Questions: Construct the confidence interval for the ratio of the population variances given the following sample statistics. Round your answers to four decimal places. n1=12, n2=10, s1^2/s2^2=1.72, α=0.01

Construct the confidence interval for the ratio of the population variances given the following sample statistics. Round your answers to four decimal places. n1=12, n2=10, s1^2/s2^2=1.72, α=0.01
Transcript text: Construct the confidence interval for the ratio of the population variances given the following sample statistics. Round your answers to four decimal places. \[ n_{1}=12, n_{2}=10, \frac{s_{1}^{2}}{s_{2}^{2}}=1.72, \alpha=0.01 \]
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Solution

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Solution Steps

Step 1: Calculate the F-Test Statistic

The F-test statistic is calculated as follows:

\[ F = \frac{s_1^2}{s_2^2} = \frac{1.72}{1} = 1.72 \]

Step 2: Determine the Critical Value

The critical value for the F-distribution at a significance level of \(\alpha = 0.01\) with degrees of freedom \(df_n = 11\) and \(df_d = 9\) is:

\[ F_{critical} = F(0.01, 11, 9) = 5.1779 \]

Step 3: Calculate the P-Value

The p-value is calculated using the formula:

\[ P = 2 \times \min(F(F_{stat}, df_n, df_d), 1 - F(F_{stat}, df_n, df_d)) = 0.4245 \]

Step 4: Calculate the Confidence Interval for the Variance

The confidence interval for the variance of the first population is given by:

\[ CI = \left(\frac{(n_1 - 1)s_1^2}{\chi^2_{\alpha/2}}, \frac{(n_1 - 1)s_1^2}{\chi^2_{1 - \alpha/2}}\right) \]

Substituting the values:

\[ CI = \left(\frac{(12 - 1) \times 1.72}{\chi^2_{0.005}}, \frac{(12 - 1) \times 1.72}{\chi^2_{0.995}}\right) \]

Calculating the endpoints yields:

\[ CI = (0.7071, 7.2679) \]

Final Answer

Lower endpoint: \(\boxed{0.7071}\)

Upper endpoint: \(\boxed{7.2679}\)

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