To find the numerator of the variance for the given test scores, follow these steps:
1. List of Scores: The scores given are:
[tex]\[
90, 75, 72, 88, 85
\][/tex]
2. Calculate the Mean: The mean (average) score, [tex]\(\mu\)[/tex], is obtained by summing all the scores and dividing by the number of scores.
[tex]\[
\mu = \frac{90 + 75 + 72 + 88 + 85}{5} = \frac{410}{5} = 82
\][/tex]
3. Subtract the Mean from Each Score and Square the Result: For each score, calculate [tex]\((x_i - \mu)^2\)[/tex]:
- For [tex]\(90\)[/tex]:
[tex]\[
(90 - 82)^2 = 8^2 = 64
\][/tex]
- For [tex]\(75\)[/tex]:
[tex]\[
(75 - 82)^2 = (-7)^2 = 49
\][/tex]
- For [tex]\(72\)[/tex]:
[tex]\[
(72 - 82)^2 = (-10)^2 = 100
\][/tex]
- For [tex]\(88\)[/tex]:
[tex]\[
(88 - 82)^2 = 6^2 = 36
\][/tex]
- For [tex]\(85\)[/tex]:
[tex]\[
(85 - 82)^2 = 3^2 = 9
\][/tex]
4. Sum the Squared Differences: Add all the squared differences together.
[tex]\[
64 + 49 + 100 + 36 + 9 = 258
\][/tex]
Therefore, the numerator of the variance calculation is [tex]\(258\)[/tex].
Thus, the value of the numerator of the variance for the given set of scores is [tex]\(\boxed{258}\)[/tex].
