Question

A simple random sample consists of four observations 7, 8, 10, 7. The point estimate of population standard deviation is :

• A. \(\sqrt{\frac{3}{2}}\)
• B. \(\sqrt{3}\)
• C. 2.5
• D. \(\sqrt{2}\)

Answer 1

The Correct Option is D

To find the point estimate of the population standard deviation from a simple random sample, we first calculate the sample standard deviation using the following steps:

1. Determine the sample mean (\(\bar{x}\)): \(\bar{x} = \frac{7 + 8 + 10 + 7}{4} = \frac{32}{4} = 8\).

2. Calculate each observation's deviation from the mean, square the deviations, and sum them:
\((7-8)^2+(8-8)^2+(10-8)^2+(7-8)^2 = 1^2+0^2+2^2+1^2 = 1+0+4+1=6\).

3. Divide by the number of observations minus one (n-1) to find the sample variance: \(\frac{6}{4-1}=\frac{6}{3}=2\).

4. The sample standard deviation is the square root of the sample variance: \(\text{Sample Standard Deviation} = \sqrt{2}\).

Thus, the point estimate of the population standard deviation is \(\sqrt{2}\).