Obligations to Vote and Serve In the General Social Survey, individuals were asked whether civic duty included voting and whether it included serving on a jury. The results of the survey are shown in the table. Is there a difference in the proportion of individuals who feel jury duty is a civic duty and the proportion of individuals who feel voting is a civic duty? Use the α = 0.05 level of significance.

Verified step by step guidance

1. Step 1: Define the hypotheses for the test. The null hypothesis (H0) states that there is no difference in the proportion of individuals who feel jury duty is a civic duty and those who feel voting is a civic duty. The alternative hypothesis (H1) states that there is a difference in these proportions.
2. Step 2: Calculate the sample proportions for each category. For voting, calculate the proportion of individuals who consider voting a duty by dividing the number of 'Duty' responses by the total number of respondents. Similarly, calculate the proportion for jury duty.
3. Step 3: Use the data from the contingency table to set up a test for the difference between two proportions. The formula for the test statistic (z) is: \[z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}\] where \(\hat{p}_1\) and \(\hat{p}_2\) are the sample proportions, \(n_1\) and \(n_2\) are the sample sizes, and \(\hat{p}\) is the pooled proportion.
4. Step 4: Calculate the pooled proportion \(\hat{p}\) by combining the successes and total observations from both groups: \[\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}\] where \(x_1\) and \(x_2\) are the counts of successes in each group.
5. Step 5: Compute the test statistic using the formula from Step 3, then compare the calculated z-value to the critical z-value for \(\alpha = 0.05\) (two-tailed test). Based on this comparison, decide whether to reject or fail to reject the null hypothesis.

Key Concepts

Hypothesis Testing for Difference in Proportions

This concept involves comparing two population proportions to determine if there is a statistically significant difference between them. The null hypothesis typically states that the proportions are equal, while the alternative suggests a difference. Using sample data, a test statistic is calculated and compared to a critical value or p-value to decide whether to reject the null hypothesis at a given significance level (α).

Contingency Table and Joint Probability

A contingency table displays the frequency distribution of variables and helps analyze the relationship between categorical variables. In this question, the table shows counts of individuals who consider voting and jury duty as civic duties. Understanding how to interpret these counts and calculate proportions or joint probabilities is essential for conducting the hypothesis test.

Significance Level (α) and Decision Rule

The significance level, α, is the threshold for rejecting the null hypothesis, commonly set at 0.05. It represents the probability of making a Type I error—incorrectly rejecting a true null hypothesis. The decision rule compares the p-value from the test to α; if p-value ≤ α, reject the null hypothesis, indicating a significant difference in proportions.