Answer :
To verify that the large counts condition has been met, let's calculate the expected counts for each of the labeled cells (A, B, C, D, and E) in the table based on the information provided.
To find the expected count for a specific cell in a two-way table, we use the formula:
[tex]\[
\text{Expected Count} = \frac{(\text{Row Total} \times \text{Column Total})}{\text{Grand Total}}
\][/tex]
Let's break it down step-by-step:
1. Determine the Total Counts:
- Total "Yes" responses: The sum of the given "Yes" responses.
- Yes Total = 76.5 (Less than HS) + 109.4 (HS Grad) + A + B + 150.4 (Postgrad Degree)
- Given "No" responses:
- Some College (No): 90.8
- College Grad (No): 163.5
- Column Totals:
- Less than HS Total: 76.5 (since "No" is not given)
- HS Grad Total: 109.4 (since "No" is not given)
- Some College Total: 90.8 (since "Yes" is not given)
- College Grad Total: 163.5 (since "Yes" is not given)
- Postgrad Degree Total: 150.4 (since "No" is not given)
- Grand Total (sum of all categories): Sum of all "Yes" and given "No" responses.
2. Calculate Expected Counts for Unknown Cells:
- A (Some College - Yes):
[tex]\[
A = \frac{(\text{Total "Yes"} \times \text{Some College Total})}{\text{Grand Total}}
\][/tex]
Result: [tex]\( A \approx 51.70 \)[/tex]
- B (College Grad - Yes):
[tex]\[
B = \frac{(\text{Total "Yes"} \times \text{College Grad Total})}{\text{Grand Total}}
\][/tex]
Result: [tex]\( B \approx 93.10 \)[/tex]
- C (Less than HS - No):
[tex]\[
C = \frac{(\text{Some College "No"} \times \text{Less than HS Total})}{\text{Grand Total}}
\][/tex]
Result: [tex]\( C \approx 11.76 \)[/tex]
- D (HS Grad - No):
[tex]\[
D = \frac{(\text{College Grad "No"} \times \text{HS Grad Total})}{\text{Grand Total}}
\][/tex]
Result: [tex]\( D \approx 30.29 \)[/tex]
- E (Postgrad Degree - No):
[tex]\[
E = \frac{(\text{College Grad "No"} \times \text{Postgrad Degree Total})}{\text{Grand Total}}
\][/tex]
Result: [tex]\( E \approx 41.64 \)[/tex]
These expected counts demonstrate that all categories have enough respondents to satisfy the large counts condition.
To find the expected count for a specific cell in a two-way table, we use the formula:
[tex]\[
\text{Expected Count} = \frac{(\text{Row Total} \times \text{Column Total})}{\text{Grand Total}}
\][/tex]
Let's break it down step-by-step:
1. Determine the Total Counts:
- Total "Yes" responses: The sum of the given "Yes" responses.
- Yes Total = 76.5 (Less than HS) + 109.4 (HS Grad) + A + B + 150.4 (Postgrad Degree)
- Given "No" responses:
- Some College (No): 90.8
- College Grad (No): 163.5
- Column Totals:
- Less than HS Total: 76.5 (since "No" is not given)
- HS Grad Total: 109.4 (since "No" is not given)
- Some College Total: 90.8 (since "Yes" is not given)
- College Grad Total: 163.5 (since "Yes" is not given)
- Postgrad Degree Total: 150.4 (since "No" is not given)
- Grand Total (sum of all categories): Sum of all "Yes" and given "No" responses.
2. Calculate Expected Counts for Unknown Cells:
- A (Some College - Yes):
[tex]\[
A = \frac{(\text{Total "Yes"} \times \text{Some College Total})}{\text{Grand Total}}
\][/tex]
Result: [tex]\( A \approx 51.70 \)[/tex]
- B (College Grad - Yes):
[tex]\[
B = \frac{(\text{Total "Yes"} \times \text{College Grad Total})}{\text{Grand Total}}
\][/tex]
Result: [tex]\( B \approx 93.10 \)[/tex]
- C (Less than HS - No):
[tex]\[
C = \frac{(\text{Some College "No"} \times \text{Less than HS Total})}{\text{Grand Total}}
\][/tex]
Result: [tex]\( C \approx 11.76 \)[/tex]
- D (HS Grad - No):
[tex]\[
D = \frac{(\text{College Grad "No"} \times \text{HS Grad Total})}{\text{Grand Total}}
\][/tex]
Result: [tex]\( D \approx 30.29 \)[/tex]
- E (Postgrad Degree - No):
[tex]\[
E = \frac{(\text{College Grad "No"} \times \text{Postgrad Degree Total})}{\text{Grand Total}}
\][/tex]
Result: [tex]\( E \approx 41.64 \)[/tex]
These expected counts demonstrate that all categories have enough respondents to satisfy the large counts condition.
