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 hypotheses are:
$\begin{array} { r }
\mathrm { H } _ { 0 } : \sigma _ { 1 } = \sigma _ { 2 } \
\mathrm { H } _ { \mathrm { a } } : \sigma _ { 1 }

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 hypotheses are:
$\begin{array} { r } 
\mathrm { H } _ { 0 } : \sigma _ { 1 } = \sigma _ { 2 } \
\mathrm { H } _ { \mathrm { a } } : \sigma _ { 1 } < \sigma _ { 2 }
\end{array}$
Which of the following would provide evidence against the null hypothesis in favor of the alternative?  
A)$\mathrm { s } _ { 1 } ^ { 2 } / \mathrm { s } _ { 2 } ^ { 2 }$ is too close to 1 .
B) $\mathrm { s } _ { 1 } ^ { 2 } / \mathrm { s } _ { 2 } ^ { 2 }$ is much bigger or much smaller than 1 .
C) $\mathrm { s } _ { 1 } ^ { 2 } / \mathrm { s } _ { 2 } ^ { 2 }$ is much bigger than 1 .
D) $\mathrm { s } _ { 1 } ^ { 2 } / \mathrm { s } _ { 2 } ^ { 2 }$ is much smaller than 1 .

Quizplus · Accepted Answer

Evidence against the null hypothesis ($\sigma_1 = \sigma_2$) in favor of the alternative hypothesis ($\sigma_1 < \sigma_2$) would be provided if the sample variance of the first population ($s_1^2$) divided by the sample variance of the second population ($s_2^2$) is much smaller than 1. This indicates that the first population's variance is significantly smaller than the second's, supporting the alternative hypothesis.

A Hypothesis Test for Two Population Standard Deviations Is to Be