Answer :
We are given the dataset
[tex]$$
[129,\,131,\,128,\,123,\,124,\,129,\,139,\,139].
$$[/tex]
We want to find the sample standard deviation, which is calculated using the formula
[tex]$$
s = \sqrt{\frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2},
$$[/tex]
where [tex]$\bar{x}$[/tex] is the sample mean and [tex]$n$[/tex] is the number of data points.
Step 1. Calculate the Sample Mean
First, compute the sample mean [tex]$\bar{x}$[/tex]:
[tex]$$
\bar{x} = \frac{129 + 131 + 128 + 123 + 124 + 129 + 139 + 139}{8} = \frac{1042}{8} = 130.25.
$$[/tex]
Step 2. Compute the Deviations from the Mean
Next, find the difference between each data point and the mean:
[tex]\[
\begin{array}{ccl}
129 - 130.25 &=& -1.25, \\
131 - 130.25 &=& 0.75, \\
128 - 130.25 &=& -2.25, \\
123 - 130.25 &=& -7.25, \\
124 - 130.25 &=& -6.25, \\
129 - 130.25 &=& -1.25, \\
139 - 130.25 &=& 8.75, \\
139 - 130.25 &=& 8.75.
\end{array}
\][/tex]
Step 3. Square Each Deviation
Square each of the differences:
[tex]\[
\begin{array}{ccl}
(-1.25)^2 &=& 1.5625, \\
(0.75)^2 &=& 0.5625, \\
(-2.25)^2 &=& 5.0625, \\
(-7.25)^2 &=& 52.5625, \\
(-6.25)^2 &=& 39.0625, \\
(-1.25)^2 &=& 1.5625, \\
(8.75)^2 &=& 76.5625, \\
(8.75)^2 &=& 76.5625.
\end{array}
\][/tex]
Step 4. Compute the Sample Variance
Add up all the squared deviations and divide by [tex]$n - 1$[/tex] (where [tex]$n=8$[/tex]):
[tex]$$
s^2 = \frac{1.5625 + 0.5625 + 5.0625 + 52.5625 + 39.0625 + 1.5625 + 76.5625 + 76.5625}{8-1} = \frac{253.5}{7} \approx 36.2143.
$$[/tex]
Step 5. Calculate the Sample Standard Deviation
Take the square root of the variance:
[tex]$$
s = \sqrt{36.2143} \approx 6.02.
$$[/tex]
Thus, the sample standard deviation, rounded to the nearest hundredth, is [tex]$\boxed{6.02}$[/tex].
[tex]$$
[129,\,131,\,128,\,123,\,124,\,129,\,139,\,139].
$$[/tex]
We want to find the sample standard deviation, which is calculated using the formula
[tex]$$
s = \sqrt{\frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2},
$$[/tex]
where [tex]$\bar{x}$[/tex] is the sample mean and [tex]$n$[/tex] is the number of data points.
Step 1. Calculate the Sample Mean
First, compute the sample mean [tex]$\bar{x}$[/tex]:
[tex]$$
\bar{x} = \frac{129 + 131 + 128 + 123 + 124 + 129 + 139 + 139}{8} = \frac{1042}{8} = 130.25.
$$[/tex]
Step 2. Compute the Deviations from the Mean
Next, find the difference between each data point and the mean:
[tex]\[
\begin{array}{ccl}
129 - 130.25 &=& -1.25, \\
131 - 130.25 &=& 0.75, \\
128 - 130.25 &=& -2.25, \\
123 - 130.25 &=& -7.25, \\
124 - 130.25 &=& -6.25, \\
129 - 130.25 &=& -1.25, \\
139 - 130.25 &=& 8.75, \\
139 - 130.25 &=& 8.75.
\end{array}
\][/tex]
Step 3. Square Each Deviation
Square each of the differences:
[tex]\[
\begin{array}{ccl}
(-1.25)^2 &=& 1.5625, \\
(0.75)^2 &=& 0.5625, \\
(-2.25)^2 &=& 5.0625, \\
(-7.25)^2 &=& 52.5625, \\
(-6.25)^2 &=& 39.0625, \\
(-1.25)^2 &=& 1.5625, \\
(8.75)^2 &=& 76.5625, \\
(8.75)^2 &=& 76.5625.
\end{array}
\][/tex]
Step 4. Compute the Sample Variance
Add up all the squared deviations and divide by [tex]$n - 1$[/tex] (where [tex]$n=8$[/tex]):
[tex]$$
s^2 = \frac{1.5625 + 0.5625 + 5.0625 + 52.5625 + 39.0625 + 1.5625 + 76.5625 + 76.5625}{8-1} = \frac{253.5}{7} \approx 36.2143.
$$[/tex]
Step 5. Calculate the Sample Standard Deviation
Take the square root of the variance:
[tex]$$
s = \sqrt{36.2143} \approx 6.02.
$$[/tex]
Thus, the sample standard deviation, rounded to the nearest hundredth, is [tex]$\boxed{6.02}$[/tex].
