Let's go through the correct steps to compute the variance to identify where Cara made her first error.
1. Compute the Mean: Cara correctly calculated the mean to be 78.
2. Calculate the Squared Differences from the Mean:
- [tex]\((87 - 78)^2 = 81\)[/tex]
- [tex]\((46 - 78)^2 = 1024\)[/tex]
- [tex]\((90 - 78)^2 = 144\)[/tex]
- [tex]\((78 - 78)^2 = 0\)[/tex]
- [tex]\((89 - 78)^2 = 121\)[/tex]
So, the squared differences from the mean are [tex]\([81, 1024, 144, 0, 121]\)[/tex].
3. Sum of Squared Differences:
[tex]\[
81 + 1024 + 144 + 0 + 121 = 1370
\][/tex]
4. Compute the Variance (using the population formula):
[tex]\[
\sigma^2 = \frac{1370}{5} = 274.0
\][/tex]
However, let's follow Cara's method to identify where she went wrong.
Cara's work:
[tex]\[
\sigma^2=\frac{(87-78)^2 + (46-78)^2 + (90-78)^2 + (78-78)^2 + (89-78)^2}{5}
\][/tex]
[tex]\[
\sigma^2=\frac{(9)^2 - (32)^2 + (12)^2 + 0^2 + (11)^2}{5}
\][/tex]
Notice that instead of correctly calculating each squared difference individually, she made an error:
- Correct calculation:
[tex]\[
(9)^2 = 81
\][/tex]
[tex]\[
(32)^2 = 1024 \quad \text{(Should have been added, not subtracted)}
\][/tex]
[tex]\[
(12)^2 = 144
\][/tex]
[tex]\[
0^2 = 0
\][/tex]
[tex]\[
(11)^2 = 121
\][/tex]
Cara calculated:
[tex]\[
\sigma^2=\frac{81 - 1024 + 144 + 0 + 121}{5}
\][/tex]
She incorrectly subtracted [tex]\((32)^2 = 1024\)[/tex] instead of adding it.
Cara's Step:
[tex]\[
\sigma^2=\frac{81 - 1024 + 144 + 0 + 121}{5} = \frac{-678}{5} = -135.6
\][/tex]
### Conclusion:
The first error Cara made was in the subtraction of [tex]\((46 - 78)^2\)[/tex], which should have been added instead. This is the reason why her calculation for the sum of squared differences resulted in an incorrect negative number.
