Question

The following data is taken from a simple random sample :
3, 7, 5, 9, 15, 11, 8, 4, 6, 2
The point estimate of the population standard deviation is :

• A. \(\frac{2}{3}\)
• B. \(\frac{4}{9}\)
• C. \(\frac{2}{3}\sqrt{35}\)
• D. \(\frac{4}{9}\sqrt{35}\)

Answer 1

The Correct Option is C

To calculate the point estimate of the population standard deviation for the given sample data \(3, 7, 5, 9, 15, 11, 8, 4, 6, 2\), follow these steps: 1. Calculate the sample mean (\(\bar{x}\)): \(\bar{x} = \frac{3 + 7 + 5 + 9 + 15 + 11 + 8 + 4 + 6 + 2}{10} = 7\). 2. Calculate the sample variance (\(s^2\)), using the formula: \[s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}\] where \(n\) is the sample size, which is 10. \[ \begin{align*} (x_1 - \bar{x})^2 = (3-7)^2 = 16, \\ (x_2 - \bar{x})^2 = (7-7)^2 = 0, \\ (x_3 - \bar{x})^2 = (5-7)^2 = 4, \\ (x_4 - \bar{x})^2 = (9-7)^2 = 4, \\ (x_5 - \bar{x})^2 = (15-7)^2 = 64, \\ (x_6 - \bar{x})^2 = (11-7)^2 = 16, \\ (x_7 - \bar{x})^2 = (8-7)^2 = 1, \\ (x_8 - \bar{x})^2 = (4-7)^2 = 9, \\ (x_9 - \bar{x})^2 = (6-7)^2 = 1, \\ (x_{10} - \bar{x})^2 = (2-7)^2 = 25 \end{align*} \] \[ \sum (x_i - \bar{x})^2 = 16 + 0 + 4 + 4 + 64 + 16 + 1 + 9 + 1 + 25 = 140 \] \[ s^2 = \frac{140}{10-1} = \frac{140}{9} \] 3. Calculate the sample standard deviation (\(s\)): \[ s = \sqrt{s^2} = \sqrt{\frac{140}{9}} = \frac{\sqrt{140}}{3} \] 4. The point estimate of the population standard deviation is \(\frac{2}{3}\sqrt{35}\). Thus, the point estimate of the population standard deviation is the simplified form \(\frac{2}{3}\sqrt{35}\), aligning with the provided data and possible answers.