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. Summer reading programs are very popular with children. At the Citizens Library, Team Ramona read an average of 23.2 books with a standard deviation of 6.1. There were 21 members on this team. Team Beezus read an average of 26.1 books with a standard deviation of 2.3 . There were 23 members on this team. Did the variances of the two teams differ? Use \(\alpha=0.05\).

Short answer

The variances of the two teams differ.

Step-by-step solution

State the Hypotheses

To determine if the variances of the two teams differ, we need to set up our null and alternative hypotheses. We are conducting an F-test for equality of variances. The null hypothesis is that the variances are equal, and the alternative hypothesis is that they are not equal. Thus: Null Hypothesis (H0): \( \sigma_1^2 = \sigma_2^2 \) Alternative Hypothesis (H1): \( \sigma_1^2 eq \sigma_2^2 \) The claim in this hypothesis test is that the variances do differ.

Find the Critical Value

The critical value is found from the F-distribution table using the degrees of freedom for the numerator and denominator. The degrees of freedom for Team Ramona (numerator) is \( df_1 = n_1 - 1 = 21 - 1 = 20 \), and for Team Beezus (denominator) is \( df_2 = n_2 - 1 = 23 - 1 = 22 \). Since we are conducting a two-tailed test with \( \alpha = 0.05 \), we split this into two tails, each with 0.025. The critical values can be found in the F-distribution table for these degrees of freedom.

Compute the Test Value

The test statistic for comparing two variances is calculated using the formula:\[ F = \frac{s_1^2}{s_2^2} \]where \( s_1 = 6.1 \) (standard deviation of Team Ramona) and \( s_2 = 2.3 \) (standard deviation of Team Beezus). Calculate the variances first:- \( s_1^2 = 6.1^2 = 37.21 \)- \( s_2^2 = 2.3^2 = 5.29 \)Then calculate the test statistic:\[ F = \frac{37.21}{5.29} \approx 7.03 \]

Make the Decision

Compare the calculated F-value to the critical values obtained from the F-distribution table. If the calculated F-value is greater than the critical value, then we reject the null hypothesis. If it is within the range of critical values, we do not reject the null hypothesis. Assuming a typical F-table critical value for \( df_1 = 20 \), \( df_2 = 22 \), and \( \alpha = 0.05 \) two-tailed, the typical critical F-values are approximately 2.69 and 0.372 (depending on tables, check specific F-table). Since \( 7.03 > 2.69 \), reject the null hypothesis.

Summarize the Results

Since we rejected the null hypothesis, we conclude there is significant evidence to say that the variances of the two teams differ. Therefore, we confirm the claim that the variances between Team Ramona and Team Beezus are not the same.

Key concepts

Null Hypothesis

The null hypothesis is a fundamental concept in hypothesis testing, functioning as a starting assumption used to determine if there is enough evidence to make a significant conclusion. In the context of variance comparison, the null hypothesis posits that the variances between two groups are equal. Mathematically, this is expressed as: \[ H_0: \sigma_1^2 = \sigma_2^2 \] where \(\sigma_1^2\) and \(\sigma_2^2\) represent the population variances of the two groups in question. The null hypothesis essentially states that any observed difference in sample variances is due to random sampling errors rather than a real difference. If testing within the framework of the null hypothesis results in its rejection, we then consider the alternative hypothesis true, suggesting that the variances do truly differ. In our exercise, the claim is oriented towards the variances differing, thus making our alternative hypothesis: \[ H_1: \sigma_1^2 eq \sigma_2^2 \] The decision to either accept or reject this null hypothesis hinges on the results of testing components such as the critical value and the test statistic.

F-test

The F-test is a type of statistical test that is employed when comparing two variances. It determines whether two datasets have significantly different variances by computing an F-statistic, which is based on the ratio of two sample variances. The formula used for calculating the F-statistic is: \[ F = \frac{s_1^2}{s_2^2} \] where

• \(s_1^2\) is the variance of the first group or sample.
• \(s_2^2\) is the variance of the second group or sample.

For the F-test to be applicable, it is crucial that the underlying data follows a normal distribution and the samples are independent. The calculated F-statistic is then compared against critical values from the F-distribution table, which depend on the degrees of freedom associated with each variance (degree of freedom is typically the number of data points minus one). In our context, the F-test assists in understanding whether the difference in variances between Team Ramona and Team Beezus is significant or not.

Critical Value

In hypothesis testing, the critical value is a threshold that the test statistic is compared against to decide whether to reject the null hypothesis. It defines the point beyond which the null hypothesis is unlikely to hold, given the significance level \(\alpha\). In our specific case of a two-tailed F-test with a significance level of \(\alpha = 0.05\), the critical values are obtained based on the degrees of freedom (df) from each sample. For instance, if Team Ramona has a df of 20, and Team Beezus has a df of 22, we use an F-distribution table to find these critical values. Typically, the critical values determine the boundaries of the acceptance region:

• If the calculated F-statistic exceeds the upper critical value, we reject the null hypothesis.
• If it falls below the lower critical value in a two-tailed test, we also reject the null hypothesis.

In our example exercise, assuming typical critical values derived from common F-tables, would place them around 2.69 and 0.372, determining the decision boundary for the test outcome.

Variances Comparison

Comparing variances between two datasets can reveal if there is statistical significance in their difference. This is often a crucial step in establishing whether changes or differences are real or mere artifacts of random sampling. When conducting an F-test for variances comparison, it is important to first standardize the assumptions. These include:

• Both data sets should follow a normal distribution.
• The samples are assumed to be independent.
• Homogeneity of variances, ensuring the variances are representative.

In practical terms, after calculating the variances of each dataset and subsequently the F-statistic, comparing this statistic against the critical values determines the outcome. If the F-statistic falls outside the range defined by these critical values, this suggests that the observed variance differences are not due to chance, thus implying a significant difference in the population variances. In the context of our exercise, applying these principles demonstrated significant differences in the variances between the books read by Team Ramona and Team Beezus, leading to the rejection of the null hypothesis.