This chapter uses the example of shoe sizes forming a bell curve to introduce the normal distribution, noting it is used, and sometimes abused, across psychology, business, economics, and the sciences. It covers recognizing and applying the normal probability distribution, the standard normal distribution, and converting between the two to compare probabilities.
A bell curve across many fields
The chapter introduces the normal distribution through the example of shoe sizes graphing into a bell-shaped curve, noting that the same pattern appears across psychology, business, economics, and the sciences, and that some instructors even use it to help set grades, while emphasizing that recognizing when it does and does not apply is itself a skill.
The standard normal distribution
Students learn to recognize and apply the standard normal probability distribution appropriately, the reference version of the normal distribution used to standardize any normally distributed value so that probabilities can be looked up or computed consistently regardless of the original mean and standard deviation.
Comparing probabilities across distributions
The chapter covers comparing normal probabilities by converting values to the standard normal distribution, a technique that lets students compare two values from different normal distributions, such as scores on two different tests, on a common scale before drawing any conclusion about which is relatively higher or lower.
This chapter uses examples such as whether lottery numbers are evenly distributed to introduce the chi-square distribution and its notation. It covers interpreting the chi-square probability distribution as sample size changes, and conducting and interpreting goodness-of-fit tests, tests of independence, tests of homogeneity, and single-variance tests.
Interpreting the chi-square distribution
The chapter teaches students to interpret the chi-square probability distribution as the sample size changes, establishing its notation and shape before applying it to four distinct hypothesis-testing situations, each suited to a different kind of categorical or variance question.
Goodness-of-fit and independence tests
Students learn to conduct and interpret chi-square goodness-of-fit hypothesis tests, which check whether data such as lottery number frequencies match an expected distribution, and chi-square tests of independence, which check whether two categorical variables, such as movie preference and age group, are related.
Homogeneity and single-variance tests
The chapter also covers chi-square homogeneity hypothesis tests, which compare distributions across several populations, and chi-square single-variance hypothesis tests, illustrated with whether a coffee machine dispenses a consistent amount, rounding out the four applications of the distribution covered in the chapter.
This chapter uses the example of the change people carry in their pockets to introduce the Central Limit Theorem, showing that a large enough sample produces a normal, bell-shaped distribution of sample means. It covers recognizing central limit theorem problems, classifying continuous word problems by distribution, and applying the theorem to both means and sums.
Recognizing when the theorem applies
The chapter teaches students to recognize central limit theorem problems and to classify continuous word problems by their distributions, using the example of pocket change to show that even though individual amounts vary unpredictably, the distribution of sample means from repeated sampling settles into a predictable, normal, bell-shaped pattern.
Applying the theorem to sample means
Students learn to apply and interpret the Central Limit Theorem for means, the case where repeated samples are drawn from a population and the sample mean is calculated each time, with the resulting distribution of those means approaching normality as the sample size grows large enough.
Applying the theorem to sums
The chapter also covers applying and interpreting the Central Limit Theorem for sums, the parallel case concerned with the total of a sample’s values rather than its average, giving students both versions of the theorem side by side for the different kinds of problems each is suited to.
This opening chapter answers the question of when and where statistics gets used, pointing to the statistical information embedded in everyday newspaper and news coverage. It covers recognizing and differentiating key statistical terms, applying various sampling methods to data collection, and creating and interpreting frequency tables.
When and where statistics is used
The chapter opens by addressing a question many students ask, when and where they will actually use statistics, and answers it by pointing to the statistical information embedded in everyday newspaper articles, television news, and internet coverage of topics such as crime, sports, education, politics, and real estate.
Key terms and sampling methods
Students learn to recognize and differentiate between key statistical terms and to apply various types of sampling methods to data collection, building the vocabulary and practical technique needed before any data can be meaningfully analyzed or discussed.
Frequency tables
The chapter also covers creating and interpreting frequency tables, a basic tool for organizing raw data into counts by category or value, giving students their first structured way to summarize a data set before moving into the graphical and numerical descriptive methods covered in later chapters.
This chapter uses the rarity of meteor showers to introduce the terminology of probability, from politicians studying polls to doctors assessing treatments. It covers using probability terminology correctly, determining whether events are mutually exclusive or independent, calculating probabilities with addition and multiplication rules, and constructing contingency tables, Venn diagrams, and tree diagrams.
The language of probability
The chapter opens by teaching students to understand and use the terminology of probability, connecting it to everyday decisions such as a politician studying polls or a doctor assessing treatment options, before formalizing the vocabulary into precise definitions used throughout the rest of the course.
Mutually exclusive and independent events
Students learn to determine whether two events are mutually exclusive, meaning they cannot both occur, and whether two events are independent, meaning one occurring does not change the probability of the other, then apply the addition and multiplication rules for calculating probabilities that depend on these relationships.
Visualizing probability with diagrams
The chapter covers three ways to visualize probability problems: constructing and interpreting contingency tables, which organize outcomes by two categories at once; Venn diagrams, which show overlaps between events; and tree diagrams, which lay out a sequence of outcomes step by step, giving students several complementary tools for the same underlying calculations.
