[DATA] Health and Happiness Are health and happiness related? The following data represent the level of happiness and level of health for a random sample of individuals from the General Social Survey.

a. Does the evidence suggest that health and happiness are related? Use the level of significance.

Verified step by step guidance

1. Step 1: Identify the type of test needed. Since we want to determine if there is a relationship between two categorical variables (Health and Happiness), we use the Chi-Square Test of Independence.
2. Step 2: Set up the hypotheses. The null hypothesis (\(H_0\)) states that health and happiness are independent (no relationship), and the alternative hypothesis (\(H_a\)) states that health and happiness are related.
3. Step 3: Calculate the expected counts for each cell in the contingency table using the formula: \[\text{Expected Count} = \frac{(\text{Row Total}) \times (\text{Column Total})}{\text{Grand Total}}\]
4. Step 4: Compute the Chi-Square test statistic using the formula: \[\chi^2 = \sum \frac{(\text{Observed} - \text{Expected})^2}{\text{Expected}}\] where the sum is over all cells in the table.
5. Step 5: Determine the degrees of freedom for the test, which is given by: \[df = (\text{number of rows} - 1) \times (\text{number of columns} - 1)\] Then, compare the calculated \(\chi^2\) statistic to the critical value from the Chi-Square distribution table at the given significance level, or find the p-value. If the test statistic exceeds the critical value or the p-value is less than the significance level, reject the null hypothesis and conclude that health and happiness are related.

Key Concepts

Contingency Table

A contingency table displays the frequency distribution of variables to analyze the relationship between categorical data. In this case, it shows counts of individuals classified by happiness levels and health status, allowing us to observe patterns or associations between these two variables.

Chi-Square Test of Independence

This statistical test determines whether two categorical variables are independent or related. It compares observed frequencies in the contingency table to expected frequencies under the assumption of independence, helping to assess if health and happiness are associated.

Level of Significance (α)

The level of significance is the threshold for deciding whether to reject the null hypothesis, commonly set at 0.05. It represents the probability of making a Type I error, or falsely concluding a relationship exists when it does not. The test result is compared to this level to draw conclusions.