Let's follow Cara's steps in detail to find the error in her computation.
### Step-by-Step Solution
1. Cara's given values: 87, 46, 90, 78, 89.
2. Mean (average) calculation:
- Cara correctly finds the mean, [tex]\(\overline{X}\)[/tex], to be 78.
3. Steps to find variance ([tex]\(\sigma^2\)[/tex]):
Cara uses the formula for variance:
[tex]\[
\sigma^2 = \frac{\sum_{i=1}^n (x_i - \overline{X})^2}{n}
\][/tex]
Substituting the given data:
[tex]\[
\sigma^2 = \frac{(87 - 78)^2 + (46 - 78)^2 + (90 - 78)^2 + (78 - 78)^2 + (89 - 78)^2}{5}
\][/tex]
4. Calculating each squared difference from the mean:
[tex]\[
(87 - 78)^2 = 9^2 = 81
\][/tex]
[tex]\[
(46 - 78)^2 = (-32)^2 = 1024
\][/tex]
[tex]\[
(90 - 78)^2 = 12^2 = 144
\][/tex]
[tex]\[
(78 - 78)^2 = 0^2 = 0
\][/tex]
[tex]\[
(89 - 78)^2 = 11^2 = 121
\][/tex]
5. Adding the squared differences:
[tex]\[
81 + 1024 + 144 + 0 + 121
\][/tex]
Sum these values:
[tex]\[
81 + 1024 + 144 + 0 + 121 = 1370
\][/tex]
6. Calculating the variance:
[tex]\[
\sigma^2 = \frac{1370}{5} = 274.0
\][/tex]
### Identifying Cara's Mistake
Cara made a mistake at this critical step:
In her solution, she subtracts some squared differences rather than summing all the squared differences, which is an incorrect approach. She incorrectly wrote the expression as:
[tex]\[
81 - 1024 + 144 + 0 + 121
\][/tex]
Instead of summing the squared differences, she subtracted [tex]\(1024\)[/tex].
### Correct Calculation:
Correctly summing the squared differences, as shown, should be:
[tex]\[
81 + 1024 + 144 + 0 + 121 = 1370
\][/tex]
Dividing this sum by the number of values (5):
[tex]\[
\sigma^2 = \frac{1370}{5} = 274.0
\][/tex]
Therefore, the first error Cara made in computing the variance was subtracting the squared differences instead of adding them all together. The correct variance should be 274.0.
