$\mathrm { s } _ { 1 } = 20.4 , \mathrm { n } _ { 1 } = 16 , \mathrm {~s} _ { 2 } = 21.9 , \mathrm { n } _ { 2 } = 13$ left-tailed test, $\alpha$ =0.10 Test the hypothesis that $\sigma _ { 1 }

Question

$\mathrm { s } _ { 1 } = 20.4 , \mathrm { n } _ { 1 } = 16 , \mathrm {~s} _ { 2 } = 21.9 , \mathrm { n } _ { 2 } = 13$ left-tailed test, $\alpha$ =0.10 Test the hypothesis that $\sigma _ { 1 } < \sigma _ { 2 } \text { at the } \alpha = 0.10$ level of significance for the given sample data. $\begin{array} { c | c c } & \text { Population 1 } & \text { Population 2 } \ \hline \mathrm { n } & 16 & 13 \ \mathrm {~s} & 20.4 & 21.9 \end{array}$ A) Test statistic: F=0.87 . Critical value =0.495 . Do not reject H₀ B) Test statistic: F=0.87 . Critical value =0.476 . Do not reject H₀ C) Test statistic: F=0.87 . Critical value =0.495 . Reject H₀ D) Test statistic: F=0.87 . Critical value =0.476 . Reject H₀

Quizplus · Accepted Answer

The test statistic F is calculated as the ratio of the two variances (squared standard deviations), so $F = \frac{s_1^2}{s_2^2} = \frac{20.4^2}{21.9^2} = 0.87$. For a left-tailed test at $\alpha = 0.10$, we compare the F value to the critical value from the F-distribution table for the given degrees of freedom ($n_1 - 1 = 15$ and $n_2 - 1 = 12$). The correct critical value might vary slightly depending on the source, but if the provided critical value is 0.495 and the test statistic is 0.87, which is greater than the critical value, we do not reject the null hypothesis that $\sigma_1$ is less than or equal to $\sigma_2$. This means there isn't enough evidence to conclude that $\sigma_1 < \sigma_2$ at the 0.10 level of significance.

s1=20.4,n1=16, s2=21.9,n2=13\mathrm { s } _ { 1 } = 20.4 , \mathrm { n } _ { 1 } = 16 , \mathrm {~s} _ { 2 } = 21.9 , \mathrm { n } _ { 2 } = 13s1​=20.4,n1​=16, s2​=21.9,n2​=13

$\mathrm { s } _ { 1 } = 20.4 , \mathrm { n } _ { 1 } = 16 , \mathrm {~s} _ { 2 } = 21.9 , \mathrm { n } _ { 2 } = 13$