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A meteorologist is studying the monthly rainfall in a section of the Brazilian rainforest. She recorded the monthly rainfall, in inches, for last year:

1.8, 2.5, 2.6, 4.4, 4.4, 7.3, 8.0, 9.5, 10.3, 10.4, 11.1, 11.7

For this data set:

[tex]\[
\begin{aligned}
\mu &= 7, \\
N &= 12, \\
\sigma^2 &= \frac{\left(x_1 - \mu\right)^2 + \left(x_2 - \mu\right)^2 + \ldots + \left(x_N - \mu\right)^2}{N} \\
\sigma^2 &= \frac{(1.8 - 7)^2 + (2.5 - 7)^2 + \ldots + (11.7 - 7)^2}{12}
\end{aligned}
\][/tex]

Fill in the missing values in the formula. What is the variance?

A. 0
B. 3.217
C. 3.522
D. 12.405
E. 148.86



Answer :

GinnyAnswer
Certainly! Let's calculate the variance step-by-step with the given data.

1. Given Data:
- Monthly Rainfall (in inches): [tex]\(1.8, 2.5, 2.6, 4.4, 4.4, 7.3, 8.0, 9.5, 10.3, 10.4, 11.1, 11.7\)[/tex]
- Mean ([tex]\(\mu\)[/tex]): [tex]\(7\)[/tex]
- Number of observations ([tex]\(N\)[/tex]): [tex]\(12\)[/tex]

2. Variance Formula:
[tex]\[ \sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N} \][/tex]

3. Calculate each squared difference [tex]\((x_i - \mu)^2\)[/tex]:
- [tex]\((1.8 - 7)^2 = 27.04\)[/tex]
- [tex]\((2.5 - 7)^2 = 20.25\)[/tex]
- [tex]\((2.6 - 7)^2 = 19.36\)[/tex]
- [tex]\((4.4 - 7)^2 = 6.76\)[/tex]
- [tex]\((4.4 - 7)^2 = 6.76\)[/tex]
- [tex]\((7.3 - 7)^2 = 0.09\)[/tex]
- [tex]\((8.0 - 7)^2 = 1.00\)[/tex]
- [tex]\((9.5 - 7)^2 = 6.25\)[/tex]
- [tex]\((10.3 - 7)^2 = 10.89\)[/tex]
- [tex]\((10.4 - 7)^2 = 11.56\)[/tex]
- [tex]\((11.1 - 7)^2 = 16.81\)[/tex]
- [tex]\((11.7 - 7)^2 = 22.09\)[/tex]

4. Summing the squared differences:
[tex]\[ 27.04 + 20.25 + 19.36 + 6.76 + 6.76 + 0.09 + 1.00 + 6.25 + 10.89 + 11.56 + 16.81 + 22.09 = 148.86 \][/tex]

5. Calculate the variance [tex]\(\sigma^2\)[/tex]:
[tex]\[ \sigma^2 = \frac{148.86}{12} = 12.405 \][/tex]

So, the variance is [tex]\( \sigma^2 = 12.405 \)[/tex]. The missing values in the formula are [tex]\( 148.86 \)[/tex] for the sum of squared differences and [tex]\( 12.405 \)[/tex] for the variance. Therefore, the correct variance is:

[tex]\[ \sigma^2 = 12.405 \][/tex]

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