Goodness of fit test: sampling distribution of the chi-squared statistic

Definition of the sampling distribution of the chi-squared statistic

Sampling distribution of $ X^2$:

As you may know, when we perform a goodness of fit test, we compute the chi-squared statistic $$ X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}} $$ based on our sample data. Now suppose that we would draw many more samples. Specifically, suppose that we would draw an infinite number of samples, each with the same sample size. In each sample, we could compute the chi-squared statistic $ X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$. Different samples will give different values for $X^2$. The distribution of all these $ X^2$ values is the sampling distribution of $ X^2$. Note that this sampling distribution is purely hypothetical. We will never really draw an infinite number of samples, but hypothetically, we could.

Sampling distribution of $ X^2$ if H0 were true:

Suppose that the assumptions of the goodness of fit test hold, and that the null hypothesis that the population proportions in each of the $ J$ conditions are $\pi_1$, $\pi_2$, $\ldots$, $\pi_J$ is true. Then the sampling distribution of $ X^2$ is approximately the chi-squared distribution with $ J - 1$ degrees of freedom. That is, most of the time we would find relatively small $ X^2$ values, and only sometimes we would find large $ X^2$ values. If we find a $ X^2$ value in our actual sample that is very large, this is a rare event if the null hypothesis were true, and is therefore considered evidence against the null hypothesis ($ X^2$ value in rejection region, small $ p$ value).

chi squared distribution