[NOW WORK] Job Satisfaction Is there an association between one’s level of education and satisfaction with work? A random sample of 5244 employed individuals were asked to disclose their highest level of education and satisfaction with their work/job. The results are shown in the table below. The data are from the General Social Survey.
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a. Compute the expected values of each cell under the assumption of independence.

Verified step by step guidance

1. Identify the variables involved: one is the level of education (categorical variable with several categories), and the other is satisfaction with work (also categorical with several categories). The goal is to test for independence between these two categorical variables.
2. Recall that under the assumption of independence, the expected frequency for each cell in a contingency table is calculated using the formula: \[ E_{ij} = \frac{(\text{Row total for row } i) \times (\text{Column total for column } j)}{\text{Grand total}} \]
3. Obtain the row totals (sum of counts for each education level), the column totals (sum of counts for each satisfaction level), and the grand total (total number of individuals, which is 5244 in this case) from the given data table.
4. For each cell in the table, multiply the corresponding row total by the column total, then divide by the grand total to find the expected count for that cell under the assumption of independence.
5. Repeat this calculation for every cell in the table to complete the expected frequency table, which will be used later for hypothesis testing (such as a chi-square test of independence).

Key Concepts

Chi-Square Test of Independence

This test evaluates whether two categorical variables are related or independent. It compares observed frequencies in a contingency table to expected frequencies calculated under the assumption of independence. A significant difference suggests an association between the variables.

Expected Cell Frequencies

Expected frequencies represent the counts we would expect in each cell of a contingency table if the two variables were independent. They are calculated by multiplying the row total by the column total and dividing by the overall sample size.

Contingency Table

A contingency table displays the frequency distribution of variables and helps summarize the relationship between categorical variables. It organizes data into rows and columns, showing counts for each combination of categories.