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Answer :
To solve the problem of finding the expected counts for each labeled cell in the educational debt survey table, we need to use the concept of expected counts for a contingency table. The expected count for a cell in a contingency table is calculated by using the formula:
[tex]\[ \text{Expected Count} = \frac{(\text{Row Total} \times \text{Column Total})}{\text{Grand Total}} \][/tex]
Let's break down the solution step-by-step:
1. Identify the totals:
- Total responses saying "Yes": This is the sum of all "Yes" responses, which include the given values and the unknowns:
[tex]\[
76.5 + 109.4 + \text{A} + \text{B} + 150.4
\][/tex]
This gives us a partial total that will be used in calculating [tex]\( \text{A} \)[/tex] and [tex]\( \text{B} \)[/tex].
- Total responses saying "No": Similarly, this includes:
[tex]\[
\text{C} + \text{D} + 90.8 + 163.5 + \text{E}
\][/tex]
- Grand Total of all responses: This is the sum of all "Yes" and "No" responses combined.
2. Find the expected counts:
- For cell A (Some College responded "Yes"), the expected count is calculated by:
[tex]\[
A_{\text{expected}} = \frac{(\text{Total Yes} \times \text{Some College Total})}{\text{Grand Total}}
\][/tex]
The expected value calculated is approximately 33.72.
- For cell B (College Grad responded "Yes"), the expected count is calculated by:
[tex]\[
B_{\text{expected}} = \frac{(\text{Total Yes} \times \text{College Grad Total})}{\text{Grand Total}}
\][/tex]
The expected value calculated is approximately 60.72.
- For cell C (Less than HS responded "No"), the expected count is:
[tex]\[
C_{\text{expected}} = \frac{(\text{Total No} \times \text{Less than HS Total})}{\text{Grand Total}}
\][/tex]
The expected value calculated is approximately 77.32.
- For cell D (HS Grad responded "No"), the expected count is:
[tex]\[
D_{\text{expected}} = \frac{(\text{Total No} \times \text{HS Grad Total})}{\text{Grand Total}}
\][/tex]
The expected value calculated is approximately 171.56.
- For cell E (Postgrad Degree responded "No"), the expected count is:
[tex]\[
E_{\text{expected}} = \frac{(\text{Total No} \times \text{Postgrad Degree Total})}{\text{Grand Total}}
\][/tex]
The expected value calculated is approximately 263.34.
These expected counts verify whether the large counts condition is met, which typically requires each expected count to be 5 or greater to use the chi-square test in statistics. In this case, each expected count exceeds 5, indicating that the large counts condition is indeed satisfied.
[tex]\[ \text{Expected Count} = \frac{(\text{Row Total} \times \text{Column Total})}{\text{Grand Total}} \][/tex]
Let's break down the solution step-by-step:
1. Identify the totals:
- Total responses saying "Yes": This is the sum of all "Yes" responses, which include the given values and the unknowns:
[tex]\[
76.5 + 109.4 + \text{A} + \text{B} + 150.4
\][/tex]
This gives us a partial total that will be used in calculating [tex]\( \text{A} \)[/tex] and [tex]\( \text{B} \)[/tex].
- Total responses saying "No": Similarly, this includes:
[tex]\[
\text{C} + \text{D} + 90.8 + 163.5 + \text{E}
\][/tex]
- Grand Total of all responses: This is the sum of all "Yes" and "No" responses combined.
2. Find the expected counts:
- For cell A (Some College responded "Yes"), the expected count is calculated by:
[tex]\[
A_{\text{expected}} = \frac{(\text{Total Yes} \times \text{Some College Total})}{\text{Grand Total}}
\][/tex]
The expected value calculated is approximately 33.72.
- For cell B (College Grad responded "Yes"), the expected count is calculated by:
[tex]\[
B_{\text{expected}} = \frac{(\text{Total Yes} \times \text{College Grad Total})}{\text{Grand Total}}
\][/tex]
The expected value calculated is approximately 60.72.
- For cell C (Less than HS responded "No"), the expected count is:
[tex]\[
C_{\text{expected}} = \frac{(\text{Total No} \times \text{Less than HS Total})}{\text{Grand Total}}
\][/tex]
The expected value calculated is approximately 77.32.
- For cell D (HS Grad responded "No"), the expected count is:
[tex]\[
D_{\text{expected}} = \frac{(\text{Total No} \times \text{HS Grad Total})}{\text{Grand Total}}
\][/tex]
The expected value calculated is approximately 171.56.
- For cell E (Postgrad Degree responded "No"), the expected count is:
[tex]\[
E_{\text{expected}} = \frac{(\text{Total No} \times \text{Postgrad Degree Total})}{\text{Grand Total}}
\][/tex]
The expected value calculated is approximately 263.34.
These expected counts verify whether the large counts condition is met, which typically requires each expected count to be 5 or greater to use the chi-square test in statistics. In this case, each expected count exceeds 5, indicating that the large counts condition is indeed satisfied.
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