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.
$\begin{array}{c|cc}
& \text { Population 1 } & \text { Population 2 } \
\hline \mathrm{n} & 25 & 17 \
\mathrm{~s} & 5.76 & 2.21
\end{array}$

A) Test statistic: $\mathrm { F } = 6.79$. Critical value: $\mathrm { F } = 3.18$. Reject $\mathrm { H } _ { 0 }$.
B) Test statistic: $\mathrm { F } = 6.79$. Critical value: $\mathrm { F } = 3.18$. Do not reject $\mathrm { H } _ { 0 }$.
C) Test statistic: $\mathrm { F } = 2.61$. Critical value $= 1.87$. Reject $\mathrm { H } _ { 0 }$.
D) Test statistic: $\mathrm { F } = 2.61$. Critical value $= 3.18$. 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.
$\begin{array}{c|cc} 
& \text { Population 1 } & \text { Population 2 } \
\hline \mathrm{n} & 25 & 17 \
\mathrm{~s} & 5.76 & 2.21
\end{array}$

A) Test statistic: $\mathrm { F } = 6.79$. Critical value: $\mathrm { F } = 3.18$. Reject $\mathrm { H } _ { 0 }$.
B) Test statistic: $\mathrm { F } = 6.79$. Critical value: $\mathrm { F } = 3.18$. Do not reject $\mathrm { H } _ { 0 }$.
C) Test statistic: $\mathrm { F } = 2.61$. Critical value $= 1.87$. Reject $\mathrm { H } _ { 0 }$.
D) Test statistic: $\mathrm { F } = 2.61$. Critical value $= 3.18$. Do not reject $\mathrm { H } _ { 0 }$.

Quizplus · Accepted Answer

The test statistic is calculated by squaring the ratio of the two sample standard deviations ($s_1^2 / s_2^2$), which gives $F = (5.76^2 / 2.21^2) = 6.79$. Given the degrees of freedom for the numerator ($n_1 - 1 = 24$) and the denominator ($n_2 - 1 = 16$), and the significance level ($\alpha = 0.01$), the critical value for a one-tailed F-test is approximately 3.18. Since the test statistic $6.79$ is greater than the critical value $3.18$, we reject the null hypothesis $\mathrm { H } _ { 0 }$ that $\sigma _ { 1 } \leq \sigma _ { 2 }$, supporting the alternative hypothesis that $\sigma _ { 1 } > \sigma _ { 2 }$.

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 $\sigma _ { 1 } > \sigma _ { 2 }$