AMC 8 2022 Problem 19
AMC8 2022 年第 19 题
统计与数据图表★★★☆☆
Mr. Ramos gave a test to his class of 20 students. The dot plot below shows the distribution of test scores. Later Mr. Ramos discovered that there was a scoring error on one of the questions. He regraded the tests, awarding some of the students 5 extra points, which increased the median test score to 85 . What is the minimum number of students who received extra points? (Note that the median test score equals the average of the 2 scores in the middle if the 20 test scores are arranged in increasing order.)
Ramos 先生给他班上的 20 名学生进行了一次测试。下面的点图显示了考试成绩的分布。后来， Ramos 先生发现有一道题的评分有误。他重新批改，给一些学生各加了 5 分，使得考试成绩的中位数增加到了 85。问最少有多少名学生获得了加分？（请注意，如果 20 个考试成绩按递增顺序排列，那么中位数等于中间 2 个考试成绩的平均值。）
- A.2
- B.3
- C.4
- D.5
- E.6
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
We notice that 13 students have scores under 85 currently, and only 5 have scores over 85. We find the median of these two numbers, getting 13-5=8 13-4=9 Thus, we realize that 4 students must have their score increased by 5. So, the correct answer is 4.
We notice that if we number each of the dots from bottom to top in each column ( 65 's, 70 's, 75 's, etc.) to find the median, the second and third dots in the " 80 's" column correspond to the tenth and eleventh values in the set (or the closest values to the median). To make the median now 85, we must add 5 to the tenth and eleventh values in the set (or the second and third dots in the " 80 's" column), as well as all the dots above it in the " 80 's" column. So, the correct answer is 4.
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