Finding Expected Frequencies
In Exercises 7–12, (a) calculate the marginal frequencies and (b) find the expected frequency for each cell in the contingency table. Assume that the variables are independent.

Verified step by step guidance

1. Step 1: Calculate the marginal frequencies for each row and column. Marginal frequencies are the totals for each row and column in the contingency table. Add the values across each row to find the row totals, and add the values down each column to find the column totals.
2. Step 2: Compute the grand total by summing all the values in the table. This represents the total number of observations in the dataset.
3. Step 3: Use the formula for expected frequency to calculate the expected frequency for each cell in the table. The formula is: $E=(\mathrm{row} \mathrm{total}\times \mathrm{column} \mathrm{total})/\mathrm{grand} \mathrm{total}$, where E is the expected frequency.
4. Step 4: Apply the formula to each cell in the table. For example, for the cell corresponding to 'Male' and 'Compact', use the row total for 'Male', the column total for 'Compact', and the grand total to calculate the expected frequency.
5. Step 5: Repeat the calculation for all cells in the table to find the expected frequencies for each cell. Ensure that the expected frequencies are rounded appropriately if needed.

Key Concepts

Marginal Frequencies

Marginal frequencies are the totals of the rows and columns in a contingency table. They provide a summary of the data by showing the total counts for each category, allowing for a quick overview of the distribution of responses. For example, in the given table, the marginal frequency for males would be the sum of all car types chosen by males.

Expected Frequencies

Expected frequencies are the theoretical counts that would occur in each cell of a contingency table if the variables were independent. They are calculated by multiplying the marginal totals of the corresponding row and column, then dividing by the total number of observations. This concept is crucial for conducting chi-square tests to determine if there is a significant association between the variables.

Independence of Variables

The independence of variables means that the occurrence of one variable does not affect the occurrence of another. In the context of a contingency table, if the variables (like gender and car type) are independent, the expected frequencies can be calculated using the marginal totals. This assumption is essential for accurately interpreting the results of statistical tests, such as the chi-square test.