Question

assume a population of 1, 2, and 12. assume that samples of size ( n = 2 ) are randomly selected with replacement from the population. listed below are the nine different samples. complete parts a through d below.
1,1 1,2 1,12 2,1 2,2 2,12 12,1 12,2 12,12

a. find the value of the population standard deviation ( sigma ).
( sigma=square )
(round to three decimal places as needed.)

Explanation:

Step1: Calculate the population mean

The population is \(1\), \(2\), \(12\). The population mean \(\mu=\frac{1 + 2+12}{3}=\frac{15}{3}=5\)

Step2: Calculate the squared - differences from the mean

For \(x = 1\): \((1 - 5)^2=( - 4)^2 = 16\)
For \(x = 2\): \((2 - 5)^2=( - 3)^2 = 9\)
For \(x = 12\): \((12 - 5)^2=7^2 = 49\)

Step3: Calculate the population variance

The population variance \(\sigma^{2}=\frac{(1 - 5)^{2}+(2 - 5)^{2}+(12 - 5)^{2}}{3}=\frac{16 + 9+49}{3}=\frac{74}{3}\approx24.667\)

Step4: Calculate the population standard deviation

The population standard deviation \(\sigma=\sqrt{\sigma^{2}}=\sqrt{\frac{74}{3}}\approx4.967\)

Answer:

\(4.967\)