Question

Random samples of size n1 = 21 and n2 = 31 were drawn from populations 1 and 2, respectively. For each of the following cases, determine the test statistic and appropriate decision for a two-tailed test of the null hypothesis \({H_0}:\sigma _1^2 = \sigma _2^2\) against \({H_a}:\sigma _1^2 \ne \sigma _2^2\,at\,\alpha = 0.05\)

1. \(s_1^2 = 500\,and\;s_2^2 = 900\)

Short answer

1. The test statistic is 1.8.
   We fail to reject the null hypothesis.

Step-by-step solution

Given Information

The hypothesis are given by

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

The level of significance is 0.05.

Compute the test statistic

The test statistic is calculated as

\(\begin{aligned}F &= \frac{{s_2^2}}{{s_1^2}}\\ &= \frac{{900}}{{500}}\\ &= \frac{9}{5}\\ &= 1.8\end{aligned}\)

Therefore, the test statistic is 1.8.

Decision-rule

The rejection region for two tailed test is

\(F > {F_{\frac{\alpha }{2}}}\)

For

\(\begin{aligned}{F_{\frac{\alpha }{2}}}\left( {{n_1} - 1,{n_2} - 1} \right)\\ = {F_{0.025}}\left( {20,30} \right)\end{aligned}\)

Critical value: 0.4258 and 2.1952

Hence, the statistic doesn’t fall in rejection region.

Therefore, we fail to reject the null hypothesis.