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Chapter 4: Problem 12

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. $$\begin{array}{lrrr} & \text { Male } & \text { Female } & \text { Total } \\\\\hline \text { Very happy } & 7,609 & 7,942 & 15,551 \\\\\hline \text { Pretty happy } & 3,738 & 4,447 & 8,185 \\\\\hline \text { Not too happy } & 259 & 460 & 719 \\\\\hline \text { Total } & 11.606 & 12.849 & 24.455\end{array}$$

Short Answer

Expert verified
Perform a Chi-square test to check if the calculated value exceeds the critical value (5.991). If it does, gender and happiness in marriage are associated.

Step by step solution

01

Set up null and alternative hypothesis

We need to determine if gender is associated with happiness in marriage. Set up the null hypothesis: \( H_0: \text{There is no association between gender and happiness in marriage.} \) and the alternative hypothesis: \( H_A: \text{There is an association between gender and happiness in marriage.} \)
02

Calculate expected frequencies

Using the formula for expected frequency \( E_{ij} = \frac{(\text{Row total}) \times (\text{Column total})}{\text{Grand total}} \), calculate the expected frequency for each cell in the table: * For Very Happy Males: \( E_{11} = \frac{(11,606)(15,551)}{24,455} = 7,369.55 \) * Continue this for all other cells.
03

Perform Chi-square test

Use the formula for Chi-square: \[ \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}} \] where \( O_{ij} \) is the observed frequency and \( E_{ij} \) is the expected frequency. Calculate \( \chi^2 \) by summing up the values for each cell.
04

Determine degrees of freedom

Calculate the degrees of freedom using the formula: \[ \text{df} = (\text{number of rows} - 1)(\text{number of columns} - 1) \] For this case: \[ \text{df} = (3-1)(2-1) = 2 \]
05

Find the critical value and compare

Use a Chi-square distribution table to find the critical value for \( \alpha = 0.05 \) and \( df = 2 \). The critical value is approximately 5.991. Compare the calculated \( \chi^2 \) value to this critical value.
06

Make a decision

If \( \chi^2 \) calculated is greater than the critical value, reject the null hypothesis. Otherwise, do not reject the null hypothesis. This will tell us if there is a statistically significant association between gender and happiness in marriage.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

hypothesis testing
The foundation of a Chi-square test for independence is hypothesis testing. We begin by setting up our hypotheses: the null hypothesis (\(H_0\)) states that there is no association between the two categorical variables (gender and happiness in marriage), while the alternative hypothesis (\(H_A\)) suggests that there is an association. By assessing these hypotheses, we aim to determine if observed differences are due to chance or indicate a real relationship. Properly forming our hypotheses is crucial because it frames our entire analysis and conclusion.
expected frequency
Once our hypotheses are set, we calculate the expected frequencies for each cell in our contingency table using the formula: \( E_{ij} = \frac{(\text{Row total}) \times (\text{Column total})}{\text{Grand total}} \). Expected frequencies are what we anticipate under the null hypothesis, implying no association between the variables. These values help us compare what we actually observe with what we would expect by chance, providing a baseline for our Chi-square calculation. Calculating expected frequencies accurately is essential as it directly impacts the Chi-square value.
degrees of freedom
Degrees of freedom (df) in a Chi-square test represent the number of values in the final calculation that are free to vary. The formula is: \( \text{df} = (\text{number of rows} - 1)(\text{number of columns} - 1) \). For our example with three happiness levels and two genders, \( \text{df} = (3-1)(2-1) = 2 \). Knowing the degrees of freedom helps us determine the correct critical value from the Chi-square distribution table, which is needed for the final comparison step. More categories generally mean more degrees of freedom and thus a higher threshold for significance.
critical value
The critical value is a threshold we compare our Chi-square statistic against to make a decision regarding the null hypothesis. Using the Chi-square distribution table, we find the critical value for a chosen significance level (\( \alpha \), often 0.05) and our calculated degrees of freedom. For \( \alpha = 0.05 \) and \( df = 2 \), the critical value is approximately 5.991. If our Chi-square statistic exceeds this value, we reject the null hypothesis. The critical value helps in determining whether the observed data differ significantly from the expected data, signifying an association between variables.
statistical significance
Statistical significance tells us whether the observed association in our sample could be due to chance alone. We achieve this by comparing our Chi-square statistic to the critical value. If our calculated statistic surpasses the critical value, we conclude that the association is statistically significant, meaning it is unlikely to have occurred by random chance. This reinforces our decision to reject or fail to reject the null hypothesis. Understanding statistical significance is key in interpreting the results of hypothesis tests and making informed conclusions about the data.

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