Question

一所学校有 $100$ 名学生和 $5$ 名老师。第一节课中，每名学生上一门课，每名老师教一门课。五门课的人数分别为 $50$ ， $20$ ， $20$ ， $5$ ，和 $5$ 。随机选一名老师，记录其班级的学生人数，所得数值的平均值记为 $t$ 。随机选一名学生，记录其所在班级的学生人数（包括这名学生本人），所得数值的平均值记为 $s$ 。 $t - s$ 是多少？

A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50,$ $20,$ $20,$ $5,$ and $5.$ Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $t - s?$

$-18.5$

$-13.5$

$0$

$13.5$

$18.5$

答案：B

知识点：平均数期望值

难度评级：1410

小提示：

$t$ 是对 $5$ 个班级的普通平均： $\dfrac{50+20+20+5+5}{5}$ 。

$t$ is a plain average over the $5$ classes: $\dfrac{50+20+20+5+5}{5}$

大提示：

对于 $s$ ，每个大小为 $n$ 的班级会被计数 $n$ 次： $s = \dfrac{\sum n^2}{100}$ 。

For $s,$ each class of size $n$ is counted $n$ times: $s = \dfrac{\sum n^2}{100}$

解答：

老师平均值为 $t = \dfrac{50 + 20 + 20 + 5 + 5}{5}$ $= \dfrac{100}{5} = 20$ 。

学生平均值按每个班级中的学生人数给班级人数加权： $\begin{aligned} s &= \small \frac{50^2 + 20^2 + 20^2 + 5^2 + 5^2}{100} \\ &= \small \frac{2500 + 400 + 400 + 25 + 25}{100} \\ &= 33.5 \end{aligned}\text{。}$

所以 $t - s = 20 - 33.5 = -13.5$ 。

所以正确答案是 B。

The teacher average is $t = \dfrac{50 + 20 + 20 + 5 + 5}{5}$ $= \dfrac{100}{5} = 20.$

The student average weights each class size by how many students are in it: $\begin{aligned} s &= \small \frac{50^2 + 20^2 + 20^2 + 5^2 + 5^2}{100} \\ &= \small \frac{2500 + 400 + 400 + 25 + 25}{100} \\ &= 33.5. \end{aligned}$

So $t - s = 20 - 33.5 = -13.5.$

Thus, the correct answer is B.

2021 AMC 12A Fall 第 7 题

答案：B

难度评级：1410

小提示：

大提示：

解答：

其他年份的第 7 题