Sure, let's solve the problem step by step.
### Part A: Finding the Conditional Relative Frequency
The conditional relative frequency of a student who does not plan to attend college, given that they plan to live at home, is calculated as follows:
1. Identify the total percentage of students who plan to live at home. This is given in the problem as [tex]\(60\%\)[/tex].
2. Identify the percentage of students who do not plan to attend college but plan to live at home. This is given as [tex]\(25\%\)[/tex].
The conditional relative frequency is calculated by dividing the percentage of students who do not plan to attend college but plan to live at home by the total percentage of students who plan to live at home:
[tex]\[
\text{Conditional Relative Frequency} = \frac{\text{Percentage of students who do not plan to attend college and live at home}}{\text{Percentage of students who plan to live at home}}
\][/tex]
[tex]\[
\text{Conditional Relative Frequency} = \frac{25\%}{60\%}
\][/tex]
Convert percentages to decimals:
[tex]\[
\text{Conditional Relative Frequency} = \frac{0.25}{0.60}
\][/tex]
Compute the frequency:
[tex]\[
\text{Conditional Relative Frequency} = \frac{0.25}{0.60} \approx 0.4167
\][/tex]
So, the conditional relative frequency is approximately [tex]\(0.4167\)[/tex] or [tex]\(41.67\%\)[/tex].
### Part B: Checking for an Association
To check for an association between not attending college and living on their own, we compare the observed probability to the expected probability assuming independence.
1. Observed Probability (Given):
- Percentage of students who do not plan to attend college and plan to live on their own is [tex]\(10\%\)[/tex].
2. Expected Probability:
- If there were no association, the events "not attending college" and "living on their own" would be independent.
- The expected probability can be calculated by multiplying the marginal probabilities:
- Probability of not attending college: [tex]\(35\%\)[/tex]
- Probability of living on their own: [tex]\(40\%\)[/tex]
[tex]\[
\text{Expected Probability} = (\text{Probability of not attending college}) \times (\text{Probability of living on their own})
\][/tex]
Convert percentages to decimals:
[tex]\[
\text{Expected Probability} = (0.35) \times (0.40) = 0.14
\][/tex]
Convert the decimal back to a percentage:
[tex]\[
\text{Expected Probability} = 14\%
\][/tex]
3. Comparison:
- Observed probability: [tex]\(10\%\)[/tex]
- Expected probability: [tex]\(14\%\)[/tex]
Since the observed probability ([tex]\(10\%\)[/tex]) is different from the expected probability ([tex]\(14\%\)[/tex]), there is an association between not attending college and living on their own. Specifically, the observed probability is lower than the expected probability, suggesting that students who do not plan to attend college are less likely to live on their own than would be expected if the two events were independent.
