Let's go through the calculations step-by-step.
1. Given Data:
- Measurements: [tex]\( x = 8 \)[/tex], [tex]\( y = 10 \)[/tex], [tex]\( z = 5 \)[/tex]
- Sample Mean ([tex]\( \bar{x} \)[/tex]): [tex]\( \bar{x} = \frac{x + y + z}{3} = 7.67 \)[/tex]
2. Calculate the squared differences from the mean for each measurement:
- For [tex]\( x \)[/tex]:
[tex]\[
(x - \bar{x})^2 = (8 - 7.67)^2 = 0.1089
\][/tex]
- For [tex]\( y \)[/tex]:
[tex]\[
(y - \bar{x})^2 = (10 - 7.67)^2 = 5.4289
\][/tex]
- For [tex]\( z \)[/tex]:
[tex]\[
(z - \bar{x})^2 = (5 - 7.67)^2 = 7.1289
\][/tex]
3. Fill in the table with these values:
[tex]\[
\begin{array}{|c|c|}
\hline a_i & (a_i - \bar{x})^2 \\
\hline x & 0.1089 \\
\hline y & 5.4289 \\
\hline z & 7.1289 \\
\hline
\end{array}
\][/tex]
4. Calculate the sample variance ([tex]\( s^2 \)[/tex]):
[tex]\[
s^2 = \frac{(x - \bar{x})^2 + (y - \bar{x})^2 + (z - \bar{x})^2}{2}
\][/tex]
Substitute the values:
[tex]\[
s^2 = \frac{0.1089 + 5.4289 + 7.1289}{2} = \frac{12.6667}{2} = 6.33335
\][/tex]
5. Calculate the sample standard deviation ([tex]\( s \)[/tex]):
[tex]\[
s = \sqrt{s^2} = \sqrt{6.33335} = 2.5166
\][/tex]
Finally, present the calculated values:
[tex]\[
\begin{array}{|c|c|}
\hline a_i & (a_i - \bar{x})^2 \\
\hline x & 0.1089 \\
\hline y & 5.4289 \\
\hline z & 7.1289 \\
\hline
\end{array}
\][/tex]
[tex]\[
s^2 = \frac{(x - \bar{x})^2 + (y - \bar{x})^2 + (z - \bar{x})^2}{2} = 6.33335
\][/tex]
[tex]\[
s = \sqrt{s^2} = 2.5166
\][/tex]
