To find the standard deviation of the given data set [tex]\(7, 5, 10, 11, 12\)[/tex] with the mean [tex]\(\bar{x} = 9\)[/tex], follow these steps:
1. Calculate the deviations from the mean:
For each data point [tex]\(x\)[/tex], subtract the mean [tex]\(\bar{x} = 9\)[/tex]:
- For [tex]\(7\)[/tex]: [tex]\(7 - 9 = -2\)[/tex]
- For [tex]\(5\)[/tex]: [tex]\(5 - 9 = -4\)[/tex]
- For [tex]\(10\)[/tex]: [tex]\(10 - 9 = 1\)[/tex]
- For [tex]\(11\)[/tex]: [tex]\(11 - 9 = 2\)[/tex]
- For [tex]\(12\)[/tex]: [tex]\(12 - 9 = 3\)[/tex]
2. Square each deviation:
- For [tex]\(-2\)[/tex]: [tex]\((-2)^2 = 4\)[/tex]
- For [tex]\(-4\)[/tex]: [tex]\((-4)^2 = 16\)[/tex]
- For [tex]\(1\)[/tex]: [tex]\(1^2 = 1\)[/tex]
- For [tex]\(2\)[/tex]: [tex]\(2^2 = 4\)[/tex]
- For [tex]\(3\)[/tex]: [tex]\(3^2 = 9\)[/tex]
3. Sum of the squared deviations:
Calculate the sum of [tex]\((x - \bar{x})^2\)[/tex]:
[tex]\[
4 + 16 + 1 + 4 + 9 = 34
\][/tex]
4. Determine [tex]\(n - 1\)[/tex]:
The number of data points [tex]\(n\)[/tex] is 5, therefore [tex]\(n - 1 = 5 - 1 = 4\)[/tex].
5. Calculate the sample standard deviation:
Use the formula [tex]\(s = \sqrt{\frac{1}{n-1} \Sigma (x - \bar{x})^2}\)[/tex]:
[tex]\[
s = \sqrt{\frac{34}{4}} = \sqrt{8.5} \approx 2.9154759474226504
\][/tex]
6. Round to the nearest tenth:
[tex]\(s\)[/tex] rounded to the nearest tenth is [tex]\(2.9\)[/tex].
So, the standard deviation for the data set is [tex]\(2.9\)[/tex]. Hence, the correct answer is:
C. 2.9
