In a chi-square test, what best describes the relationship between the critical value and the calculated value?

• A. The critical value is the threshold needed to reject the null hypothesis; the calculated value is computed from the data.
• B. The critical value equals the calculated value if the null is true.
• C. The calculated value must be smaller than the critical value to reject.
• D. The critical value is always less than the calculated value.

Answer: (A)

In a chi-square test, the crucial idea is comparing the statistic you compute from the data to a fixed cutoff called the critical value, which comes from the chi-square distribution for the chosen significance level and degrees of freedom. The statistic is calculated from observed and expected frequencies, and the critical value is the threshold that separates likely from unlikely deviations under the null. If the calculated value exceeds this threshold, the result is unlikely under the null hypothesis, so you reject it; if it does not exceed the threshold, you do not reject the null.

This is why describing the critical value as the threshold needed to reject the null and the calculated value as derived from the data is the best characterization. The other statements aren’t the rule: the critical value does not have to equal the calculated value, and rejection occurs when the calculated value is larger than the critical value, not smaller, and the critical value is not inherently always less than the calculated value.