What is the conditional relative frequency that a student prefers a cat as a pet, given that the student is in high school?

[tex]\[
\begin{tabular}{|l|c|c|c|c|}
\hline & \multicolumn{4}{|c|}{ Type of Pet } \\
\hline School & Dog & Cat & Other & Total \\
\hline \begin{tabular}{l}
Middle \\
School
\end{tabular} & 0.26 & 0.18 & 0.10 & 0.54 \\
\hline \begin{tabular}{l}
High \\
School
\end{tabular} & 0.25 & 0.15 & 0.06 & 0.46 \\
\hline Total & 0.51 & 0.33 & 0.16 & 1.00 \\
\hline
\end{tabular}
\][/tex]

[tex]$\square$[/tex]



Answer :

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].