How Do You Perform a Chi-Squared Goodness of Fit Test?

Perform a chi-squared goodness of fit test by comparing observed frequencies with the frequencies expected under a proposed categorical distribution. Use a GDC to calculate the test statistic and p-value, then compare the p-value with the significance level.

This is SL 4.11 Hypothesis testing and may appear in GDC-active Paper 1 or Paper 2 questions.

The mechanism is based on discrepancies between observed and expected counts. Each category contributes $\frac{(O_i-E_i)^2}{E_i}$, so larger discrepancies relative to the expected count increase $\chi^2$. A large statistic therefore produces a small p-value and stronger evidence against $H_0$.

On the GDC, enter the observed and expected lists in the goodness-of-fit routine. Report both $\chi^2$ and $p$, keeping full precision until the final answer.

For example, suppose the observed frequencies are $18,22,30$, while the expected frequencies are $20,25,25$. Then

$\chi^2=\frac{(18-20)^2}{20}+\frac{(22-25)^2}{25}+\frac{(30-25)^2}{25}=1.56.$

With $df=3-1=2$, the GDC gives $p\approx0.458$. Since $0.458>0.05$, there is insufficient evidence at the significance level to reject ; the data is consistent with the proposed distribution.

Exam technique: Show the hypotheses, expected frequencies, degrees of freedom, p-value, comparison with $\alpha$, and a contextual conclusion. A common misconception is to say “accept $H_0$.” Instead, write “insufficient evidence to reject $H_0$.”