Question
The median test score earned by 8 sixth graders on their math exam is 85. The range of the test scores earned by the same sixth graders on the same exam is 12.
Which statement MUST be true based on the given information?
A At least one sixth grader scored an 85.
B The highest test score is less than 90.
C If the greatest test score is 97, then the lowest test score is 85.
D The mean test score is greater than 85.
Which statement MUST be true based on the given information?
A At least one sixth grader scored an 85.
B The highest test score is less than 90.
C If the greatest test score is 97, then the lowest test score is 85.
D The mean test score is greater than 85.
Answers
GPT-4o mini
To analyze the statements based on the information that the median test score is 85 and the range of the test scores is 12, let's break it down:
1. **Median of 85**: This means that when the scores are arranged in order, the average of the 4th and 5th scores (middle scores in an ordered list of 8) is 85. Thus, at least two of the scores must be 85 or higher.
2. **Range of 12**: The range is defined as the difference between the highest score and the lowest score. This means that if the highest score is X and the lowest score is Y, then X - Y = 12.
Now let's evaluate each statement:
**A. At least one sixth grader scored an 85.**
- This statement must be true, because the median being 85 implies that there are scores at least equal to 85.
**B. The highest test score is less than 90.**
- This statement is not necessarily true. The highest score could be 90 or more, as long as the lowest score is adjusted to maintain a range of 12.
**C. If the greatest test score is 97, then the lowest test score is 85.**
- This statement is also incorrect because if the highest score is 97, the lowest score would need to be 97 - 12 = 85. However, it doesn't follow that the lowest score cannot be lower than 85 while still allowing a median of 85 with sufficient other scores.
**D. The mean test score is greater than 85.**
- This statement does not have to be true. The mean could be equal to, less than, or greater than 85 depending on the actual distribution of scores.
Given this analysis, the only statement that **MUST** be true based on the provided information is:
**A. At least one sixth grader scored an 85.**
1. **Median of 85**: This means that when the scores are arranged in order, the average of the 4th and 5th scores (middle scores in an ordered list of 8) is 85. Thus, at least two of the scores must be 85 or higher.
2. **Range of 12**: The range is defined as the difference between the highest score and the lowest score. This means that if the highest score is X and the lowest score is Y, then X - Y = 12.
Now let's evaluate each statement:
**A. At least one sixth grader scored an 85.**
- This statement must be true, because the median being 85 implies that there are scores at least equal to 85.
**B. The highest test score is less than 90.**
- This statement is not necessarily true. The highest score could be 90 or more, as long as the lowest score is adjusted to maintain a range of 12.
**C. If the greatest test score is 97, then the lowest test score is 85.**
- This statement is also incorrect because if the highest score is 97, the lowest score would need to be 97 - 12 = 85. However, it doesn't follow that the lowest score cannot be lower than 85 while still allowing a median of 85 with sufficient other scores.
**D. The mean test score is greater than 85.**
- This statement does not have to be true. The mean could be equal to, less than, or greater than 85 depending on the actual distribution of scores.
Given this analysis, the only statement that **MUST** be true based on the provided information is:
**A. At least one sixth grader scored an 85.**