To determine which table correctly represents the data from the survey, we need to fill in a two-way frequency table with the information given and inferred:
1. Total number of students surveyed: 50
2. Number of students who play an instrument: 25
3. Number of students who are in band: 20
4. Number of students who are not in band: 30
From the information, we can infer the following:
1. Number of students who play an instrument and are in band: 20
2. Number of students who play an instrument and are not in band: [tex]\( 25 - 20 = 5 \)[/tex]
3. Number of students who do not play an instrument and are in band: [tex]\( 20 - 20 = 0 \)[/tex]
4. Number of students who do not play an instrument and are not in band: [tex]\( 50 - 25 - 0 = 25 \)[/tex]
Now we can construct the correct table based on these calculations.
| | Band | Not in band | Total |
|-------------------|------|-------------|-------|
| Play instrument | 20 | 5 | 25 |
| Don't play instrument | 0 | 25 | 25 |
| Total | 20 | 30 | 50 |
Now let's match this with the provided options:
Option A:
\begin{tabular}{|l|c|c|c|}
\hline & Band & Not in Band & Total \\
\hline Play instrument & 20 & 5 & 25 \\
\hline Don't play instrument & 0 & 25 & 25 \\
\hline Total & 20 & 30 & 50 \\
\hline
\end{tabular}
This table exactly matches the data and our calculated values.
Option B:
\begin{tabular}{|l|c|c|c|}
\hline & Band & Not in band & Total \\
\hline Play instrument & 20 & 0 & 20 \\
\hline Don't play instrument & 5 & 25 & 30 \\
\hline Total & 25 & 25 & 50 \\
\hline
\end{tabular}
This table does not match the provided data or calculations as it misplaces the values.
Option C:
\begin{tabular}{|l|c|c|c|}
\hline & \begin{tabular}{l}
Band and\\play\\instrument
\end{tabular} & \begin{tabular}{c}
Not in band\\and play\\instrument
\end{tabular} & Total \\
\hline \begin{tabular}{l}
Not in band and\\don't play\\instrument
\end{tabular} & 20 & 0 & 20 \\
\hline \begin{tabular}{l}
Band and don't\\play instrument
\end{tabular} & 5 & 25 & 30 \\
\hline Total & 25 & 25 & 50 \\
\hline
\end{tabular}
This table arrangement and values are incorrect compared to our calculated values.
Option D:
\begin{tabular}{|l|c|c|c|}
\hline & Band & \begin{tabular}{l}
Don't play\\instrument
\end{tabular} & Total \\
\hline Not in band & 20 & 5 & 25 \\
\hline Play instrument & 0 & 25 & 25 \\
\hline Total & 20 & 30 & 50 \\
\hline
\end{tabular}
The labels and values are mixed up in this table compared to our calculations.
Therefore, the correct table is:
Option A:
\begin{tabular}{|l|c|c|c|}
\hline & Band & Not in Band & Total \\
\hline Play instrument & 20 & 5 & 25 \\
\hline Don't play instrument & 0 & 25 & 25 \\
\hline Total & 20 & 30 & 50 \\
\hline
\end{tabular}
