### 第二部分：统计学 [60 分] 在上次选举中，在计票完成之前出现了来自非官方渠道的猜测。为了防止本次选举中出现…

第二部分：统计学 [60 分]

在上次选举中，在计票完成之前出现了来自非官方渠道的猜测。为了防止本次选举中出现不必要的猜测，阳光岛的选举办公室将在投票结束后在每个选区进行抽样计票。每个选区的登记选民人数不同，每个选区将抽取 400 张选票作为样本。

(i)
确定并描述一种获取该样本的合适方法。
(ii)
指出在第 (i) 部分中所使用的抽样方法的一个优点和一个缺点。

总体与简单随机样本 English

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完整解法

5(i)

1. Identify the appropriate sampling method.

Method: Systematic Sampling

2. Define the sampling interval.

Let $N$ be the total number of votes cast in the electoral division.
Calculate the sampling interval $k = \lfloor \frac{N}{400} \rfloor$ .

3. Describe the selection process.

Randomly choose a number $r$ between $1$ and $k$ inclusive.
Select the $r$ -th vote, and subsequently pick every $k$ -th vote (i.e., $r+k$ , $r+2k$ , ...) until $400$ votes have been sampled.

答案

Method: Systematic Sampling

Description:

Let $N$ be the total number of votes in the electoral division.
Determine the sampling interval $k = \lfloor \frac{N}{400} \rfloor$ .
Randomly select a starting vote from the 1st to the $k$ -th vote, let this be the $r$ -th vote.
Select every $k$ -th vote thereafter (i.e., the $(r+k)$ -th, $(r+2k)$ -th vote, etc.) until exactly 400 votes are obtained.

5(ii)

1. State an advantage of the sampling method (Simple Random Sampling)

Advantage: The sample is unbiased because every vote in the electoral division has an equal chance of being selected.

2. State a disadvantage of the sampling method

Disadvantage: It is time-consuming and tedious to implement, as every single vote in the electoral division needs to be assigned a number in order to randomly select 400 of them.

答案

Advantage: The sample is unbiased as every vote has an equal chance of being selected.
Disadvantage: It is time-consuming and tedious to number every single vote in the electoral division to perform the random selection.

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