If the following data are from a simple random sample : 1, 4, 6, 9, 10, 12, then which of the following statements are correct ?
(A). Sample mean = 7
(B). Sample standard deviation = 4
(C). The standard deviation has 5 degrees of freedom.
(D). Sample standard deviation = \( \sqrt{16.8} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (C) and (D) only
The mean is
\( \displaystyle \bar{x}=\frac{1+4+6+9+10+12}{6}=7 \)
The squared deviations from the mean are \( 36,\,9,\,1,\,4,\,9,\,25 \), with sum 84. For a sample the divisor is \( n-1=5 \):
\( \displaystyle s^{2}=\frac{84}{5}=16.8 \)
\( s=\sqrt{16.8}\approx 4.10 \)
So the sample mean is 7, the sample standard deviation is \( \sqrt{16.8}\approx 4.10 \), and it has \( n-1=5 \) degrees of freedom.
Checking the statements
(A) Sample mean = 7 — Correct. \( \frac{42}{6}=7 \).
(B) Sample standard deviation = 4 — Incorrect. \( s=\sqrt{16.8}\approx 4.10 \), not 4.
(C) The standard deviation has 5 degrees of freedom. — Correct. \( n-1=6-1=5 \).
(D) Sample standard deviation = \( \sqrt{16.8} \) — Correct. \( s^{2}=\frac{84}{5}=16.8 \).