Cara made a mistake when calculating the squared differences from the mean in the formula for variance. Let's break down the correct calculation step-by-step.
1. Mean Calculation:
Cara correctly identified the mean, [tex]\( \mu \)[/tex], of the dataset as 78.
2. Squared Differences Calculation:
We need to calculate the squared differences from the mean for each data point:
[tex]\[
(87 - 78)^2, (46 - 78)^2, (90 - 78)^2, (78 - 78)^2, (89 - 78)^2
\][/tex]
- For 87: [tex]\((87 - 78)^2 = (9)^2 = 81\)[/tex]
- For 46: [tex]\((46 - 78)^2 = (-32)^2 = 1024\)[/tex]
- For 90: [tex]\((90 - 78)^2 = (12)^2 = 144\)[/tex]
- For 78: [tex]\((78 - 78)^2 = 0^2 = 0\)[/tex]
- For 89: [tex]\((89 - 78)^2 = (11)^2 = 121\)[/tex]
So, the squared differences are [tex]\( 81, 1024, 144, 0, \)[/tex] and [tex]\( 121 \)[/tex].
3. Summing the Squared Differences:
Next, we sum these squared differences:
[tex]\[
81 + 1024 + 144 + 0 + 121 = 1370
\][/tex]
4. Variance Calculation:
Finally, we divide this sum by the number of data points (5) to find the variance:
[tex]\[
\sigma^2 = \frac{1370}{5} = 274.0
\][/tex]
Cara's first error was in her calculation where she included a negative sign in the squared differences, particularly in her expression:
[tex]\[
\sigma^2 = (9)^2 - (32)^2 + (12)^2 + 0^2 + (11)^2
\][/tex]
She should have added all the squared differences, not subtracted them. Thus, the correct calculation of variance should be:
[tex]\[
\sigma^2 = \frac{81 + 1024 + 144 + 0 + 121}{5} = 274.0
\][/tex]
