Hypothesis Test vs Chi-Square Test

A significance test decides whether the evidence in a sample is strong enough to reject a default claim about a parameter. You state a null and an alternative hypothesis, compute a test statistic, and find its p-value under the null. For example, to test whether a coin is fair you compare the observed number of heads to what a fair coin predicts. If the p-value is at or below the significance level, you reject the null hypothesis; otherwise you fail to reject it.

A chi-square test works with counts in categories rather than means, checking goodness of fit, independence, or homogeneity. It adds up the squared gaps between observed and expected counts, each scaled by the expected count, into one statistic. The formula is $\chi^2 = \sum \frac{(O - E)^2}{E}$, where $O$ is an observed count and $E$ an expected count. For example, rolling a die 60 times and comparing each face's tally to the expected 10 tests whether the die is fair, and a large $\chi^2$ gives a small p-value.