To solve this problem, we'll create a two-way frequency table that summarizes the survey results based on whether students play an instrument and whether they are in band.
We know the following:
1. There are 40 students in total.
2. 25 students play an instrument.
3. 20 students are in band.
4. 20 students are not in band (hence, the total students in band and not in band is 40).
To build the table, we should determine:
- The number of students who play an instrument and are in band.
- The number of students who play an instrument but are not in band.
- The number of students who do not play an instrument but are in band.
- The number of students who do not play an instrument and are not in band.
Let's break it down step-by-step:
1. Number of students who do not play an instrument:
[tex]\[
\text{Total students} - \text{Students who play an instrument} = 40 - 25 = 15
\][/tex]
2. Number of students who play an instrument and are in band:
[tex]\[
\text{Students who play an instrument} - \left( \text{Total students} - \text{Students not in band} \right)
= 25 - (40 - 20) = 25 - 20 = 5
\][/tex]
3. Number of students who play an instrument but are not in band:
[tex]\[
\text{Students who play an instrument} - \text{Students who play an instrument and are in band} = 25 - 5 = 20
\][/tex]
4. Number of students who do not play an instrument and are in band:
[tex]\[
\text{Total students in band} - \text{Students who play an instrument and are in band} = 20 - 5 = 15
\][/tex]
5. Number of students who do not play an instrument and are not in band:
[tex]\[
\text{Total students not in band} - \text{Students who play an instrument and are not in band} = 20 - 20 = 0
\][/tex]
Now, we will use these results to construct the table:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline & Band & \begin{tabular}{c}
Not In \\
band
\end{tabular} & Total \\
\hline \begin{tabular}{c}
Play \\
Instrument
\end{tabular} & 5 & 20 & 25 \\
\hline \begin{tabular}{c}
Don't play \\
Instrument
\end{tabular} & 15 & 0 & 15 \\
\hline Total & 20 & 20 & 40 \\
\hline
\end{tabular}
\][/tex]
Comparing this with the provided options, we see that the correct table is different from option A. Here is the correct summary in a two-way frequency table format:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline & Band & \begin{tabular}{c}
Not In \\
band
\end{tabular} & Total \\
\hline \begin{tabular}{c}
Play \\
Instrument
\end{tabular} & 5 & 20 & 25 \\
\hline \begin{tabular}{c}
Don't play \\
Instrument
\end{tabular} & 15 & 0 & 15 \\
\hline Total & 20 & 20 & 40 \\
\hline
\end{tabular}
\][/tex]
So, the table given in option A does not correctly represent the data. The correctly structured table is shown above.
