Question

Specify the appropriate rejection region for testing \({H_0}:\sigma _1^2 = \sigma _2^2\) in each of the following situations:

1. \({H_a}:\sigma _1^2 > \sigma _2^2\,\alpha = .05,\,{n_1} = 25,{n_2} = 20\)

Short answer

1. The rejection region for right tailed test is given by \(F > 2.1141\)

Step-by-step solution

Given Information

The hypothesis are given by

\(\begin{aligned}{H_0}:\sigma _1^2 = \sigma _2^2\\{H_a}:\sigma _1^2 > \sigma _2^2\end{aligned}\)

The level of significance is 0.05

Compute degrees of freedom

\(\begin{aligned}{v_1} &= {n_1} - 1\\ &= 25 - 1\\ &= 24\\{v_2} &= {n_2} - 1\\ &= 20 - 1\\ &= 19\end{aligned}\)

Test statistic

The test statistic is computed as

\(F = \frac{{s_1^2}}{{s_2^2}}\)

Critical value

For,

\(\alpha = 0.05\,and\,{v_1} = 24\;and\,{v_2} = 19\)

The critical value is

\({F^ \bullet } = {F_{0.05}}\left( {24,19} \right) = 2.1141\)

Decision-rule

The rejection region for right tailed test is given by

\(\begin{aligned}F > {F^ \bullet }\\F > 2.1141\end{aligned}\)