To solve the problem, we need to determine the correct equation that shows the variance for the number of miles Fiona biked last week.
Given data:
Recorded miles: [tex]\(4, 7, 4, 10, 5\)[/tex]
Mean ([tex]\(\mu\)[/tex]): 6
The formula for the population variance ([tex]\(\sigma^2\)[/tex]) is:
[tex]\[
\sigma^2 = \frac{1}{N}\sum_{i=1}^{N} (x_i - \mu)^2
\][/tex]
where [tex]\(N\)[/tex] is the number of data points, [tex]\(x_i\)[/tex] represents each data point, and [tex]\(\mu\)[/tex] is the mean.
In this case:
1. Calculate the squared differences from the mean for each data point:
[tex]\[
(4-6)^2, (7-6)^2, (4-6)^2, (10-6)^2, (5-6)^2
\][/tex]
[tex]\[
4, 1, 4, 16, 1
\][/tex]
2. Sum these squared differences:
[tex]\[
4 + 1 + 4 + 16 + 1 = 26
\][/tex]
3. Divide by the number of data points ([tex]\(N = 5\)[/tex]) to get the variance ([tex]\(\sigma^2\)[/tex]):
[tex]\[
\sigma^2 = \frac{26}{5} = 5.2
\][/tex]
Given the choices, the only correct formula for the population variance [tex]\(\sigma^2\)[/tex] is:
[tex]\[
\sigma^2 = \frac{(4-6)^2 + (7-6)^2 + (4-6)^2 + (10-6)^2 + (5-6)^2}{5}
\][/tex]
Therefore, the correct equation showing the variance for the number of miles Fiona biked last week is:
\[
\sigma^2 = \frac{(4-6)^2 + (7-6)^2 + (4-6)^2 + (10-6)^2 + (5-6)^2}{5}
