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 score will increase the most?
Dana: 68,74,83,80,81,82
Jose: 91,93,97,96,96,96
Ara: 100,98,99,97,100,100
Theo: 84,88,81,85,77,76

Answer 1

To determine whose mean score will increase the most after receiving a score of 100 on their latest test, we first need to calculate the mean of each individual's previous scores, then add the new score and calculate the new mean.

### Step 1: Calculate the mean for each student from their previous six tests.

#### Dana:
- Previous scores: 68, 74, 83, 80, 81, 82
- Mean = (68 + 74 + 83 + 80 + 81 + 82) / 6 = 468 / 6 = 78

#### Jose:
- Previous scores: 91, 93, 97, 96, 96, 96
- Mean = (91 + 93 + 97 + 96 + 96 + 96) / 6 = 569 / 6 = 94.8333 (approximately 94.83)

#### Ara:
- Previous scores: 100, 98, 99, 97, 100, 100
- Mean = (100 + 98 + 99 + 97 + 100 + 100) / 6 = 594 / 6 = 99

#### Theo:
- Previous scores: 84, 88, 81, 85, 77, 76
- Mean = (84 + 88 + 81 + 85 + 77 + 76) / 6 = 491 / 6 = 81.8333 (approximately 81.83)

### Step 2: Calculate new mean after adding the score of 100 for each student.

#### Dana:
- New total score = 468 + 100 = 568
- New mean = 568 / 7 = 81.1429 (approximately 81.14)
- Increase in mean = 81.14 - 78 = 3.14

#### Jose:
- New total score = 569 + 100 = 669
- New mean = 669 / 7 = 95.5714 (approximately 95.57)
- Increase in mean = 95.57 - 94.83 = 0.74

#### Ara:
- New total score = 594 + 100 = 694
- New mean = 694 / 7 = 99.1429 (approximately 99.14)
- Increase in mean = 99.14 - 99 = 0.14

#### Theo:
- New total score = 491 + 100 = 591
- New mean = 591 / 7 = 84.4286 (approximately 84.43)
- Increase in mean = 84.43 - 81.83 = 2.60

### Step 3: Compare the increases in mean.
- Dana: Increase of 3.14
- Jose: Increase of 0.74
- Ara: Increase of 0.14
- Theo: Increase of 2.60

### Conclusion:
Dana's mean score will increase the most by **3.14 points**.