2021 AMC 12A Fall 第 7 题

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：

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