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Cara computes the mean and variance for the set \{87, 46, 90, 78, 89\}. She finds the mean to be 78. Her steps for finding the variance are shown below.

[tex]\[
\begin{array}{l}
\sigma^2 = \frac{(87-78)^2 + (46-78)^2 + (90-78)^2 + (78-78)^2 + (89-78)^2}{5} \\
\sigma^2 = \frac{(9)^2 + (-32)^2 + (12)^2 + 0^2 + (11)^2}{5} \\
\sigma^2 = \frac{81 + 1024 + 144 + 0 + 121}{5} \\
\sigma^2 = \frac{1370}{5} = 274
\end{array}
\][/tex]

What is the first error Cara made in computing the variance?

Sagot :

GinnyAnswer
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].
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