Question

Theo, Ara, Jose, and Dana all got 100 percent on their latest math test. Their scores on the previous six tests are listed. Whose mean test score will increase the most? (1 point)
Responses

Ara: 100, 98, 99, 97, 100, 100
Ara: 100, 98, 99, 97, 100, 100

Theo: 84, 88, 81, 85, 77, 76
Theo: 84, 88, 81, 85, 77, 76

Jose: 91, 93, 97, 96, 96, 96
Jose: 91, 93, 97, 96, 96, 96

Dana: 68, 74, 83, 80, 81, 82
Dana: 68, 74, 83, 80, 81, 82

Answer 1

To determine whose mean test score will increase the most after scoring 100 percent on their latest test, we first need to calculate the current mean scores for each student, then find the new mean after adding the score of 100.

### Step 1: Calculate current mean scores

1. **Ara's scores:** 100, 98, 99, 97, 100, 100
\[
\text{Mean} = \frac{100 + 98 + 99 + 97 + 100 + 100}{6} = \frac{594}{6} = 99
\]

2. **Theo's scores:** 84, 88, 81, 85, 77, 76
\[
\text{Mean} = \frac{84 + 88 + 81 + 85 + 77 + 76}{6} = \frac{491}{6} \approx 81.83
\]

3. **Jose's scores:** 91, 93, 97, 96, 96, 96
\[
\text{Mean} = \frac{91 + 93 + 97 + 96 + 96 + 96}{6} = \frac{579}{6} = 96.5
\]

4. **Dana's scores:** 68, 74, 83, 80, 81, 82
\[
\text{Mean} = \frac{68 + 74 + 83 + 80 + 81 + 82}{6} = \frac{468}{6} = 78
\]

### Step 2: Calculate new mean scores

Now, we will add the 100 percent score for each student and calculate the new mean.

1. **Ara's new mean:**
\[
\text{New Mean} = \frac{594 + 100}{7} = \frac{694}{7} \approx 99.14
\]
\[
\text{Increase} = 99.14 - 99 = 0.14
\]

2. **Theo's new mean:**
\[
\text{New Mean} = \frac{491 + 100}{7} = \frac{591}{7} \approx 84.43
\]
\[
\text{Increase} = 84.43 - 81.83 \approx 2.60
\]

3. **Jose's new mean:**
\[
\text{New Mean} = \frac{579 + 100}{7} = \frac{679}{7} \approx 97
\]
\[
\text{Increase} = 97 - 96.5 = 0.5
\]

4. **Dana's new mean:**
\[
\text{New Mean} = \frac{468 + 100}{7} = \frac{568}{7} \approx 81.14
\]
\[
\text{Increase} = 81.14 - 78 = 3.14
\]

### Summary of Increases
- Ara: Increase of 0.14
- Theo: Increase of 2.60
- Jose: Increase of 0.5
- Dana: Increase of 3.14

### Conclusion
Dana’s mean test score will increase the most by approximately **3.14 points**.