题目

Problem

The owner of a very large youth club has designed a new method for allocating people to teams. Before introducing the method he decided to find out how the members of the youth club might react.

(a) Explain why the owner decided to take a random sample of the youth club members rather than ask all the youth club members.

(1)

(b) Suggest a suitable sampling frame.

(1)

The new method uses a bag containing a large number of balls. Each ball is numbered either 20, 50 or 70 When a ball is selected at random, the random variable X represents the number on the ball where P(X = 20) = p, P(X = 50) = q, P(X = 70) = r

A youth club member takes a ball from the bag, records its number and replaces it in the bag. He then takes a second ball from the bag, records its number and replaces it in the bag. The random variable M is the mean of the 2 numbers recorded.

Given that P(M = 20) = 1/16, show that P(M = 50) = 25/64, P(M = 60) = 1/16 and q > r

(7)

(Total for Question 6 is 10 marks)

题目中文翻译

一位大型青年俱乐部的负责人设计了一种新的分组方法。在引入该方法之前，他决定了解俱乐部成员可能的反应。

(a) 解释为什么负责人决定抽取俱乐部成员的随机样本，而不是询问所有成员。

(b) 提出一个合适的抽样框。

新的方法使用一个装有许多球的袋子。每个球上写着 20、50 或 70 之一。

随机选取一个球时，随机变量 $X$ 表示球上的数字，其中

$P(X=20)=p,\; P(X=50)=q,\; P(X=70)=r$ 。

一名俱乐部成员从袋中取出一个球，记下数字后放回袋中。然后再取出第二个球，记下数字后再放回袋中。随机变量 $M$ 表示记录下的两个数字的平均数。

已知 $P(M=20)=1/16$ ，证明 $P(M=50)=25/64$ ， $P(M=60)=1/16$ ，且 $q>r$ 。

（第 6 题共 10 分）

IAL 2021 Jan S2 Q6

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