Certainly! Let's solve the given problem step-by-step.
The problem asks for the conditional relative frequency that a student prefers a cat as a pet given that the student is in high school.
First, let's define our notation and key values from the provided data:
- The total proportion of high school students is \(0.46\). This means that \(46\%\) of the students are in high school.
- The proportion of high school students who prefer cats as pets is \(0.15\). This means that \(15\%\) of high school students prefer cats as pets.
The conditional relative frequency \(P(\text{Cat}|\text{High School})\) is calculated using the formula:
[tex]\[
P(\text{Cat}|\text{High School}) = \frac{P(\text{Cat} \cap \text{High School})}{P(\text{High School})}
\][/tex]
Where:
- \(P(\text{Cat} \cap \text{High School})\) is the proportion of students who are both in high school and prefer cats (\(0.15\)).
- \(P(\text{High School})\) is the total proportion of high school students (\(0.46\)).
Thus, we have:
[tex]\[
P(\text{Cat}|\text{High School}) = \frac{0.15}{0.46}
\][/tex]
When performing this division, we find:
[tex]\[
P(\text{Cat}|\text{High School}) \approx 0.3261
\][/tex]
Therefore, the conditional relative frequency that a student prefers a cat as a pet, given that the student is in high school, is approximately [tex]\(0.3261\)[/tex] or [tex]\(32.61\%\)[/tex].
