Question

A survey asked eight students about weekly reading hours and whether they play musical instruments. The table shows the results of the survey.


Weekly Reading Hours

Hours of Reading if Student Plays an Instrument
Hours of Reading if Student Does Not Play an Instrument
Student 1
16

Student 2
18

Student 3
15

Student 4
15

Student 5

2
Student 6

2
Student 7

4
Student 8

8


Which statements about the data sets are true? Check all that apply.
The data for the group that plays an instrument are more spread out than the data for the group that did not play an instrument.
The data for the group that plays an instrument are more clustered around the mean than the data for the group that did not play an instrument.
The mean absolute deviation for students who play an instrument is 1.
The data for the group that does not play an instrument are more spread out than the data for the group that does play an instrument
The mean absolute deviation for the group of students who do not play an instrument is 2.
The data for the group that does not play an instrument are more clustered around the mean than the data for the group that does play an instrument.

Answer 1

To analyze the data sets, we first need to fill in the table based on the provided hours of reading for each group:

1. **Students who play an instrument:**
- Student 1: 16
- Student 2: 18
- Student 3: 15
- Student 4: 15
- **Average for this group:**
\[
\text{Mean} = \frac{16 + 18 + 15 + 15}{4} = \frac{64}{4} = 16
\]

2. **Students who do not play an instrument:**
- Student 5: 2
- Student 6: 2
- Student 7: 4
- Student 8: 8
- **Average for this group:**
\[
\text{Mean} = \frac{2 + 2 + 4 + 8}{4} = \frac{16}{4} = 4
\]

Next, we need to see the spread of each data set, which can be determined by the mean absolute deviation (MAD).

### Calculating Mean Absolute Deviation (MAD)

**For students who play an instrument:**
1. Mean: 16
2. Deviations:
- |16 - 16| = 0
- |18 - 16| = 2
- |15 - 16| = 1
- |15 - 16| = 1
3. MAD:
\[
\text{MAD} = \frac{0 + 2 + 1 + 1}{4} = \frac{4}{4} = 1
\]

**For students who do not play an instrument:**
1. Mean: 4
2. Deviations:
- |2 - 4| = 2
- |2 - 4| = 2
- |4 - 4| = 0
- |8 - 4| = 4
3. MAD:
\[
\text{MAD} = \frac{2 + 2 + 0 + 4}{4} = \frac{8}{4} = 2
\]

### Evaluating the Statements

1. **The data for the group that plays an instrument are more spread out than the data for the group that did not play an instrument.**
- **False** (Playing instruments: MAD = 1; Not playing instruments: MAD = 2)

2. **The data for the group that plays an instrument are more clustered around the mean than the data for the group that did not play an instrument.**
- **True** (MAD for those playing music is less than those not playing; thus, they are more clustered)

3. **The mean absolute deviation for students who play an instrument is 1.**
- **True** (Calculated MAD = 1)

4. **The data for the group that does not play an instrument are more spread out than the data for the group that does play an instrument.**
- **True** (MAD for those not playing instruments is greater)

5. **The mean absolute deviation for the group of students who do not play an instrument is 2.**
- **True** (Calculated MAD = 2)

6. **The data for the group that does not play an instrument are more clustered around the mean than the data for the group that does play an instrument.**
- **False** (The group that plays an instrument is more clustered as indicated by a lower MAD)

### Conclusion of True Statements
- The true statements are:
- The data for the group that plays an instrument are more clustered around the mean than the data for the group that did not play an instrument.
- The mean absolute deviation for students who play an instrument is 1.
- The data for the group that does not play an instrument are more spread out than the data for the group that does play an instrument.
- The mean absolute deviation for the group of students who do not play an instrument is 2.