In a chi-square goodness-of-fit test, which statement about the null hypothesis is correct?

In a chi-square goodness-of-fit test, which statement about the null hypothesis is correct?

• A. The observed frequencies match exactly the expected frequencies
• B. The observed frequencies are identical across categories
• C. The observed frequencies follow the specified distribution
• D. The observed frequencies differ from the specified distribution

Answer: (C)

The main idea being tested is whether the data come from a specified distribution. In a chi-square goodness-of-fit test, the null hypothesis states that the population distribution matches the specified distribution, so the expected frequencies are the counts you’d predict under that distribution. We compare the observed frequencies to these expected frequencies, and deviations are evaluated against sampling variability. If the observed counts align with those expectations, we do not have evidence to reject the null; if they differ beyond what chance would explain, we reject it. So, saying that the observed frequencies follow the specified distribution captures what the null is asserting. Exact equality of counts isn’t required because sampling variability can lead to differences; saying they are identical across categories is a special case and not the general null. Saying they differ from the specified distribution describes the alternative, not the null.

The main idea being tested is whether the data come from a specified distribution. In a chi-square goodness-of-fit test, the null hypothesis states that the population distribution matches the specified distribution, so the expected frequencies are the counts you’d predict under that distribution. We compare the observed frequencies to these expected frequencies, and deviations are evaluated against sampling variability. If the observed counts align with those expectations, we do not have evidence to reject the null; if they differ beyond what chance would explain, we reject it.

So, saying that the observed frequencies follow the specified distribution captures what the null is asserting. Exact equality of counts isn’t required because sampling variability can lead to differences; saying they are identical across categories is a special case and not the general null. Saying they differ from the specified distribution describes the alternative, not the null.