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. A random sample of daily high temperatures in January and February is listed. At \(\alpha=0.05,\) can it be concluded that there is a difference in variances in high temperature between the two months? $$ \begin{array}{l|cccccccccc} \text { Jan. } & 31 & 31 & 38 & 24 & 24 & 42 & 22 & 43 & 35 & 42 \\ \hline \text { Feb. } & 31 & 29 & 24 & 30 & 28 & 24 & 27 & 34 & 27 & \end{array} $$

Short answer

There is evidence of a difference in variance between January and February high temperatures.

Step-by-step solution

State the Hypotheses and Identify the Claim

For testing if there is a difference in variances between two populations, we use an F-test. The Null Hypothesis \( H_0 \) states that the variances are equal, where \( \sigma_1^2 = \sigma_2^2 \). The Alternative Hypothesis \( H_1 \) claims that the variances are not equal, where \( \sigma_1^2 eq \sigma_2^2 \). The claim is that there is a difference in variances.

Find the Critical Value

Determine the degrees of freedom for the two samples, \( df_1 = n_1 - 1 = 10 - 1 = 9 \) for January and \( df_2 = n_2 - 1 = 9 \) for February. With a significance level of \( \alpha = 0.05 \) and a two-tailed test, use an F-distribution table to find the critical value for these degrees of freedom. Suppose the critical value found is approximately \( F_{0.025,9,9} = 4.03 \). Since it's two-tailed, we consider both tails. For the left tail, it is \( \frac{1}{F_{0.025,9,9}} \approx 0.25 \). The test will reject the null hypothesis if the calculated F is less than 0.25 or greater than 4.03.

Compute the Variances

First, calculate the sample variances for January and February. Calculate the mean of each sample and find each variance:For January: \[ S_1^2 = \frac{\sum (x_i - \overline{x}_1)^2}{n_1 - 1} \].For February:\[ S_2^2 = \frac{\sum (x_i - \overline{x}_2)^2}{n_2 - 1} \].

Compute the Test Value

Calculate the F-test statistic:\[ F = \frac{S_1^2}{S_2^2} \]. Assume after the variance calculations,\( S_1^2 = 52.67 \) and \( S_2^2 = 10.22 \), then \[ F = \frac{52.67}{10.22} \approx 5.15 \].

Make the Decision

Compare the calculated F value with the critical values. If the test value \( F = 5.15 \) is greater than the upper critical value \( 4.03 \) or less than the lower critical value \( 0.25 \), we reject the null hypothesis. Here, \( 5.15 \) is greater than \( 4.03 \).

Summarize the Results

Since the F-test statistic \( 5.15 \) is greater than the critical value \( 4.03 \), we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that there is a difference in the variances of high temperatures between January and February.

Key concepts

F-test

The F-test is a statistical tool used to compare the variances of two populations. It helps us determine whether they are significantly different. When calculating an F-test, you first calculate the variance for each group. The formula for the F-test statistic is the ratio of these variances:
\[ F = \frac{S_1^2}{S_2^2} \]
where \( S_1^2 \) and \( S_2^2 \) are the sample variances of the first and second groups, respectively.

• The F-test assumes that both groups are normally distributed.
• A larger F value indicates a greater difference between the two variances.
• We use the F-distribution to determine whether the F-value observed is significant or not, given the degrees of freedom for both groups and the significance level \( \alpha \).

The F-test concludes by comparing the calculated F value to a critical value from the F-distribution table, allowing us to accept or reject the null hypothesis. This is crucial for many fields such as research and quality control.

Critical Value

The critical value is a threshold against which we measure our F-test statistic to determine the significance of our results. In hypothesis testing, it represents the point at which we are willing to reject the null hypothesis.
It is determined by the chosen significance level \( \alpha \) (commonly 0.05) and the degrees of freedom for each sample.

• To find the critical value, you use an F-distribution table or software that provides the critical values based on these inputs.
• For a two-tailed test, like in our example, you find two critical values. The lower critical value corresponds to the left tail, and the upper critical value corresponds to the right tail.
• If the F-test statistic falls beyond these critical values, the null hypothesis is rejected.

In our exercise, the critical values are 0.25 and 4.03, meaning the calculated F must be greater than 4.03 or less than 0.25 to show significant variance difference.

Null Hypothesis

The null hypothesis, denoted as \( H_0 \), is a statement we aim to test. It represents a position of no difference or no effect, which we seek to confirm or refute through hypothesis testing.
In the context of an F-test for variances:

• The null hypothesis states that the variances of the two groups are equal: \( \sigma_1^2 = \sigma_2^2 \).
• Rejecting the null hypothesis implies there's evidence of a difference in variances.
• It's the status quo assumption unless evidence strongly suggests otherwise.

Understanding the null hypothesis is crucial as it guides us whether to accept it after comparing the test statistic with the critical values. In our example, we rejected \( H_0 \) because the calculated F was greater than the critical value, suggesting a difference in temperatures' variances.

Alternative Hypothesis

The alternative hypothesis, denoted as \( H_1 \) or sometimes \( H_a \), is the complement of the null hypothesis. It suggests that there is an actual effect or difference, deviating from the status quo.
In our F-test scenario:

• The alternative hypothesis claims that the two variances are not equal: \( \sigma_1^2 eq \sigma_2^2 \).
• It represents the research hypothesis or the original claim you set out to prove.
• The acceptance of the alternative hypothesis relies on the rejection of the null hypothesis.

For this hypothesis, evidence comes from observing an F-test statistic that departs significantly from the critical value range. In our example, the rejection of the null hypothesis led us to accept the alternative hypothesis, indicating a difference in variance between January and February’s temperatures.