Question

Ramos 老师给 $20$ 名学生进行了一次测试。下面的点图显示了成绩分布。

后来 Ramos 老师发现某题评分有误。他重新评分，给一些学生额外加了 $5$ 分，使得测试成绩的中位数提高到 $85$ 。至少有多少名学生得到了额外分数？

（注意，若 $20$ 个成绩按从小到大排列，中位数等于中间 $2$ 个成绩的平均数。）

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.)

$2$

$3$

$4$

$5$

$6$

答案：C

知识点：中位数（数据）数据与图表解读

难度评级：1370

解答：

所有成绩都是 $5$ 的倍数，加上 $5$ 分后仍然如此。如果第 $10$ 和第 $11$ 个成绩的平均数是 $85$ ，但两者不都是 $85$ ，那么第 $11$ 个成绩至少为 $90$ 。这就要求至少有 $10$ 个成绩不低于 $90$ 。

可是重新评分后，只有原来不低于 $85$ 的成绩才可能达到 $90$ 。点图中这样的成绩只有 $7$ 个，所以中间两个成绩必须都是 $85$ 。

点图中原来有 $7$ 个不低于 $85$ 的成绩。要使第 $10$ 和第 $11$ 个成绩都为 $85$ ，至少要有 $11$ 个成绩不低于 $85$ 。把四个 $80$ 分提高到 $85$ 分既是必要的，也是充分的。

所以正确答案是 C。

All scores are multiples of $5,$ and adding $5$ preserves that fact. If the $10$ th and $11$ th scores average $85$ but are not both $85,$ then the $11$ th score must be at least $90.$ That would require at least $10$ scores of $90$ or more.

However, even after regrading, only the original scores of $85$ or more can reach $90.$ The plot has only $7$ such scores, so the two middle scores must both be $85.$

The plot initially has $7$ scores of at least $85.$ To make the $10$ th and $11$ th scores both $85,$ there must be at least $11$ scores of at least $85.$ Raising four of the $80$ 's to $85$ is both necessary and sufficient.

Thus, the correct answer is C.

2022 AMC 8 第 19 题

答案：C

难度评级：1370

解答：

其他年份的第 19 题