To construct a two-way frequency table based on the provided information, we need to correctly categorize the students into four groups:
1. Students who play an instrument and are in the band.
2. Students who play an instrument but are not in the band.
3. Students who do not play an instrument but are in the band.
4. Students who do not play an instrument and are not in the band.
Given the survey results:
- There are 60 students in total.
- 35 students play an instrument.
- 30 students are in the band.
- 30 students are not in the band.
We know that the total students equals the sum of students in the band and not in the band:
[tex]\[ 30 (\text{in band}) + 30 (\text{not in band}) = 60 (\text{total students}) \][/tex]
We need to determine:
- The number of students who play an instrument and are in the band.
- The number of students who play an instrument but are not in the band.
- The number of students who do not play an instrument but are in the band.
- The number of students who do not play an instrument and are not in the band.
Step-by-Step Solution:
1. Calculate students who play an instrument and are in the band:
Let [tex]\( x \)[/tex] be the number of students who play an instrument and are in the band.
We know that 35 students play an instrument, and 30 students are in the band. Therefore:
[tex]\[ x + (\text{students who play an instrument but are not in the band}) = 35 \][/tex]
2. Calculate students who play an instrument but are not in the band:
Given there are 30 students in the band and [tex]\( x \)[/tex] students who play an instrument and are in the band:
[tex]\[ 30 (\text{total in band}) - x = (\text{students who do not play an instrument but are in the band}) \][/tex]
3. Calculate students who do not play an instrument and are in the band:
With [tex]\( x = 35 \)[/tex]:
[tex]\[ 30 - 35 = -5 \][/tex]
4. Calculate students who do not play an instrument and are not in the band:
We know there are 60 total students, 35 who play an instrument, so the number of students who do not play an instrument is:
[tex]\[ 60 - 35 = 25 \][/tex]
Now subtract students who do not play an instrument and are in the band:
[tex]\[ 25 - (\text{students who do not play an instrument but are in the band}) = 25 - (-5) = 30 \][/tex]
Summarizing these values into a two-way frequency table:
[tex]\[
\begin{array}{|l|c|c|c|}
\hline & \text{Band} & \text{Not in band} & \text{Total} \\
\hline \text{Play instrument} & 35 & 0 & 35 \\
\hline \text{Don't play instrument} & -5 & 30 & 25 \\
\hline \text{Total} & 30 & 30 & 60 \\
\hline
\end{array}
\][/tex]
This shows the data correctly in a two-way frequency table.
