A systematic random sample of 20 shops is to be drawn from the 400 shops of a market, numbered 1 to 400. Which of the following statements are correct?
(A) The sampling interval is 20.
(B) The first shop is chosen at random from the shops numbered 1 to 20.
(C) If the first shop chosen is shop number 13, the 10th shop in the sample is shop number 213.
(D) If the first shop chosen is shop number 13, shop number 393 is also in the sample.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (D) only
The sampling interval is
\( \displaystyle k = \frac{400}{20} = 20 \)
The first shop is chosen at random from 1 to 20, and then every 20th shop is taken. With first shop 13, the r-th shop in the sample is
\(13 + (r - 1) \times 20\)
So the 10th shop is \(13 + 9 \times 20 = 193\), and the 20th (last) shop is \(13 + 19 \times 20 = 393\).
So k = 20, the start is random from 1 to 20, the 10th shop is 193 and shop 393 is in the sample.
Checking the statements
(A) The sampling interval is 20. — Correct. k = 400 ÷ 20 = 20.
(B) The first shop is chosen at random from the shops numbered 1 to 20. — Correct. The random start lies within the first interval of 20 shops.
(C) If the first shop chosen is shop number 13, the 10th shop in the sample is shop number 213. — Incorrect. The 10th shop is \(13 + 9 \times 20 = 193\).
(D) If the first shop chosen is shop number 13, shop number 393 is also in the sample. — Correct. \(393 = 13 + 19 \times 20\), the 20th shop.