91Ó°ÊÓ

From two normal populations with respective variances \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2},\) we observe independent sample variances \(S_{1}^{2}\) and \(S_{2}^{2}\), with corresponding degrees of freedom \(\nu_{1}=n_{1}-1\) and \(\nu_{2}=n_{2}-1 .\) We wish to test \(H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2}\) versus \(H_{a}: \sigma_{1}^{2} \neq \sigma_{2}^{2}\) a. Show that the rejection region given by $$\left\\{F>F_{\nu_{2}, \alpha / 2}^{\nu_{1}} \quad \text { or } \quad F<\left(F_{\nu_{1}, \alpha / 2}^{\nu_{2}}\right)^{-1}\right\\}$$ where \(F=S_{1}^{2} / S_{2}^{2},\) is the same as the rejection region given by $$\left\\{S_{1}^{2} / S_{2}^{2}>F_{\nu_{2}, \alpha / 2}^{\nu_{2}} \text { or } S_{2}^{2} / S_{1}^{2}>F_{\nu_{1}, \alpha / 2}^{\nu_{2}}\right\\}$$ b. Let \(S_{L}^{2}\) denote the larger of \(S_{1}^{2}\) and \(S_{2}^{2}\) and let \(S_{S}^{2}\) denote the smaller of \(S_{1}^{2}\) and \(S_{2}^{2} .\) Let \(\nu_{L}\) and \(\nu_{S}\) denote the degrees of freedom associated with \(S_{L}^{2}\) and \(S_{S}^{2}\), respectively. Use part (a) to show that, under \(H_{0}\) $$\mathrm{P}\left(S_{L}^{2} / S_{S}^{2}>F_{\nu_{S}, \alpha / 2}^{\mu_{L}}\right)=\alpha$$ Notice that this gives an equivalent method for testing the equality of two variances.

Step by step solution

01

Understand the Rejection Region

The rejection region for testing \(H_0: \sigma_1^2 = \sigma_2^2\) is given by two conditions: \(F > F_{u_2, \alpha/2}^{u_1}\) or \(F < (F_{u_1, \alpha/2}^{u_2})^{-1}\), where \(F = \frac{S_1^2}{S_2^2}\). This tests for the variability by considering the ratio of sample variances.

02

Demonstrate Equivalence of the Rejection Regions

The given condition \(F > F_{u_2, \alpha/2}^{u_1}\) implies that the sample variance \(S_1^2\) is significantly larger than \(S_2^2\). Alternatively, \(F < (F_{u_1, \alpha/2}^{u_2})^{-1}\) implies \(S_1^2\) is much smaller than \(S_2^2\). Thus, testing the ratios \(S_1^2 / S_2^2\) and \(S_2^2 / S_1^2\) lead to rejection conditions: \(S_1^2/S_2^2 > F_{u_2, \alpha/2}^{u_2}\) or \(S_2^2/S_1^2 > F_{u_1, \alpha/2}^{u_2}\).

03

Set Up the Problem for Part (b)

Given \(S_L^2\) as the larger variance and \(S_S^2\) as the smaller, we need to consider \(F = \frac{S_L^2}{S_S^2}\). Under \(H_0\), the expectation is that the larger variance divided by the smaller still falls under the null hypothesis bounds.

04

Show Equivalence for Larger and Smaller Sample Variances

Since \(S_L^2 = \max(S_1^2, S_2^2)\) and \(S_S^2 = \min(S_1^2, S_2^2)\), part (a)'s rejection region \(\left\{F > F_{u_2, \alpha/2}^{u_1} \text{ or } F < (F_{u_1, \alpha/2}^{u_2})^{-1}\right\}\) is used to test \(F = S_L^2 / S_S^2\). The critical value \(F_{u_S, \alpha/2}^{u_L}\) shifts the decision threshold for \(S_L^2 / S_S^2 > F_{u_S, \alpha/2}^{u_L}\) when \(\alpha\) is the level of significance.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

F-test

The F-test is a type of statistical hypothesis test that is used to compare the variances of two populations. Variance measures the spread or dispersion of a set of data. The F-test specifically evaluates whether two population variances are equal.

When performing an F-test, you'll usually start with formulating the null hypothesis (\(H_0\) ): that the two variances are equal \(\sigma_{1}^{2} = \sigma_{2}^{2}\). The alternative hypothesis (\(H_a\) ) would state that the variances are not equal, \(\sigma_{1}^{2} eq \sigma_{2}^{2}\). This setup helps in understanding whether the observed sample variances provide enough evidence to infer that population variances are different.

The F-statistic is calculated by taking the ratio of two sample variances: \(F = \frac{S_{1}^{2}}{S_{2}^{2}}\). Under the null hypothesis assumption, if the two variances are equal, the F-statistic should be close to 1.

For significance testing, we compare this calculated F-value with a critical value from the F-distribution table, based on the specified level of significance (\(\alpha\)) and degrees of freedom.

sample variance

Sample variance is a statistic used to estimate the variance of a population. It's calculated from a sample rather than the entire population, making it a more practical measure when dealing with large data sets. To compute the sample variance, we take each data value's deviation from the sample mean, square it, then find the average of these squared deviations.

Mathematically, the formula for sample variance (\(S^2\) ) is:
\[S^{2} = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2\]where:

• \(x_i\) is each individual data point
• \(\bar{x}\) is the sample mean
• \(n\) is the number of data points in the sample

The subtraction of 1 from \(n\) in the denominator is due to Bessel's correction, which corrects the bias in the estimation of the population variance and thus results in a more accurate estimate.

Understanding sample variance is crucial in executing the F-test, as it forms the foundation for calculating the F-statistic.

degrees of freedom

Degrees of freedom refer to the number of independent values that can vary in an analysis without breaking any constraints. In statistical tests, they are crucial in determining the shape of probability distributions and are often used to locate the critical value threshold for tests like the F-test.

In the context of sample variance, degrees of freedom is calculated as:\(u = n - 1\), where \(n\) is the sample size. Essentially, degrees of freedom adjust the final variance estimate, accounting for the fact that the sample mean estimation constrains one piece of data.

Using degrees of freedom \(u_1\) and \(u_2\) for two sample variances \(S_1^{2}\) and \(S_2^{2}\) is significant in determining the F-distribution for the F-test. Each variance has its own degrees of freedom since it is estimated from separate datasets. These values help in finding the appropriate critical values from the F-table, specific to the test being conducted. Without accurately accounting for degrees of freedom, statistical tests could lead to incorrect conclusions.