Question

the numbers of regular - season wins for 10 football teams in a given season are given below. determine the range, mean, variance, and standard deviation of the population data set. 2, 8, 15, 2, 15, 6, 11, 8, 5, 8
the range is 13
(simplify your answer.)
the population mean is 8
(simplify your answer. round to the nearest tenth as needed.)
the population variance is
(simplify your answer. round to the nearest tenth as needed.)

Explanation:

Step1: Recall variance formula

The formula for population variance $\sigma^{2}=\frac{\sum_{i = 1}^{N}(x_{i}-\mu)^{2}}{N}$, where $x_{i}$ are the data - points, $\mu$ is the population mean, and $N$ is the number of data - points. Here $N = 10$ and $\mu=8$.

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

For $x_1 = 2$: $(2 - 8)^{2}=(-6)^{2}=36$
For $x_2 = 8$: $(8 - 8)^{2}=0^{2}=0$
For $x_3 = 15$: $(15 - 8)^{2}=7^{2}=49$
For $x_4 = 2$: $(2 - 8)^{2}=(-6)^{2}=36$
For $x_5 = 15$: $(15 - 8)^{2}=7^{2}=49$
For $x_6 = 6$: $(6 - 8)^{2}=(-2)^{2}=4$
For $x_7 = 11$: $(11 - 8)^{2}=3^{2}=9$
For $x_8 = 8$: $(8 - 8)^{2}=0^{2}=0$
For $x_9 = 5$: $(5 - 8)^{2}=(-3)^{2}=9$
For $x_{10}=8$: $(8 - 8)^{2}=0^{2}=0$

Step3: Calculate the sum $\sum_{i = 1}^{10}(x_{i}-\mu)^{2}$

$\sum_{i = 1}^{10}(x_{i}-\mu)^{2}=36 + 0+49+36+49+4+9+0+9+0=201$

Step4: Calculate the population variance

$\sigma^{2}=\frac{\sum_{i = 1}^{10}(x_{i}-\mu)^{2}}{10}=\frac{201}{10}=20.1$

Answer:

$20.1$