 Types of outliers in linear regression . . . . . . . . . . . . . . . . . . . . . . . 457
8.2.8 Categorical predictors with two levels (special topic) . . . . . . . . . . . . . . 459
8.3 Inference for the slope of a regression line . . . . . . . . . . . . . . . . . . . . . . . . 465
8.3.1 The role of inference for regression parameters . . . . . . . . . . . . . . . . . 465
8.3.2 Conditions for the least squares line . . . . . . . . . . . . . . . . . . . . . . . 466
8.3.3 Constructing a confidence interval for the slope of a regression line . . . . . . 467
8.3.4 Calculator: the t-interval for the slope . . . . . . . . . . . . . . . . . . . . . . 471
8.3.5 Midterm elections and unemployment . . . . . . . . . . . . . . . . . . . . . . 471
8.3.6 Understanding regression output from software . . . . . . . . . . . . . . . . . 473
8.3.7 Calculator: the t-test for the slope . . . . . . . . . . . . . . . . . . . . . . . . 477
8.3.8 Which inference procedure to use for paired data? . . . . . . . . . . . . . . . 478
8.4 Transformations for skewed data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
8.4.1 Introduction to transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 484
8.4.2 Transformations to achieve linearity . . . . . . . . . . . . . . . . . . . . . . . 486
Chapter highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
Chapter exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
A Exercise solutions 495
B Data sets within the text 514
C Distribution tables 519
D Calculator reference, Formulas, and Inference guide 532
8
Preface
Advanced High School Statistics covers a first course in statistics, providing an introduction to
appliedstatisticsthatisclear,concise,andaccessible. ThisbookwaswrittentoalignwiththeAP®
Statistics Course Description1, but it’s also popular in non-AP courses and community colleges.
This book may be downloaded as a free PDF at openintro.org/ahss.
We hope readers will take away three ideas from this book in addition to forming a foundation of
statistical thinking and methods.
(1) Statistics is an applied field with a wide range of practical applications.
(2) You don’t have to be a math guru to learn from real, interesting data.
(3) Data are messy, and statistical tools are imperfect. But, when you understand the strengths
and weaknesses of these tools, you can use them to learn about the real world.
Textbook overview
The chapters of this book are as follows:
1. Data collection. Data structures, variables, and basic data collection techniques.
2. Summarizing data. Data summaries and graphics.
3. Probability. The basic principles of probability.
4. Distributions of random variables. Introduction to key distributions, and how the normal
model applies to the sample mean and sample proportion.
5. Foundations for inference. Generalideasforstatisticalinferenceinthecontextofestimating
the population proportion.
6. Inference for categorical data. Inference for proportions and contingency tables using the
normal and chi-square distributions.
7. Inference for numerical data. Inferenceforoneortwosamplemeansusingthet-distribution.
8. Introduction to linear regression. An introduction to regression with two variables, and in-
ference on the slope of the regression line.
Online resources
OpenIntroisfocusedonincreasingaccesstoeducationbydevelopingfree,high-qualityeducationma-
terials. In addition to textbooks, we provide the following accompanying resources to help teachers
and students be successful.
• Video overviews for each section of the textbook
• Lecture slides for each section of the textbook
• Casio and TI calculator tutorials
• Video solutions for selected section and chapter exercises
1AP® is a trademark registered and owned by the College Board, which was not involved in the production of,
anddoesnotendorse,thisproduct. apcentral.collegeboard.org/pdf/ap-statistics-course-description.pdf
TABLE OF CONTENTS 9
• Statistical software labs
• A small but growing number of Desmos activities2
• Quizlet sets for each chapter3
• A Tableau public page to further interact with data sets4
• Online, interactive version of textbook5
• Complete companion course with the learning management software MyOpenMath6
• Complete Canvas course accessible through Canvas Commons7
All of these resources can be found at:
openintro.org/ahss
We also have improved the ability to access data in this book through the addition of Appendix B,
whichprovidesadditionalinformationforeachofthedatasetsusedinthemaintextandisnewinthe
Second Edition. Online guides to each of these data sets are also provided at openintro.org/data
and through a companion R package.
Examples and exercises
Many examples are provided to establish an understanding of how to apply methods.
EXAMPLE0.1
This is an example.
Full solutions to examples are provided here, within the example.
When we think the reader should be ready to do an example problem on their own, we frame it as
Guided Practice.
GUIDEDPRACTICE0.2
The reader may check or learn the answer to any Guided Practice problem by reviewing the full
solution in a footnote.8
Exercises are also provided at the end of each section and each chapter for practice or homework
assignments. Solutions for odd-numbered exercises are given in Appendix A.
Getting involved
We encourage anyone learning or teaching statistics to visit openintro.org and get involved. We
value your feedback. Please send any questions or comments to [email protected]. You can also
provide feedback, report typos, and review known typos at
openintro.org/ahss/feedback
Acknowledgements
This project would not be possible without the passion and dedication of all those involved. The
authors would like to thank the OpenIntro Staff for their involvement and ongoing contributions.
We are also very grateful to the hundreds of students and instructors who have provided us with
valuable feedback since we first started working on this project in 2009. A special thank you to
Catherine Ko for proofreading the second edition of AHSS.
2openintro.org/ahss/desmos
3quizlet.com/openintro-ahss
4public.tableau.com/profile/openintro
5DevelopedbyEmilianoVegaandRalfYoutzofPortlandCommunityCollegeusingPreTeXt.
6myopenmath.com/course/public.php?cid=11774
7sfuhs.instructure.com/courses/1068
8GuidedPracticesolutionsarealwayslocateddownhere!
10
Chapter 1
Data collection
1.1 Case study
1.2 Data basics
1.3 Overview of data collection principles
1.4 Observational studies and sampling strategies
1.5 Experiments