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贷款、房屋贷款和助学贷款三类。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 的系统抽样样本。
(ii)
解释为什么系统抽样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，解释为什么分层抽样可能比系统抽样更可取。

它不会直接告诉你答案，而是问你卡在哪一步看看 AI 是怎么陪你一步步弄懂这道题的

完整解法

6(i)

1. Determine the sampling interval and describe the random start.

Number the loan applications from $1$ to $810$ .
The sampling interval $k = \frac{810}{30} = 27$ .
Randomly choose an integer $r$ between $1$ and $27$ inclusive.

2. Describe the systematic selection of the remaining items.

Select the $r$ -th application and every $27$ th application thereafter (i.e., applications numbered $r, r+27, r+54, \dots$ ) until a sample of $30$ applications is obtained.

答案

Number the applications from $1$ to $810$ . Randomly select an integer $r$ from $1$ to $27$ inclusive. Select the $r$ -th application, and every $27$ th application thereafter until $30$ applications are chosen.

6(ii)

1. Explain how the order and sampling interval in systematic sampling might miss an entire category.

Systematic sampling relies on selecting items at regular intervals (every 27th application, as $810 \div 30 = 27$ ). If the applications are arranged by category, and one category has fewer than 27 applications or is clustered entirely between the selected intervals, it will not be represented in the sample.

答案

In systematic sampling, the selection depends heavily on the ordering of the list. If a particular category of loans has a small number of applications that are clustered together, they might all fall between the regular sampling intervals, causing that category to be completely skipped.

6(iii)

1. State the advantage of stratified sampling relating to the categories.

A stratified sample guarantees that all three categories of loan applications (business, house, and study loans) are represented in the sample, whereas a systematic sample might miss a category.

答案

A stratified sample guarantees that all three categories of loan applications (business, house, and study loans) are represented in the sample, whereas a systematic sample might miss a category.

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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贷款、房屋贷款和助学贷款三类。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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