To determine the expected frequency of students who failed at the West Campus, we follow these steps:
1. Gather the Observed Frequencies:
- Passed at East Campus: 48
- Passed at West Campus: 37
- Failed at East Campus: 52
- Failed at West Campus: 63
2. Calculate the Total Number of Students at Each Campus:
- Total students at East Campus [tex]\( = 48 + 52 = 100 \)[/tex]
- Total students at West Campus [tex]\( = 37 + 63 = 100 \)[/tex]
3. Calculate the Total Number of Students Who Passed or Failed:
- Total students who passed [tex]\( = 48 + 37 = 85 \)[/tex]
- Total students who failed [tex]\( = 52 + 63 = 115 \)[/tex]
4. Calculate the Total Number of Students Overall:
- Total students overall [tex]\( = 100 + 100 = 200 \)[/tex]
5. Use the Formula for Expected Frequency:
The expected frequency for a cell in a contingency table can be calculated using the formula:
[tex]\[
\text{Expected Frequency} = \frac{(\text{Row Total} \times \text{Column Total})}{\text{Grand Total}}
\][/tex]
Here, we need the expected frequency of students who failed at the West Campus.
- Row Total (Total Failures): [tex]\( 115 \)[/tex]
- Column Total (Total students at West Campus): [tex]\( 100 \)[/tex]
- Grand Total (Total Students Overall): [tex]\( 200 \)[/tex]
Plugging in these values:
[tex]\[
\text{Expected Frequency of Failed at West Campus} = \frac{(115 \times 100)}{200} = 57.5
\][/tex]
Therefore, the expected frequency of students who failed at the West Campus is 57.5.
