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 introduces the chi-square distribution through examples such as whether a coffee machine dispenses a consistent amount each time, framing it as the tool behind three major hypothesis tests. It covers the facts and notation of the chi-square distribution, the test of a single variance, and the goodness-of-fit test.
Three applications of one distribution
The chapter introduces the chi-square distribution as the basis for three distinct hypothesis tests: the goodness-of-fit test, which checks whether data fit a particular distribution, such as evenly distributed lottery numbers; the test of independence, which checks whether two categorical variables such as movie preference and age group are related; and the test of a single variance, illustrated with whether a coffee machine’s dispensed amount is consistent.
Facts about the chi-square distribution
The chi-square distribution is nonsymmetrical and skewed to the right, with a different curve for each value of its degrees of freedom, and the chapter notes that its test statistic is always greater than or equal to zero and that, for a sufficiently large number of degrees of freedom, the chi-square curve begins to approximate the normal distribution.
Testing a single variance and goodness of fit
Where earlier chapters focused on means and proportions, the chi-square test of a single variance shifts attention to variability itself, while the goodness-of-fit test determines whether an observed distribution of categorical data is consistent with a claimed or expected distribution, both worked through with the degrees of freedom appropriate to each specific use.
The chi-square distribution is used to test claims about categorical data and variances. This chapter describes the properties of the chi-square distribution and applies it to goodness-of-fit tests, tests of independence between two variables, tests of homogeneity across populations, and tests about a single population variance.
Facts About the Chi-Square Distribution
The chi-square distribution is a family of right-skewed curves that take only non-negative values, because the statistic is built from squared quantities. Its shape depends on a single degrees-of-freedom value, and as that value grows the curve becomes more symmetric and bell-like. The distribution is used to compare observed counts with the counts expected under a hypothesis, and to draw conclusions about categorical data and about the variance of a population.
Goodness-of-Fit and Independence Tests
A goodness-of-fit test checks whether observed frequencies across categories match a claimed distribution, such as whether lottery numbers occur equally often. A test of independence uses a contingency table to decide whether two categorical variables are related or independent. In each case the test compares observed counts with expected counts, sums the standardised squared differences into a chi-square statistic, and compares it with the distribution to obtain a p-value and a decision.
Homogeneity and Single-Variance Tests
A test of homogeneity checks whether two or more populations share the same distribution across categories, extending the same counting logic to several groups at once. The chi-square distribution is also used to test a claim about a single population variance or standard deviation. Across all these uses the pattern is the same: measure how far the data depart from what a hypothesis predicts, express that gap as a chi-square value, and judge whether it is larger than chance would explain.
