SECTION 9.2 SUMMARY

• • The chi-square goodness-of-fit test is used to compare the sample distribution of a categorical variable from a population with a hypothesized distribution. The data for $n$ observations with $k$ possible outcomes are summarized as observed counts, $n_1{,n}_2,\dots {,n}_k,$ in $k$ cells. The null hypothesis specifies probabilities $p_1{,p}_2,\dots {,p}_k$ for the possible outcomes.

• • The analysis of these data is similar to the analyses of two-way tables discussed in Section 9.1. For each cell, the expected count is determined by multiplying the total number of observations $n$ by the specified probability$p_i$. The null hypothesis is tested by the usual chi-square statistic, which compares the observed counts, $n_i$, with the expected counts. Under the null hypothesis, $X^2$ has approximately the ${\chi }^2$ distribution with $\text{df}=k-1$.