Assume a population of 3,5 , and 10 . 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 $\mathrm{d}$ below
\[
3,3
\]
\[
3,5
\]
3,10
5,3
5,5
5,10
10,3
10,5
10,10
a. Find the value of the population standard deviation $\sigma$
\[
\sigma=\square
\]
(Round to three decimal places as needed)

Step 1 ：The population in this case is \([3, 5, 10]\).

Step 2 ：First, calculate the mean of the population. The mean is the sum of all the numbers in the population divided by the count of numbers in the population. In this case, the mean is \((3+5+10)/3 = 6.0\).

Step 3 ：Next, subtract the mean from each number in the population and square the result. The results are \((3-6)^2 = 9\), \((5-6)^2 = 1\), and \((10-6)^2 = 16\).

Step 4 ：Then, find the average of these squared differences. This is called the variance. The variance is \((9+1+16)/3 = 8.666666666666666\).

Step 5 ：Finally, take the square root of the variance to get the standard deviation. The standard deviation is \(\sqrt{8.666666666666666} = 2.944\).

Step 6 ：Final Answer: The value of the population standard deviation \(\sigma\) is \(\boxed{2.944}\).