Chi-Square Goodness-of-Fit Tests for Categorical Counts

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Chi-square goodness-of-fit test

A hypothesis test comparing observed categorical counts with counts expected under a specified distribution.

Observed and expected counts

Observed counts are recorded in the data; expected counts are predicted by the null hypothesis.

Goodness-of-fit test statistic

$\chi^2=\sum_{i=1}^{k}\frac{(O_i-E_i)^2}{E_i}$. It summarizes the discrepancies between observed and expected counts.

Degrees of freedom

For a goodness-of-fit test with $k$ categories, $df=k-1$.

Expected-count condition

Every expected count must be at least five for the usual chi-square goodness-of-fit procedure.

Quiz questions

1. What does the null hypothesis state in a goodness-of-fit test?Answer: The population distribution follows the specified pattern.
2. All expected counts in a goodness-of-fit test are at least five, but one observed count is below five. What follows from the expected-count condition?Answer: The expected-count condition is met; a low observed count alone does not violate it.
3. At significance level $\alpha$, when should a goodness-of-fit test reject the null hypothesis based on its p-value?Answer: When the p-value is at most $\alpha$.
4. If observed counts are close to their expected counts across categories, what is generally true of the chi-square statistic?Answer: It tends to be small.
5. A goodness-of-fit test has an expected count below five. What is an appropriate response?Answer: Do not use the usual procedure as is; consider justified category combining or another method.