Let's follow the steps to calculate the sample variance and the sample standard deviation for the given data set:
[tex]\[ 7, 53, 13, 51, 38, 23, 32, 30, 30, 27 \][/tex]
1. Calculate the mean (average) of the data:
[tex]\[
\text{Mean} \, (\bar{x}) = \frac{\sum x_i}{n}
\][/tex]
where [tex]\(\sum x_i\)[/tex] is the sum of all the data points and [tex]\(n\)[/tex] is the number of data points.
[tex]\[
\bar{x} = \frac{7 + 53 + 13 + 51 + 38 + 23 + 32 + 30 + 30 + 27}{10} = \frac{304}{10} = 30.4
\][/tex]
2. Calculate the squared differences from the mean:
For each data point [tex]\(x_i\)[/tex], calculate [tex]\((x_i - \bar{x})^2\)[/tex]:
[tex]\[
\begin{align*}
(7 - 30.4)^2 &= (-23.4)^2 = 547.56 \\
(53 - 30.4)^2 &= (22.6)^2 = 510.76 \\
(13 - 30.4)^2 &= (-17.4)^2 = 302.76 \\
(51 - 30.4)^2 &= (20.6)^2 = 424.36 \\
(38 - 30.4)^2 &= (7.6)^2 = 57.76 \\
(23 - 30.4)^2 &= (-7.4)^2 = 54.76 \\
(32 - 30.4)^2 &= (1.6)^2 = 2.56 \\
(30 - 30.4)^2 &= (-0.4)^2 = 0.16 \\
(30 - 30.4)^2 &= (-0.4)^2 = 0.16 \\
(27 - 30.4)^2 &= (-3.4)^2 = 11.56 \\
\end{align*}
\][/tex]
3. Calculate the sample variance:
[tex]\[
s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} \\
\][/tex]
[tex]\[
s^2 = \frac{547.56 + 510.76 + 302.76 + 424.36 + 57.76 + 54.76 + 2.56 + 0.16 + 0.16 + 11.56}{10 - 1} \\
s^2 = \frac{1912.4}{9} = 212.49 \\
\][/tex]
Thus, the sample variance is [tex]\(s^2 = 212.49\)[/tex].
4. Calculate the sample standard deviation:
[tex]\[
s = \sqrt{s^2} = \sqrt{212.49} \approx 14.6
\][/tex]
Thus, the sample standard deviation is [tex]\(s = 14.6\)[/tex].
With these calculations, we can fill in the boxes:
- For the sample variance:
[tex]\[
B. \boxed{s^2=212.49}
\][/tex]
- For the sample standard deviation:
[tex]\[
B. \boxed{s=14.6}
\][/tex]
