Question

Jorge asked each of the 24 students in his class if they play a musical instrument. He also asked each student if they play a sport. He gathered the following results:

6 students play both a musical instrument and a sport.
3 students play neither a musical instrument nor a sport.
14 students in total play a sport.
If you were to construct a two-way table from this information, what value would be in the intersection of "plays a sport" and "does not play a musical instrument"?

Question 3 options:

7


6


14


8

Answer 1

To solve the problem, let's define the following sets:

- Let A be the set of students who play a musical instrument.
- Let B be the set of students who play a sport.
- We know that:
- |A ∩ B| = 6 (students who play both an instrument and a sport)
- |A^c ∩ B^c| = 3 (students who play neither)
- |B| = 14 (students who play a sport)

Now, we can also find out the number of students that play only a sport (not a musical instrument).

1. Start with the total number of students who play sports:
- We have 14 students who play a sport. This includes those who play both an instrument and a sport.
- Therefore, the number of students who play only a sport is:
\[
|B| - |A ∩ B| = 14 - 6 = 8
\]

2. Next, determine the total number of students in the class:
- There are 24 students in total.
- We already know that 3 students play neither a musical instrument nor a sport (|A^c ∩ B^c|).

3. Thus, the number of students who play either (or both) a musical instrument or a sport is:
\[
24 - 3 = 21
\]
This means there are 21 students who play either a musical instrument, a sport, or both.

4. Next, we can break down those students who play musical instruments further:
- From those 21 students, we have:
\[
\text{Students who only play a musical instrument} + \text{Students who play both} + \text{Students who only play a sport} = 21
\]
- Let \( x \) be the number of students who play only a musical instrument. We can now set up the equation:
\[
x + 6 + 8 = 21
\]
- Simplifying this gives:
\[
x + 14 = 21 \implies x = 7
\]
So, there are 7 students who play only a musical instrument.

5. Finally, to get the values for the two-way table:
- The number of students who play a musical instrument but not a sport is 7.
- The number of students who play a sport but not a musical instrument (which we computed earlier) is 8.

Thus, the value in the intersection of "plays a sport" and "does not play a musical instrument" is **8**.

So the answer is:

**8**