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A survey asked 40 students if they play an instrument and if they are in the band.
1. 25 students play an instrument.
2. 20 students are in the band.
3. 20 students are not in the band.

Which table shows these data correctly entered in a two-way frequency table?

A.
\begin{tabular}{|l|c|c|c|}
\hline & \begin{tabular}{c}
In 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
\end{tabular}



Answer :

GinnyAnswer
Let's carefully break down the given problem and construct the two-way frequency table step by step.

1. First, let's understand the variables:
- The total number of students is 40.
- The number of students who play an instrument is 25.
- The number of students who are in the band is 20.
- The number of students who are not in the band is 20.

2. We need to determine the number of students who:
- Play an instrument and are in the band.
- Play an instrument but are not in the band.
- Do not play an instrument but are in the band.
- Do not play an instrument and are not in the band.

3. Since the total number of students is 40:
- The number of students who do not play an instrument is [tex]\( 40 - 25 = 15 \)[/tex].
- The total number of students in the band is 20, and the total number of students not in the band is 20, confirming that 40 students are accounted for.

4. Using the given totals:
- Total number of students in the band, whether they play an instrument or not: 20.
- Total number of students not in the band, whether they play an instrument or not: 20.
- Total number of students who play an instrument: 25.
- Total number of students who do not play an instrument: 15.

5. Analyzing intersections:
- From the given data, we find that the number of students who play an instrument and are in the band is 5.
- Therefore, the number of students who play an instrument and are not in the band is [tex]\( 25 - 5 = 20 \)[/tex].
- The number of students who do not play an instrument but are in the band is [tex]\( 20 - 5 = 15 \)[/tex].
- Finally, the number of students who do not play an instrument and are not in the band is 0 (because all other students are accounted for).

6. Constructing the two-way frequency table:

[tex]\[ \begin{array}{|l|c|c|c|} \hline & \text{In Band} & \text{Not in Band} & \text{Total} \\ \hline \text{Play Instrument} & 5 & 20 & 25 \\ \hline \text{Don't Play Instrument} & 15 & 0 & 15 \\ \hline \text{Total} & 20 & 20 & 40 \\ \hline \end{array} \][/tex]

So, the table that correctly represents the data is:

[tex]\[ \begin{array}{|l|c|c|c|} \hline & \text{In Band} & \text{Not in Band} & \text{Total} \\ \hline \text{Play Instrument} & 5 & 20 & 25 \\ \hline \text{Don't Play Instrument} & 15 & 0 & 15 \\ \hline \text{Total} & 20 & 20 & 40 \\ \hline \end{array} \][/tex]

This table accurately reflects all the provided data.

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