Happy in Your Marriage? The General Social Survey asks questions about one’s happiness in marriage. Is there an association between gender and happiness in marriage? Use the data in the table to determine if gender is associated with happiness in marriage. Treat gender as the explanatory variable.

Verified step by step guidance

1. Step 1: Identify the variables and the type of test needed. Here, gender (Male, Female) is the explanatory variable, and happiness in marriage (Very happy, Pretty happy, Not too happy) is the response variable. Since both variables are categorical, we will use a Chi-square test of independence to check for an association.
2. Step 2: State the hypotheses. The null hypothesis (H0) is that gender and happiness in marriage are independent (no association). The alternative hypothesis (H1) is that there is an association between gender and happiness in marriage.
3. Step 3: Calculate the expected counts for each cell in the table under the assumption that gender and happiness are independent. Use the formula for each cell: \(E_{ij} = \frac{(\text{row total}_i)(\text{column total}_j)}{\text{grand total}}\) where \(E_{ij}\) is the expected count for the cell in row \(i\) and column \(j\).
4. Step 4: Compute the Chi-square test statistic using the formula: \(\chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}\), where \(O_{ij}\) is the observed count and \(E_{ij}\) is the expected count for each cell.
5. Step 5: Determine the degrees of freedom for the test, which is \((\text{number of rows} - 1) \times (\text{number of columns} - 1)\), and then compare the calculated Chi-square statistic to the critical value from the Chi-square distribution table or use a p-value to decide whether to reject the null hypothesis.

Key Concepts

Contingency Tables

A contingency table displays the frequency distribution of variables and helps examine the relationship between categorical variables. In this case, the table shows counts of males and females across different happiness levels, allowing us to observe patterns or associations between gender and marital happiness.

Chi-Square Test of Independence

This statistical test determines whether there is a significant association between two categorical variables. By comparing observed counts in the table to expected counts under the assumption of independence, we can assess if gender and happiness in marriage are related or independent.

Explanatory and Response Variables

In analyzing associations, the explanatory variable is the one that may influence or explain changes in another variable, called the response variable. Here, gender is the explanatory variable, and happiness in marriage is the response variable, guiding how we interpret the relationship.