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?
Ara: 100, 98, 99, 97, 100, 100
Jose: 91, 93, 97, 96, 96, 96
Dana: 68, 74, 83, 80, 81, 82
Theo: 84, 88, 81, 85, 77, 76
Theo's mean (average) test score will increase the most because his previous scores are significantly lower than 100 and therefore have a greater potential to be improved by a perfect score.
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To determine whose mean test score will increase the most, we need to calculate the mean (average) test score for each of the four students before and after the latest math test.
For Ara, the mean test score before the latest math test is (100 + 98 + 99 + 97 + 100 + 100) / 6 = 99.
For Ara, the mean test score after the latest math test is (100 + 100 + 98 + 99 + 97 + 100 + 100) / 7 = 99.14.
Therefore, Ara's mean test score increased by 99.14 - 99 = 0.14.
For Jose, the mean test score before the latest math test is (91 + 93 + 97 + 96 + 96 + 96) / 6 = 95.
For Jose, the mean test score after the latest math test is (100 + 91 + 93 + 97 + 96 + 96 + 96) / 7 = 94.86.
Therefore, Jose's mean test score increased by 94.86 - 95 = -0.14.
For Dana, the mean test score before the latest math test is (68 + 74 + 83 + 80 + 81 + 82) / 6 = 77.6.
For Dana, the mean test score after the latest math test is (100 + 68 + 74 + 83 + 80 + 81 + 82) / 7 = 82.71.
Therefore, Dana's mean test score increased by 82.71 - 77.6 = 5.11.
For Theo, the mean test score before the latest math test is (84 + 88 + 81 + 85 + 77 + 76) / 6 = 82.83.
For Theo, the mean test score after the latest math test is (100 + 84 + 88 + 81 + 85 + 77 + 76) / 7 = 86.57.
Therefore, Theo's mean test score increased by 86.57 - 82.83 = 3.74.
Comparing the increase in mean test scores, we find that:
- Ara's mean test score increased by 0.14.
- Jose's mean test score decreased by 0.14.
- Dana's mean test score increased by 5.11.
- Theo's mean test score increased by 3.74.
Therefore, Dana's mean test score will increase the most.
To determine whose mean test score will increase the most, we need to calculate the mean test score for each person and compare the differences between their current mean and previous mean scores.
Let's start by calculating the mean test score for each person:
For Ara:
Mean = (100 + 98 + 99 + 97 + 100 + 100) / 6 = 99.33
For Jose:
Mean = (91 + 93 + 97 + 96 + 96 + 96) / 6 = 95.83
For Dana:
Mean = (68 + 74 + 83 + 80 + 81 + 82) / 6 = 77.67
For Theo:
Mean = (84 + 88 + 81 + 85 + 77 + 76) / 6 = 81.83
Now, let's calculate the differences between their current mean and previous mean scores:
For Ara: Current mean - Previous mean = 99.33 - 99.33 = 0
For Jose: Current mean - Previous mean = 95.83 - 95.83 = 0
For Dana: Current mean - Previous mean = 77.67 - 77.67 = 0
For Theo: Current mean - Previous mean = 81.83 - 81.83 = 0
From the calculations, we can see that all four students have the same mean test scores for their previous and current tests. Therefore, their mean test scores will all increase by the same amount, which is 0.
Hence, no one's mean test score will increase more than the others.