A hypothesis test for two population standard deviations is to be performed. Independent random samples of sizes $\mathrm { n } _ { 1 }$ and $\mathrm { n } _ { 2 }$ are drawn from the two populations. The variable under consideration is normally distributed on each of the two populations. The test statistic is $\mathrm { s } _ { 1 } ^ { 2 } / \mathrm { s } _ { 2 } ^ { 2 }$. If the null hypothesis, $\mathrm { H } _ { 0 } : \sigma _ { 1 } = \sigma _ { 2 }$, is true, what is the distribution of this test statistic?
A)$\chi ^ { 2 }$-distribution with $\mathrm { df } = \mathrm { n } _ { 1 } + \mathrm { n } _ { 2 } - 1$
B) $\mathrm { F }$-distribution with df $= \left( \mathrm { n } _ { 1 } , \mathrm { n } _ { 2 } \right)$
C) F-distribution with $\mathrm { df } = \left( \mathrm { n } _ { 2 } - 1 , \mathrm { n } _ { 1 } - 1 \right)$
D) $F$-distribution with $\mathrm { df } = \left( \mathrm { n } _ { 1 } - 1 , \mathrm { n } _ { 2 } - 1 \right)$

Question

A hypothesis test for two population standard deviations is to be performed. Independent random samples of sizes $\mathrm { n } _ { 1 }$ and $\mathrm { n } _ { 2 }$ are drawn from the two populations. The variable under consideration is normally distributed on each of the two populations. The test statistic is $\mathrm { s } _ { 1 } ^ { 2 } / \mathrm { s } _ { 2 } ^ { 2 }$. If the null hypothesis, $\mathrm { H } _ { 0 } : \sigma _ { 1 } = \sigma _ { 2 }$, is true, what is the distribution of this test statistic?  
A)$\chi ^ { 2 }$-distribution with $\mathrm { df } = \mathrm { n } _ { 1 } + \mathrm { n } _ { 2 } - 1$
B) $\mathrm { F }$-distribution with df $= \left( \mathrm { n } _ { 1 } , \mathrm { n } _ { 2 } \right)$
C) F-distribution with $\mathrm { df } = \left( \mathrm { n } _ { 2 } - 1 , \mathrm { n } _ { 1 } - 1 \right)$
D) $F$-distribution with $\mathrm { df } = \left( \mathrm { n } _ { 1 } - 1 , \mathrm { n } _ { 2 } - 1 \right)$

Quizplus · Accepted Answer

The test statistic for comparing two population variances (or standard deviations) under the assumption of normality follows an F-distribution. The degrees of freedom for the F-distribution are determined by the sample sizes from each population, with the formula for each being the sample size minus one ($n_1 - 1$ and $n_2 - 1$).

A Hypothesis Test for Two Population Standard Deviations Is to Be