The point estimate of the population standard deviation for the rando…

The correct answer is Option (3) → 3.1

The point estimate of the population standard deviation uses the divisor n − 1, since \(s^2\) with this divisor is the unbiased estimate of the population variance:

    \( \displaystyle s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}} \)

Here \(n = 6\):

    \( \displaystyle \bar{x} = \frac{5 + 8 + 10 + 7 + 10 + 14}{6} = \frac{54}{6} = 9 \)

Squared deviations from 9:

    \(16,\ 1,\ 1,\ 4,\ 1,\ 25\)
\(\sum (x - \bar{x})^2 = 48\)

    \( \displaystyle s = \sqrt{\frac{48}{5}} = \sqrt{9.6} \approx 3.098 \)

So the estimate is about 3.1. (With divisor n the value would be \(\sqrt{8} \approx 2.83\), which is not among the options.)