To determine the variance of the sample, we will follow a step-by-step approach:
1. Identify the sample mean ( [tex]\(\bar{x} \)[/tex]):
The sample mean ( [tex]\(\bar{x} \)[/tex]) is given as 9.
2. Identify the data values and their deviations squared ( [tex]\((x_i - \bar{x})^2 \)[/tex]):
The data values are [tex]\(7, 15, 11, 1, 11\)[/tex].
Their corresponding squared deviations from the sample mean are [tex]\(4, 36, 4, 64, 4\)[/tex].
3. Count the number of data values ( [tex]\(n \)[/tex]):
There are 5 data values.
4. Calculate the sample variance ( [tex]\(s^2 \)[/tex]):
The formula for sample variance is:
[tex]\[
s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}
\][/tex]
Here,
[tex]\[
\sum (x_i - \bar{x})^2 = 4 + 36 + 4 + 64 + 4 = 112
\][/tex]
So, we will divide this sum by [tex]\(n - 1\)[/tex], where [tex]\(n\)[/tex] is the number of data values:
[tex]\[
n - 1 = 5 - 1 = 4
\][/tex]
5. Calculate the variance:
[tex]\[
s^2 = \frac{112}{4} = 28
\][/tex]
Hence, the variance of the sample is 28.
The correct option is:
D. 28
