To find the variance of the given population, follow these steps:
1. Understand the data provided:
- The mean [tex]\(\mu\)[/tex] of the population is given as 6.
- The data values [tex]\(x\)[/tex] are: 1, 5, 7, 11.
- The squared differences from the mean, [tex]\((x - \mu)^2\)[/tex], are provided for each [tex]\(x\)[/tex]:
- For 1: [tex]\((1 - 6)^2 = 25\)[/tex]
- For 5: [tex]\((5 - 6)^2 = 1\)[/tex]
- For 7: [tex]\((7 - 6)^2 = 1\)[/tex]
- For 11: [tex]\((11 - 6)^2 = 25\)[/tex]
2. Sum the squared differences:
[tex]\[
\sum (x - \mu)^2 = 25 + 1 + 1 + 25 = 52
\][/tex]
3. Find the population size (N):
[tex]\[
N = 4
\][/tex]
4. Calculate the variance of the population using the formula:
[tex]\[
\sigma^2 = \frac{\sum (x - \mu)^2}{N}
\][/tex]
Substituting the values:
[tex]\[
\sigma^2 = \frac{52}{4} = 13
\][/tex]
Therefore, the variance of the population is:
[tex]\[
\boxed{13}
\][/tex]
The correct answer is (A) 13.
