Sure, let's go through the solution step-by-step:
Part A: Find the conditional relative frequency of a student who plans to live at home, given that they plan to attend college.
To find the conditional relative frequency, we need to determine the probability of a student living at home given that they plan to attend college. This is found using the formula:
[tex]\[ P(\text{Live at Home} | \text{Attend College}) = \frac{P(\text{Live at Home and Attend College})}{P(\text{Attend College})} \][/tex]
From the table, we have the following data:
- \( P(\text{Attend College}) = 65\% = 0.65 \)
- \( P(\text{Live at Home and Attend College}) = 35\% = 0.35 \)
Using the formula:
[tex]\[ P(\text{Live at Home} | \text{Attend College}) = \frac{0.35}{0.65} \][/tex]
Performing the division:
[tex]\[ P(\text{Live at Home} | \text{Attend College}) = 0.538 \][/tex]
Thus, the conditional relative frequency is approximately \( 0.538 \) or \( 53.8\% \).
Part B: Is there an association between attending college and living at home?
To determine if there is an association between attending college and living at home, we can compare the conditional probability found in Part A with the overall probability of living at home.
From the table, we have:
- \( P(\text{Live at Home}) = 50\% = 0.50 \)
We know:
- \( P(\text{Live at Home} | \text{Attend College}) = 0.538 \)
There is an association if the probability of living at home given that a student attends college is different from the overall probability of living at home.
Since \( P(\text{Live at Home} | \text{Attend College}) \approx 0.538 \) is not equal to \( P(\text{Live at Home}) = 0.50 \), this indicates that the probability is indeed different, confirming an association.
Conclusion:
1. The conditional relative frequency of a student who plans to live at home, given that they plan to attend college, is approximately \( 0.538 \) or \( 53.8\% \) (Part A).
2. There is an association between attending college and living at home since [tex]\( 0.538 \)[/tex] is different from [tex]\( 0.50 \)[/tex] (Part B).
