Answer :
To solve this problem, we need to fill in the table with the expected counts for each labeled cell (A, B, C, D, E) using the information provided about the responses to the question on student loan/educational debt.
### Step-by-Step Solution:
1. Understand the Problem:
- We have a contingency table showing responses (Yes/No) to whether people have educational debt, broken down by the level of education.
- Some expected counts are missing (A, B, C, D, E) which we need to calculate using the overall totals.
2. Identify the Totals:
- We do not have explicit column totals, so we will derive these from the information we have and using the conditions that row and column totals must add up to the same grand total.
3. Set Up Known Observed Counts:
- From the table:
- Yes: Sum of observed ‘Yes’ counts we know: [tex]\(76.5 + 109.4 + 150.4\)[/tex].
- No: Sum of observed ‘No’ counts we know: [tex]\(90.8 + 163.5\)[/tex].
4. Calculate Marginal Totals:
- Total number of 'Yes' respondents = [tex]\(76.5 + 109.4 + 150.4 = 336.3\)[/tex].
- Total number of 'No' respondents = [tex]\(90.8 + 163.5 = 254.3\)[/tex].
- Grand total number of respondents: [tex]\(336.3 + 254.3 = 590.6\)[/tex].
5. Derive Column Totals:
- Since [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are in the row for 'Yes', and C, D, and E are in the row for 'No', calculate probabilities and expected values as follows:
- Each column total would be the sum of the 'Yes' and 'No' for that particular educational level.
- For example, assuming the total for "Less than HS" column is X, then:
[tex]\[
76.5 + C = X
\][/tex]
6. Set Up Equation for Each Cell:
- Using the formula for expected values [tex]\(\text{Expected count} = \frac{\text{Row total} \times \text{Column total}}{\text{Grand total}}\)[/tex]:
- For cell [tex]\(A\)[/tex] under "Some College":
[tex]\[
A = \frac{\text{Yes total} \times \text{Some College column total}}{\text{Grand total}}
\][/tex]
- Similarly, calculate for [tex]\(B, C, D, E\)[/tex] using their respective row and column totals.
7. Compute Expected Counts:
- Compute the expected counts for each cell using the row and column totals you've derived. This involves some assumptions or additional data regarding the column totals shared across the columns.
8. Verification:
- Ensure the expected counts in each row sum to the respective row totals, and similarly, for the columns.
Since we don’t have complete details on column totals here, solving the question as is necessitates either assumption-based filling or you have the original column’s sum data. This step applies otherwise will need additional context or figures that reflect the incomplete table.
This explanation assumes you’re filling these values in an applied scenario where precise data-sharing precedes such filling.
### Step-by-Step Solution:
1. Understand the Problem:
- We have a contingency table showing responses (Yes/No) to whether people have educational debt, broken down by the level of education.
- Some expected counts are missing (A, B, C, D, E) which we need to calculate using the overall totals.
2. Identify the Totals:
- We do not have explicit column totals, so we will derive these from the information we have and using the conditions that row and column totals must add up to the same grand total.
3. Set Up Known Observed Counts:
- From the table:
- Yes: Sum of observed ‘Yes’ counts we know: [tex]\(76.5 + 109.4 + 150.4\)[/tex].
- No: Sum of observed ‘No’ counts we know: [tex]\(90.8 + 163.5\)[/tex].
4. Calculate Marginal Totals:
- Total number of 'Yes' respondents = [tex]\(76.5 + 109.4 + 150.4 = 336.3\)[/tex].
- Total number of 'No' respondents = [tex]\(90.8 + 163.5 = 254.3\)[/tex].
- Grand total number of respondents: [tex]\(336.3 + 254.3 = 590.6\)[/tex].
5. Derive Column Totals:
- Since [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are in the row for 'Yes', and C, D, and E are in the row for 'No', calculate probabilities and expected values as follows:
- Each column total would be the sum of the 'Yes' and 'No' for that particular educational level.
- For example, assuming the total for "Less than HS" column is X, then:
[tex]\[
76.5 + C = X
\][/tex]
6. Set Up Equation for Each Cell:
- Using the formula for expected values [tex]\(\text{Expected count} = \frac{\text{Row total} \times \text{Column total}}{\text{Grand total}}\)[/tex]:
- For cell [tex]\(A\)[/tex] under "Some College":
[tex]\[
A = \frac{\text{Yes total} \times \text{Some College column total}}{\text{Grand total}}
\][/tex]
- Similarly, calculate for [tex]\(B, C, D, E\)[/tex] using their respective row and column totals.
7. Compute Expected Counts:
- Compute the expected counts for each cell using the row and column totals you've derived. This involves some assumptions or additional data regarding the column totals shared across the columns.
8. Verification:
- Ensure the expected counts in each row sum to the respective row totals, and similarly, for the columns.
Since we don’t have complete details on column totals here, solving the question as is necessitates either assumption-based filling or you have the original column’s sum data. This step applies otherwise will need additional context or figures that reflect the incomplete table.
This explanation assumes you’re filling these values in an applied scenario where precise data-sharing precedes such filling.
