Given the following information about a hypothesis, and the test of the difference between two variances based on independent random samples, what is the critical value of the test statistic at a significance level of 0.05 ? Assume that the samples are obtained from normally distributed populations. \( H_{A}: \sigma^{2} \mathrm{~A}>\sigma^{2} \mathrm{~B}, \overline{\mathrm{X}_{1}}=12, \overline{\mathrm{X}_{2}}=9, \mathrm{~S}_{1}=5, \mathrm{~S}_{2}=3 \), \( =13, \mathrm{n}_{2}=10 \).

To determine the critical value of the test statistic for comparing two variances with a significance level of 0.05 under the hypothesis \( H_{A}: \sigma^{2} \mathrm{~A}>\sigma^{2} \mathrm{~B} \), you will need to calculate the degrees of freedom for both samples and then refer to an F-distribution table. Given: - Sample 1: \( n_{1} = 13 \), \( S_{1}^{2} = 5^{2} = 25 \) - Sample 2: \( n_{2} = 10 \), \( S_{2}^{2} = 3^{2} = 9 \) The degrees of freedom are calculated as follows: - \( df_{1} = n_{1} - 1 = 13 - 1 = 12 \) - \( df_{2} = n_{2} - 1 = 10 - 1 = 9 \) Looking up the critical value for \( F(df_{1}, df_{2}) \) at a significance level of 0.05 in an F-distribution table, you will find \( F(12, 9) \). The critical value generally found in tables or calculated using statistical software will be approximately 3.063. Thus, the critical value of the test statistic at a significance level of 0.05 is around **3.063**. Remember that since this is a one-tailed test, you are specifically looking for values in the upper tail of the distribution.