Question

the ages (in years) of a random sample of shoppers at a gaming store are shown. determine the range, mean, variance, and standard deviation of the sample data set. 12, 18, 23, 13, 14, 17, 20, 18, 14, 18
the range is 11. (simplify your answer.)
the mean is 16.7. (simplify your answer. round to the nearest tenth as needed.)
the variance is . (simplify your answer. round to the nearest tenth as needed.)

Explanation:

Step1: Recall variance formula

The formula for the variance $s^{2}$ of a sample is $s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}$, where $x_{i}$ are the data - points, $\bar{x}$ is the sample mean, and $n$ is the number of data - points. Here, $n = 10$ and $\bar{x}=16.7$.

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

For $x_1 = 12$: $(12 - 16.7)^{2}=(-4.7)^{2}=22.09$
For $x_2 = 18$: $(18 - 16.7)^{2}=(1.3)^{2}=1.69$
For $x_3 = 23$: $(23 - 16.7)^{2}=(6.3)^{2}=39.69$
For $x_4 = 13$: $(13 - 16.7)^{2}=(-3.7)^{2}=13.69$
For $x_5 = 14$: $(14 - 16.7)^{2}=(-2.7)^{2}=7.29$
For $x_6 = 17$: $(17 - 16.7)^{2}=(0.3)^{2}=0.09$
For $x_7 = 20$: $(20 - 16.7)^{2}=(3.3)^{2}=10.89$
For $x_8 = 18$: $(18 - 16.7)^{2}=(1.3)^{2}=1.69$
For $x_9 = 14$: $(14 - 16.7)^{2}=(-2.7)^{2}=7.29$
For $x_{10}=18$: $(18 - 16.7)^{2}=(1.3)^{2}=1.69$

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

$\sum_{i = 1}^{10}(x_{i}-\bar{x})^{2}=22.09+1.69 + 39.69+13.69+7.29+0.09+10.89+1.69+7.29+1.69 = 106.1$

Step4: Calculate the variance

$s^{2}=\frac{\sum_{i = 1}^{10}(x_{i}-\bar{x})^{2}}{10 - 1}=\frac{106.1}{9}\approx11.8$

Answer:

$11.8$