A simple random sample consists of five observations: 2, 4, 6, 7, 11, then which of the following are true?
A. Sample mean = 7.5
B. Sample mean = 6
C. Sample standard deviation = \( \sqrt{11} \)
D. Sample standard deviation = \( \sqrt{11.5} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → B and D only
The sample mean is
\( \displaystyle \bar{x}=\frac{2+4+6+7+11}{5} \)
\( \displaystyle \bar{x}=\frac{30}{5}=6 \)
Squared deviations from the mean:
\( \displaystyle (2-6)^2+(4-6)^2+(6-6)^2+(7-6)^2+(11-6)^2 \)
\( \displaystyle =16+4+0+1+25=46 \)
The sample standard deviation uses the divisor n − 1 = 4:
\( \displaystyle s=\sqrt{\frac{\sum(x-\bar{x})^2}{n-1}} \)
\( \displaystyle s=\sqrt{\frac{46}{4}}=\sqrt{11.5} \)
So the sample mean is 6 and the sample standard deviation is \( \sqrt{11.5} \).
Checking the statements
(A) Sample mean = 7.5 — Incorrect. The mean is 6.
(B) Sample mean = 6 — Correct. 30 ÷ 5 = 6.
(C) Sample standard deviation = \( \sqrt{11} \) — Incorrect. 46 ÷ 4 = 11.5, not 11.
(D) Sample standard deviation = \( \sqrt{11.5} \) — Correct. 46 ÷ 4 = 11.5.
Tip: For a sample standard deviation divide by n − 1, not n; dividing by 5 here would give \( \sqrt{9.2} \).