Answer :
Let's carefully analyze the information given and how it should be organized in a two-way frequency table.
We know the following:
1. There are a total of 60 students.
2. 35 students play an instrument.
3. 30 students are in the band.
4. 30 students are not in the band.
Let's break down the steps to understand how these numbers should fit into the two-way frequency tables.
### Step-by-Step Solution:
Step 1: Determine the total number of students who play an instrument and are in the band.
Since the total number of students who are in the band is given as 30, the students who are in the band and play an instrument is directly given or can be calculated as equivalent to this number, i.e., 30.
Step 2: Calculate the number of students who play an instrument but are not in the band.
We know there are 35 students who play an instrument in total. Out of these, 30 are in the band (as calculated or given directly). Therefore:
[tex]\[ \text{Students who play an instrument but are not in the band} = 35 - 30 = 5 \][/tex]
Step 3: Calculate students who are not in the band and do not play an instrument.
The total number of students not in the band is given as 30. From these, 5 students play an instrument (calculated in Step 2). Therefore:
[tex]\[ \text{Students who do not play an instrument and are not in the band} = 30 - 5 = 25 \][/tex]
Step 4: Determine students who are in the band and do not play any instrument.
We previously calculated that out of 60 total students, 35 play an instrument, so 60 - 35 = 25 do not play an instrument in total. From this, we need to subtract those who are not in the band and do not play an instrument (25, calculated in Step 3):
[tex]\[ \text{Students in the band who do not play an instrument} = 25 - 25 = 0 \][/tex]
Step 5: Construct the two-way frequency table A:
- Band and play instrument: 30
- Not in band and play instrument: 5
- Total students who play an instrument: 35
- Not in band and don't play instrument: 25
- Band and don't play instrument: 0
- Total students who don't play an instrument: 25
### Frequency Table A:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{Band and play instrument} & \text{Not in band and play instrument} & \text{Total} \\ \hline \text{Not in band and don't play instrument} & 25 & 0 & 30 \\ \hline \text{Band and don't play instrument} & 0 & 5 & 25 \\ \hline \text{Total} & 35 & 30 & 60 \\ \hline \end{array} \][/tex]
### Frequency Table B:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{Band} & \text{Not in band} & \text{Total} \\ \hline \text{Play instrument} & 30 & 5 & 35 \\ \hline \end{array} \][/tex]
### Conclusion:
The correct data entered into the two-way frequency tables are matched as follows:
- Table A correctly matches:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{Band and play instrument} & \text{Not in band and play instrument} & \text{Total} \\ \hline \text{Not in band and don't play instrument} & 25 & 0 & 30 \\ \hline \text{Band and don't play instrument} & 0 & 5 & 30 \\ \hline \text{Total} & 35 & 30 & 60 \\ \hline \end{array} \][/tex]
- Table B correctly matches:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{Band} & \text{Not in band} & \text{Total} \\ \hline \text{Play instrument} & 30 & 5 & 35 \\ \hline \end{array} \][/tex]
We know the following:
1. There are a total of 60 students.
2. 35 students play an instrument.
3. 30 students are in the band.
4. 30 students are not in the band.
Let's break down the steps to understand how these numbers should fit into the two-way frequency tables.
### Step-by-Step Solution:
Step 1: Determine the total number of students who play an instrument and are in the band.
Since the total number of students who are in the band is given as 30, the students who are in the band and play an instrument is directly given or can be calculated as equivalent to this number, i.e., 30.
Step 2: Calculate the number of students who play an instrument but are not in the band.
We know there are 35 students who play an instrument in total. Out of these, 30 are in the band (as calculated or given directly). Therefore:
[tex]\[ \text{Students who play an instrument but are not in the band} = 35 - 30 = 5 \][/tex]
Step 3: Calculate students who are not in the band and do not play an instrument.
The total number of students not in the band is given as 30. From these, 5 students play an instrument (calculated in Step 2). Therefore:
[tex]\[ \text{Students who do not play an instrument and are not in the band} = 30 - 5 = 25 \][/tex]
Step 4: Determine students who are in the band and do not play any instrument.
We previously calculated that out of 60 total students, 35 play an instrument, so 60 - 35 = 25 do not play an instrument in total. From this, we need to subtract those who are not in the band and do not play an instrument (25, calculated in Step 3):
[tex]\[ \text{Students in the band who do not play an instrument} = 25 - 25 = 0 \][/tex]
Step 5: Construct the two-way frequency table A:
- Band and play instrument: 30
- Not in band and play instrument: 5
- Total students who play an instrument: 35
- Not in band and don't play instrument: 25
- Band and don't play instrument: 0
- Total students who don't play an instrument: 25
### Frequency Table A:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{Band and play instrument} & \text{Not in band and play instrument} & \text{Total} \\ \hline \text{Not in band and don't play instrument} & 25 & 0 & 30 \\ \hline \text{Band and don't play instrument} & 0 & 5 & 25 \\ \hline \text{Total} & 35 & 30 & 60 \\ \hline \end{array} \][/tex]
### Frequency Table B:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{Band} & \text{Not in band} & \text{Total} \\ \hline \text{Play instrument} & 30 & 5 & 35 \\ \hline \end{array} \][/tex]
### Conclusion:
The correct data entered into the two-way frequency tables are matched as follows:
- Table A correctly matches:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{Band and play instrument} & \text{Not in band and play instrument} & \text{Total} \\ \hline \text{Not in band and don't play instrument} & 25 & 0 & 30 \\ \hline \text{Band and don't play instrument} & 0 & 5 & 30 \\ \hline \text{Total} & 35 & 30 & 60 \\ \hline \end{array} \][/tex]
- Table B correctly matches:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{Band} & \text{Not in band} & \text{Total} \\ \hline \text{Play instrument} & 30 & 5 & 35 \\ \hline \end{array} \][/tex]
