The numbers of defective items found in a simple random sample of six boxes are 12, 8, 9, 6, 10, 9. The point estimate of the population standard deviation is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 2
The point estimate of the population standard deviation is the sample standard deviation
\( \displaystyle s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}} \)
The sample mean:
\( \displaystyle \bar{x} = \frac{12 + 8 + 9 + 6 + 10 + 9}{6} = \frac{54}{6} = 9 \)
The deviations from 9 are 3, −1, 0, −3, 1, 0, so
\(\sum (x - \bar{x})^2 = 9 + 1 + 0 + 9 + 1 + 0 = 20\)
With n − 1 = 5:
\( \displaystyle s = \sqrt{\frac{20}{5}} = \sqrt{4} = 2 \)
Why the other options are incorrect
Option (1) 1.83 — Incorrect. This is \(\sqrt{20/6}\), dividing by n = 6 instead of n − 1 = 5.
Option (2) 4 — Incorrect. This is \(s^2\), the estimate of the variance; its square root is needed.
Option (3) 3.33 — Incorrect. This is 20 ÷ 6, a variance found with the wrong divisor and no square root.