To form a confidence interval for the ratio of variances between two populations, which statistic is used?

To form a confidence interval for the ratio of variances between two populations, which statistic is used?

• A. z statistic
• B. t statistic
• C. chi-square statistic
• D. F statistic

Answer: (D)

The F distribution is the right tool because a ratio of two independent variance estimates, each scaled by its own true variance, follows an F distribution under normality. Specifically, with two independent samples, the quantities (n1−1)s1^2/σ1^2 and (n2−1)s2^2/σ2^2 are chi-square distributed with v1 = n1−1 and v2 = n2−1, and their ratio forms an F(v1, v2) variable. This makes it possible to build a confidence interval for the true variance ratio σ1^2/σ2^2 using the observed ratio s1^2/s2^2 and the F distribution quantiles. The other statistics aren’t appropriate here: z is for known variance or large-sample means, t is for means with unknown variance, and chi-square alone doesn’t yield a ratio-based interval without the F framework.

The F distribution is the right tool because a ratio of two independent variance estimates, each scaled by its own true variance, follows an F distribution under normality. Specifically, with two independent samples, the quantities (n1−1)s1^2/σ1^2 and (n2−1)s2^2/σ2^2 are chi-square distributed with v1 = n1−1 and v2 = n2−1, and their ratio forms an F(v1, v2) variable. This makes it possible to build a confidence interval for the true variance ratio σ1^2/σ2^2 using the observed ratio s1^2/s2^2 and the F distribution quantiles. The other statistics aren’t appropriate here: z is for known variance or large-sample means, t is for means with unknown variance, and chi-square alone doesn’t yield a ratio-based interval without the F framework.