某办公室在特定的一天共收到了 810 份贷款申请。这些申请被分为商业、住房和助学贷款。由于人手不足，该办公室在当天只能处理 30 份申请。

((i))
描述你将如何从这些贷款申请中抽取一个容量为 30 的系统样本。
((ii))
解释为什么系统样本可能无法确保当天处理所有类别的贷款申请。
((iii))
结合本题背景，解释为什么分层样本可能比系统样本更优。

总体与简单随机样本 English

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完整解法

6((i))

1. Determine the sampling interval and number the applications.

Number the $810$ loan applications from $1$ to $810$ .
The sampling interval is $k = \frac{810}{30} = 27$ .

2. Describe the random start and the selection of subsequent items.

Randomly select a starting number $r$ from $1$ to $27$ (inclusive) using a random number generator.
Then, select every $27$ th application, i.e., select the applications numbered $r$ , $r+27$ , $r+54$ , $\dots$ until $30$ applications have been chosen.

答案

Number the $810$ loan applications from $1$ to $810$ . Since the sampling interval is $\frac{810}{30} = 27$ , randomly select an integer $r$ from $1$ to $27$ (inclusive) using a random number generator. Then select the applications numbered $r$ , $r+27$ , $r+54$ , $\dots$ until $30$ applications are chosen.

6((ii))

1. Explain why systematic sampling might miss a specific category.

Systematic sampling does not explicitly consider the categories of the applications. If the applications are ordered in a specific way or if a category has very few applications, they might fall entirely between the selected sample intervals, resulting in that category being excluded from the sample.

答案

Systematic sampling does not take the different categories into account. Depending on how the applications are ordered in the list, or if a category has very few applications, it is possible that none of the applications from that category fall on the selected intervals (e.g., every $27^{\text{th}}$ application), causing the category to be completely missed.

6((iii))

1. State the advantage of a stratified sample over a systematic sample in this context.

A stratified sample guarantees that all three categories (business, house, and study loans) are represented in the sample, proportional to their frequency in the total 810 applications, whereas a systematic sample might omit a category.

答案

A stratified sample ensures that the applications for all three categories (business, house, and study loans) are proportionally represented in the sample of 30.

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某办公室在特定的一天共收到了 810 份贷款申请。这些申请被分为商业、住房和助学贷款。由于人手不足，该办公室在当天只能处理 30 份申请。

完整解法

6((i))

6((ii))

6((iii))

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