An office received a total of 810 loan applications on a particular day. The applications were categorised into business, house and study loans. Due to a shortage of staff, the office was only able to process 30 applications on that day.

((i))
Describe how you would choose a systematic sample of size 30 from the loan applications.
((ii))
Explain why a systematic sample may not ensure that all categories of loan applications were processed within that day.
((iii))
Explain why a stratified sample may be preferred over a systematic sample in the context of this question.

Population and simple random sample 中文

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Worked solution

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.

Answer

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.

Answer

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.

Answer

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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An office received a total of 810 loan applications on a particular day. The applications were categorised into business, house and study loans. Due to a shortage of staff, the office was only able to process 30 applications on that day.

Worked solution

6((i))

6((ii))

6((iii))

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