94, 73, 68, 83, 100, 68, these are the test scores.
The student takes 2 more 100 point tests Before the grading period closes. The addition of these 2 test scores has no effect on the mean, but the median score of all the tests increased by 2 points. What was the scores of the 2 most recent tests?
To solve this problem, we need to first find the median score of the original 7 tests.
Arranging the scores in order:
68, 68, 73, 83, 94, 100
The median score is 83.
Since the median score increased by 2 points after adding the 2 new test scores, the new median score must be 85.
We can add the 2 new test scores (let's call them x and y) to the original scores and find the new median:
68, 68, 73, 83, 94, 100, x, y
To find the new median, we need to find the middle two numbers since there are now 8 scores.
The middle two numbers are the original 83 and the new median of 85.
So, we can set up an equation:
(83 + 85) / 2 = (68 + 68 + 73 + 83 + 94 + 100 + x + y) / 8
Simplifying this equation:
84 = (536 + x + y) / 8
672 = 536 + x + y
x + y = 136
We also know that the mean (average) of the 9 scores is the same as the mean of the original 7 scores.
The sum of the original 7 scores is:
68 + 68 + 73 + 83 + 94 + 100 + 68 = 554
The sum of all 9 scores is:
68 + 68 + 73 + 83 + 94 + 100 + x + y + 100 = 686 + x + y
We can set up another equation:
(686 + x + y) / 9 = 554 / 7
Simplifying:
686 + x + y = 791.43
x + y = 105.43
Now we have two equations:
x + y = 136
x + y = 105.43
Subtracting the second equation from the first:
x + y - x - y = 136 - 105.43
0 = 30.57
This is impossible, so our original assumption (that the median increased by 2 points after adding two 100-point tests) must be incorrect.
Therefore, there is no solution to this problem.
To start, let's calculate the mean score and the median score using the given test scores:
Mean score = (94 + 73 + 68 + 83 + 100 + 68) / 6 = 86.0
Now, let's arrange the test scores in ascending order: 68, 68, 73, 83, 94, 100.
Since the median score increased by 2 points, the median score before the addition of the two new scores was 83. Therefore, the median score after the addition of the two new scores is 85.
Now, let's list all the possible scores for the two most recent tests, and calculate the median for each scenario:
1. Scenario: (100, 100)
Median score: (73, 83, 94, 100, 100) = 94
2. Scenario: (99, 101)
Median score: (73, 83, 94, 99, 100, 101) = 96.5
3. Scenario: (98, 102)
Median score: (73, 83, 94, 98, 100, 102) = 96
4. Scenario: (97, 103)
Median score: (73, 83, 94, 97, 100, 103) = 96.5
From the above scenarios, we can see that the only possibility where the median score increases by 2 points to 85 is in Scenario 4, where the scores of the two most recent tests are 97 and 103.