From 11 positive integer scores on a 10-point quiz, the mean is 8, the median is 8, and the mode is 7. Find the maximum number of perfect scores possible on this test.
To find the maximum number of perfect scores possible on this test, we need to analyze the information given.
Let's start by understanding what each of the statistical measures means:
- Mean: It is the average of the scores. In this case, the mean is given as 8.
- Median: It is the middle value of the scores when arranged in ascending order. In this case, the median is given as 8.
- Mode: It is the value that appears most frequently in the set of scores. In this case, the mode is given as 7.
We know that the median is 8, which means that exactly half of the scores are equal to or less than 8, and the other half are equal to or greater than 8.
Since the mode is 7 and appears most frequently, there must be at least one score of 7.
Now, let's analyze the number of perfect scores. A perfect score is 10, and we want to find the maximum number of them.
Since the mean is 8, the sum of the scores must be 11 * 8 = 88. Let's denote the sum of the scores as S.
If we want to maximize the number of perfect scores, we should aim to minimize the other scores. Since we already have at least one score of 7, we need to distribute the remaining 10 - 7 = 3 points among the other 10 - 1 = 9 scores.
To minimize these 9 scores, each of them should be 1. This would give us 9 scores of 1, one score of 7, and one perfect score of 10. The sum of these scores would be 9 * 1 + 7 + 10 = 26.
Now, we can calculate the remaining points needed to reach the target sum of 88:
Remaining points = S - Sum of obtained scores = 88 - 26 = 62.
Since the maximum number of perfect scores is desired, we will distribute these remaining points in the form of 10s. Since each perfect score is worth 10, it is possible to have a maximum of 62 / 10 = 6.2 perfect scores.
However, since we are dealing with positive integer scores, we can only have a whole number of perfect scores. Therefore, the maximum number of perfect scores possible is 6.
In conclusion, the maximum number of perfect scores possible on this test is 6.
total points = 11*8 = 88
So, with median 8, the scores are
x x x x x 8 x x x x x
If the mode is 7, then for max scores of 10, we want the fewest under 8:
x x x 7 7 8 x x x x x
Now we cannot have duplicate scores below 7 (only one mode), so
1 2 3 7 7 8 x x x x x
They have to add up to 88, so since the bottom 6 scores add to 28, that leaves 60 points for the top 5 scores. If they were all 10's, that would only be 50 points, so we need another 10 points among the low scores. If we add 3 to the bottom scores, that is only 9 more, so let's have a 3rd 7 score, giving us
4 5 7 7 7 8 10 10 10 10 10