Let's create a two-way frequency table based on the given data:
1. Total number of students: 50
2. Students who play an instrument: 25
3. Students who are in the band: 20
4. Students who are not in the band: 30
Since the total number of students is 50, there are two mutually exclusive groups: those in the band and those not in the band.
Considering the students who play an instrument:
- Those who play an instrument and are in the band: Let's denote this number as [tex]\( x \)[/tex].
- Those who play an instrument but are not in the band: [tex]\( 25 - x \)[/tex].
Considering students who do not play an instrument:
- Total students not in the band: 30
- Thus, those who do not play an instrument and are not in the band: [tex]\( 30 - (25 - x) = 5 + x \)[/tex].
The key is finding [tex]\( x \)[/tex]:
- Students who are in the band: 20
- Those who play an instrument and are in the band [tex]\( x \)[/tex] + those who do not play an instrument and are in the band [tex]\( 20 - x \)[/tex].
We solve:
[tex]\[ x + (20 - x) = 20 \][/tex]
[tex]\[ 0 + 20 = 20 \][/tex]
Thus:
- Students who play an instrument and are in the band: 20
- Students who play an instrument but are not in the band must be [tex]\( 25 - 20 = 5 \)[/tex].
Now we fill out the two-way table:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
& \text{Band} & \text{Not in band} & \text{Total} \\
\hline
\text{Play instrument} & 20 & 5 & 25 \\
\hline
\text{Don't play instrument} & 0 & 25 & 25 \\
\hline
\text{Total} & 20 & 30 & 50 \\
\hline
\end{array}
\][/tex]
Thus, the table that shows these data correctly entered in a two-way frequency table is:
Answer: A
