Cara's first error in computing the variance occurs in the step where she calculates the sum of squared differences. Let's correct this step-by-step:
1. Calculate the differences from the mean:
- [tex]\( 87 - 78 = 9 \)[/tex]
- [tex]\( 46 - 78 = -32 \)[/tex]
- [tex]\( 90 - 78 = 12 \)[/tex]
- [tex]\( 78 - 78 = 0 \)[/tex]
- [tex]\( 89 - 78 = 11 \)[/tex]
2. Square each of these differences:
- [tex]\( 9^2 = 81 \)[/tex]
- [tex]\( (-32)^2 = 1024 \)[/tex]
- [tex]\( 12^2 = 144 \)[/tex]
- [tex]\( 0^2 = 0 \)[/tex]
- [tex]\( 11^2 = 121 \)[/tex]
3. Sum the squared differences:
- [tex]\( 81 + 1024 + 144 + 0 + 121 = 1370 \)[/tex]
Cara made a mistake in the calculation of the squared differences sum.
In her calculation:
[tex]\[ σ^2 = \frac{81 - 1024 + 144 + 0 + 121}{5} = \frac{-678}{5} = -135.6 \][/tex]
She incorrectly subtracted [tex]\( 1024 \)[/tex] rather than adding it. The correct step should be:
[tex]\[ σ^2 = \frac{81 + 1024 + 144 + 0 + 121}{5} = \frac{1370}{5} = 274.0 \][/tex]
Therefore, Cara's first error was incorrectly computing the sum of the squared differences. She should have added all the squared differences instead of subtracting [tex]\( 1024 \)[/tex].
