The sample standard deviations and sample sizes are given for independent simple random samples from twopopulations. Use the two-standard-deviations F-test to conduct the required hypothesis test.
-$\mathrm { s } _ { 1 } = 19.8 , \mathrm { n } _ { 1 } = 16 , \mathrm {~s} _ { 2 } = 22.2 , \mathrm { n } _ { 2 } = 13$; left-tailed test, $\alpha = 0.10$
A)Test statistic: $\mathrm { F } = 0.8$. Critical value $= 0.495$. Reject $\mathrm { H } _ { 0 }$.
B) Test statistic: $\mathrm { F } = 0.8$. Critical value $= 0.495$. Do not reject $\mathrm { H } _ { 0 }$.
C) Test statistic: $\mathrm { F } = 0.8$. Critical value $= 0.476$. Do not reject $\mathrm { H } _ { 0 }$.
D) Test statistic: $\mathrm { F } = 0.8$. Critical value $= 0.476$. Reject $\mathrm { H } _ { 0 }$.

Question

The sample standard deviations and sample sizes are given for independent simple random samples from twopopulations. Use the two-standard-deviations F-test to conduct the required hypothesis test.
-$\mathrm { s } _ { 1 } = 19.8 , \mathrm { n } _ { 1 } = 16 , \mathrm {~s} _ { 2 } = 22.2 , \mathrm { n } _ { 2 } = 13$; left-tailed test, $\alpha = 0.10$  
A)Test statistic: $\mathrm { F } = 0.8$. Critical value $= 0.495$. Reject $\mathrm { H } _ { 0 }$.
B) Test statistic: $\mathrm { F } = 0.8$. Critical value $= 0.495$. Do not reject $\mathrm { H } _ { 0 }$.
C) Test statistic: $\mathrm { F } = 0.8$. Critical value $= 0.476$. Do not reject $\mathrm { H } _ { 0 }$.
D) Test statistic: $\mathrm { F } = 0.8$. Critical value $= 0.476$. Reject $\mathrm { H } _ { 0 }$.

Quizplus · Accepted Answer

The test statistic F is calculated by dividing the variance (square of the standard deviation) of one sample by the variance of the other. Here, $F = \frac{s_1^2}{s_2^2} = \frac{19.8^2}{22.2^2} = 0.8$. The critical value for a left-tailed test with $\alpha = 0.10$ and degrees of freedom $df_1 = 15$ and $df_2 = 12$ (not directly given but can be inferred from sample sizes $n_1$ and $n_2$) would need to be looked up in an F-distribution table. The provided critical values in the options are not directly verifiable without the table, but the decision to reject or not reject $H_0$ depends on whether the test statistic is less than the critical value for a left-tailed test. Since the test statistic is greater than the critical value given in option B, we do not reject $H_0$, making option B correct.

The Sample Standard Deviations and Sample Sizes Are Given for Independent