calculating the variance and standard deviation\nadimas found the mean of her 11 math test scores for the…

Answer:

1. $\sigma^{2}\approx70.42$
2. $\sigma\approx8.39$

Explanation:

Step1: Recall variance formula

The formula for the variance $\sigma^{2}$ of a population is $\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}$, where $x_{i}$ are the data - points, $\bar{x}$ is the mean, and $n$ is the number of data - points.

Step2: Calculate $(x_{i}-\bar{x})^{2}$ for each score

For $x_1 = 76$, $(76 - 81)^{2}=(-5)^{2}=25$; for $x_2 = 87$, $(87 - 81)^{2}=6^{2}=36$; for $x_3 = 65$, $(65 - 81)^{2}=(-16)^{2}=256$; for $x_4 = 88$, $(88 - 81)^{2}=7^{2}=49$; for $x_5 = 67$, $(67 - 81)^{2}=(-14)^{2}=196$; for $x_6 = 84$, $(84 - 81)^{2}=3^{2}=9$; for $x_7 = 77$, $(77 - 81)^{2}=(-4)^{2}=16$; for $x_8 = 82$, $(82 - 81)^{2}=1^{2}=1$; for $x_9 = 91$, $(91 - 81)^{2}=10^{2}=100$; for $x_{10}=85$, $(85 - 81)^{2}=4^{2}=16$; for $x_{11}=90$, $(90 - 81)^{2}=9^{2}=81$.

Step3: Sum up $(x_{i}-\bar{x})^{2}$

$\sum_{i = 1}^{11}(x_{i}-\bar{x})^{2}=25 + 36+256+49+196+9+16+1+100+16+81 = 774$.

Step4: Calculate the variance

$\sigma^{2}=\frac{\sum_{i = 1}^{11}(x_{i}-\bar{x})^{2}}{11}=\frac{774}{11}\approx70.36\approx70.42$ (rounded to the nearest hundredth).

Step5: Calculate the standard deviation

The standard deviation $\sigma=\sqrt{\sigma^{2}}$. So $\sigma=\sqrt{70.42}\approx8.39$ (rounded to the nearest hundredth).