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.

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Worked solution

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.

Answer

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.

Answer

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.

Answer

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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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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