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The following data values represent a sample. What is the variance of the sample? [tex]\bar{x}=7[/tex]. Use the information in the table to help you.

[tex]\[
\begin{tabular}{|c|r|r|r|r|r|}
\hline
$x$ & 9 & 5 & 3 & 7 & 11 \\
\hline
$\left(x_j-\bar{x}\right)^2$ & 4 & 4 & 16 & 0 & 16 \\
\hline
\end{tabular}
\][/tex]

A. 10
B. 4.5
C. 5.2
D. 8


Sagot :

To determine the variance of the sample, we can follow these steps:

1. Determine the number of data points [tex]\( n \)[/tex]:
- The sample data values are [tex]\( 9, 5, 3, 7, 11 \)[/tex].
- The number of data points [tex]\( n \)[/tex] is 5.

2. Calculate the mean (sample mean) [tex]\( \bar{x} \)[/tex]:
- We are given that [tex]\( \bar{x} = 7 \)[/tex].

3. Compute the squared differences [tex]\((x_j - \bar{x})^2\)[/tex] for each data point:
- These values are provided in the table:
- For [tex]\( x = 9 \)[/tex], [tex]\((9 - 7)^2 = 4\)[/tex]
- For [tex]\( x = 5 \)[/tex], [tex]\((5 - 7)^2 = 4\)[/tex]
- For [tex]\( x = 3 \)[/tex], [tex]\((3 - 7)^2 = 16\)[/tex]
- For [tex]\( x = 7 \)[/tex], [tex]\((7 - 7)^2 = 0\)[/tex]
- For [tex]\( x = 11 \)[/tex], [tex]\((11 - 7)^2 = 16\)[/tex]

4. Sum the squared differences:
- Sum of squared differences = [tex]\( 4 + 4 + 16 + 0 + 16 \)[/tex].
- This sum = 40.

5. Calculate the sample variance [tex]\( s^2 \)[/tex]:
- The formula for the sample variance is:
[tex]\[ s^2 = \frac{\sum (x_j - \bar{x})^2}{n - 1} \][/tex]
- Plugging in the values we have:
[tex]\[ s^2 = \frac{40}{5 - 1} = \frac{40}{4} = 10 \][/tex]

Therefore, the variance of the sample is [tex]\( 10 \)[/tex]. The correct answer is:
A. 10
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