Question

Perform the following steps. Assume that all variables are normally distributed. a. State the hypotheses and identify the claim. b. Find the critical value. c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. The yearly tuition costs in dollars for random samples of medical schools that specialize in research and in primary care are listed. At \(\alpha=0.05,\) can it be concluded that a difference between the variances of the two groups exists? $$ \begin{array}{lll|lll} &&{\text { Research }} & {\text { Primary care }} \\ \hline 30,897 & 34,280 & 31,943 & 26,068 & 21,044 & 30,897 \\ 34,294 & 31,275 & 29,590 & 34,208 & 20,877 & 29,691 \\ 20,618 & 20,500 & 29,310 & 33,783 & 33,065 & 35,000 \\ 21,274 & & & 27,297 & & \end{array} $$

Short answer

No conclusive evidence for a variance difference at \(\alpha=0.05\).

Step-by-step solution

State the Hypotheses and Identify the Claim

We want to test if there is a difference between the variances of the tuition costs for medical schools specializing in research and primary care. The null hypothesis \(H_0\) is that there is no difference in variances, \( \sigma_1^2 = \sigma_2^2 \), and the alternative hypothesis \(H_1\) is that there is a difference, \( \sigma_1^2 eq \sigma_2^2 \). This is a two-tailed test.

Find the Critical Value

Since we are conducting a test for the equality of variances, we use the F-distribution. We need the degrees of freedom for each group: \( df_1 = n_1 - 1 \) and \( df_2 = n_2 - 1 \). The critical value is found using a significance level of \( \alpha = 0.05 \) for a two-tailed test. We look up the F-distribution table for the appropriate \( df \) and \( \alpha/2 \).

Calculate the Degrees of Freedom and Sample Sizes

First, compute the sample sizes: \( n_1 = 8 \) for research and \( n_2 = 9 \) for primary care. Therefore, the degrees of freedom are \( df_1 = 8 - 1 = 7 \) and \( df_2 = 9 - 1 = 8 \).

Calculate the Sample Variances

Calculate the sample variances for each group. For Research (\( s_1^2 \)): Find the mean and then the variance of the given values. Repeat the same for Primary Care (\( s_2^2 \)). Use the formula for variance: \[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} \]

Compute the Test Value

The test statistic for the F-test is given by \( F = \frac{s_1^2}{s_2^2} \). Calculate the F-value using the sample variances from the previous step, ensuring that the larger variance is in the numerator.

Make the Decision

Compare the calculated F-value with the critical F-value from Step 2. If the test statistic exceeds the critical value or is less than the inverse of the critical value, reject the null hypothesis. Otherwise, do not reject \(H_0\).

Summarize the Results

Based on the decision in Step 6, draw a conclusion about the hypotheses. If \(H_0\) is rejected, it suggests that there is evidence of a difference in variances of tuition costs between the two groups. Otherwise, there is not enough evidence to suggest a difference.

Key concepts

Understanding the Two-Tailed Test

A two-tailed test is used in hypothesis testing when we want to detect differences that could exist in both directions between two groups. In the context of this example, we want to test if there is a difference in variances of tuition costs between research-focused and primary care-focused medical schools.

This test checks both possibilities: that one variance could be larger or smaller than the other. The null hypothesis (\(H_0\)) suggests no difference in variances between the groups (\(\sigma_1^2 = \sigma_2^2\)), while the alternative hypothesis (\(H_1\)) suggests a difference (\(\sigma_1^2 eq \sigma_2^2\)).

Because it entertains both directions of difference, the significance level (\(\alpha\)) is typically split between the two tails of the distribution.

• Useful when differences can occur on either side of the model or prediction.
• Provides a more comprehensive evaluation, ensuring that differences are not overlooked.
• Aids in identifying either an increase or a decrease across the analyzed groups.

The Role of the F-Distribution in Variance Testing

The F-distribution is crucial when comparing variances between two groups, as it helps determine whether there is a statistically significant difference between them. It's especially useful when the data involve ratios, such as comparing variances.

In our exercise, since we are comparing variances, we use the F-test. The F-distribution is asymmetrical, skewing right, and depends on the degrees of freedom of both datasets compared (df₁ and df₂).

The critical value from the F-distribution table assists us in deciding whether to reject or not reject the null hypothesis.

• Tailored for testing differences in variance.
• Helps in drawing reliable conclusions from the data.
• It requires a specific critical value for decision-making.
• Used in combination with degrees of freedom for each group.

Calculating Sample Variance

Sample variance is a measure of how much the data points in a sample differ from the sample mean. In hypothesis testing, especially when using the F-distribution, knowing the sample variance is essential.

To calculate sample variance (\(s^2\)), we find the mean of all sample values, subtract this mean from each data point, square the result, sum these squared differences, and then divide by the number of data points minus one: \[s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}\]

Accurately computing variance ensures the reliability of our hypothesis test steps.

• Essential for analyzing variability within a sample.
• Provides a foundation for the F-test calculation.
• Vital in determining if differences are notably significant.
• Ensures that hypothesis test results are based on sound statistical practice.

Degrees of Freedom Explained

Degrees of freedom (df) refer to the number of independent values or quantities that can vary in an analysis without breaking constraints. In the context of performing an F-test, degrees of freedom help in locating the correct critical value from the F-distribution table.

For each group being compared, the degrees of freedom are calculated as the number of data points minus one (\(df = n - 1\)). In the given example:

• Research group: \(df_1 = 8 - 1 = 7\)
• Primary care group: \(df_2 = 9 - 1 = 8\)

Choosing the correct degrees of freedom is crucial for accurately finding the critical value.

Degrees of freedom reflect the amount of independent statistical information available. More degrees of freedom typically mean more reliable statistical analysis.

• Influence the precision of statistical estimates.
• Ensure valid critical value retrieval for hypothesis testing.
• Indicative of sample size robustness.
• Underlying calculations affect result interpretation significantly.