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The test scores of a geometry class are given below:

[tex]\[ 90, 75, 72, 88, 85 \][/tex]

The teacher wants to find the variance for the class population. What is the value of the numerator in the calculation of the variance?

Variance: [tex]\[ \sigma^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N} \][/tex]

A. [tex]\(-160\)[/tex]
B. [tex]\(-6\)[/tex]
C. [tex]\(16\)[/tex]
D. [tex]\(258\)[/tex]


Sagot :

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].