Question

A survey asked eight students about their scores on a history test and whether they studied for the test. The table shows the results of the survey.


Test Scores
Studied
Did Not Study
88
34
90
67
86
80
96
59


Which statements about the data sets are true? Check all that apply.
The mean absolute deviation for the group of students who studied is 12.
The mean absolute deviation for the group of students who did not study is 13.5.
The data for the group that studied are more spread out than the data for the group that did not study.
The data for the group that did not study are more spread out than the data for the group that studied.
The data for the group that studied are more clustered around the mean than the data for the group that did not study.
The data for the group that did not study are more clustered around the mean than the data for the group that did study.

Answer 1

To determine which statements about the data sets are true, we first need to calculate the mean and the mean absolute deviation (MAD) for both groups: those who studied and those who did not study.

### Group 1: Studied
Test Scores: 88, 90, 86, 96

**Mean:**
\[
\text{Mean} = \frac{88 + 90 + 86 + 96}{4} = \frac{360}{4} = 90
\]

**Mean Absolute Deviation (MAD):**
- Deviations from the mean:
- |88 - 90| = 2
- |90 - 90| = 0
- |86 - 90| = 4
- |96 - 90| = 6
- Sum of absolute deviations:
\[
2 + 0 + 4 + 6 = 12
\]
- MAD = \(\frac{12}{4} = 3\)

### Group 2: Did Not Study
Test Scores: 34, 67, 80, 59

**Mean:**
\[
\text{Mean} = \frac{34 + 67 + 80 + 59}{4} = \frac{240}{4} = 60
\]

**Mean Absolute Deviation (MAD):**
- Deviations from the mean:
- |34 - 60| = 26
- |67 - 60| = 7
- |80 - 60| = 20
- |59 - 60| = 1
- Sum of absolute deviations:
\[
26 + 7 + 20 + 1 = 54
\]
- MAD = \(\frac{54}{4} = 13.5\)

### Summary of Findings
From these calculations:
- **The MAD for the group that studied is 3.**
- **The MAD for the group that did not study is 13.5.**
- The scores for the group that studied are less spread out than the scores of the group that did not study, meaning:
- The group that studied has more clustered scores around their mean.
- The group that did not study has scores that are more spread out around their mean.

### Checking the Statements
1. **The mean absolute deviation for the group of students who studied is 12.** - **False**
2. **The mean absolute deviation for the group of students who did not study is 13.5.** - **True**
3. **The data for the group that studied are more spread out than the data for the group that did not study.** - **False**
4. **The data for the group that did not study are more spread out than the data for the group that studied.** - **True**
5. **The data for the group that studied are more clustered around the mean than the data for the group that did not study.** - **True**
6. **The data for the group that did not study are more clustered around the mean than the data for the group that did study.** - **False**

### True Statements
- The mean absolute deviation for the group of students who did not study is 13.5.
- The data for the group that did not study are more spread out than the data for the group that studied.
- The data for the group that studied are more clustered around the mean than the data for the group that did not study.