Test Hypotheses Regarding Two Population Standard Deviations
-Test the hypothesis that $\sigma _ { 1 } \sigma _ { 2 }$ at the $\alpha = 0.01$ level of significance for the given sample data.
$\begin{array}{c|cc}
& \text { Population 1 } & \text { Population 2 } \
\hline \mathrm{n} & 61 & 31 \
\mathrm{~s} & 27.5 & 26.3
\end{array}$

A) Test statistic: $\mathrm { F } = 1.09$. Critical value $= 2.21$. Do not reject $\mathrm { H } _ { 0 }$.
B) Test statistic: $\mathrm { F } = 1.09$. Critical value $= 2.03$. Reject $\mathrm { H } _ { 0 }$.
C) Test statistic: $\mathrm { F } = 28.75$. Critical value $= 2.03$. Reject $\mathrm { H } _ { 0 }$.
D) Test statistic: $\mathrm { F } = 1.05$. Critical value $= 2.21$. Do not reject $\mathrm { H } _ { 0 }$.

Question

Test Hypotheses Regarding Two Population Standard Deviations
-Test the hypothesis that $\sigma _ { 1 } > \sigma _ { 2 }$ at the $\alpha = 0.01$ level of significance for the given sample data.
$\begin{array}{c|cc} 
& \text { Population 1 } & \text { Population 2 } \
\hline \mathrm{n} & 61 & 31 \
\mathrm{~s} & 27.5 & 26.3
\end{array}$

A) Test statistic: $\mathrm { F } = 1.09$. Critical value $= 2.21$. Do not reject $\mathrm { H } _ { 0 }$.
B) Test statistic: $\mathrm { F } = 1.09$. Critical value $= 2.03$. Reject $\mathrm { H } _ { 0 }$.
C) Test statistic: $\mathrm { F } = 28.75$. Critical value $= 2.03$. Reject $\mathrm { H } _ { 0 }$.
D) Test statistic: $\mathrm { F } = 1.05$. Critical value $= 2.21$. Do not reject $\mathrm { H } _ { 0 }$.

Quizplus · Accepted Answer

The test statistic for comparing two variances (or standard deviations) is the F-test, calculated as the ratio of the two sample variances. Here, the test statistic F is calculated as $F = \frac{s_1^2}{s_2^2} = \frac{27.5^2}{26.3^2} \approx 1.09$. The decision to reject or not reject the null hypothesis depends on whether the test statistic exceeds the critical value at the given level of significance. Since the test statistic does not exceed the critical value, we do not reject the null hypothesis.

Test Hypotheses Regarding Two Population Standard Deviations -Test the Hypothesis σ1>σ2\sigma _ { 1 } > \sigma _ { 2 }σ1​>σ2​

Test Hypotheses Regarding Two Population Standard Deviations
-Test the Hypothesis $\sigma _ { 1 } > \sigma _ { 2 }$