the numbers of regular season wins for 10 football teams in a given season are given below. determine the…

Explanation:

Step1: Find the maximum and minimum values

The data set is (2, 10, 15, 2, 13, 6, 13, 8, 4, 6). The maximum value (x_{max}=15) and the minimum value (x_{min} = 2).

Step2: Calculate the range

The formula for the range (R) of a data - set is (R=x_{max}-x_{min}). Substitute (x_{max}=15) and (x_{min}=2) into the formula: (R = 15 - 2=13).

Step3: Calculate the mean (\mu)

The formula for the mean of a population (\mu=\frac{\sum_{i = 1}^{n}x_{i}}{n}), where (n = 10) and (x_{i}) are the data - points. (\sum_{i=1}^{10}x_{i}=2 + 10+15+2+13+6+13+8+4+6=79). So, (\mu=\frac{79}{10}=7.9).

Step4: Calculate the variance (\sigma^{2})

The formula for the population variance (\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\mu)^{2}}{n}). ((2 - 7.9)^{2}=(-5.9)^{2}=34.81), ((10 - 7.9)^{2}=(2.1)^{2}=4.41), ((15 - 7.9)^{2}=(7.1)^{2}=50.41), ((2 - 7.9)^{2}=(-5.9)^{2}=34.81), ((13 - 7.9)^{2}=(5.1)^{2}=26.01), ((6 - 7.9)^{2}=(-1.9)^{2}=3.61), ((13 - 7.9)^{2}=(5.1)^{2}=26.01), ((8 - 7.9)^{2}=(0.1)^{2}=0.01), ((4 - 7.9)^{2}=(-3.9)^{2}=15.21), ((6 - 7.9)^{2}=(-1.9)^{2}=3.61). (\sum_{i = 1}^{10}(x_{i}-7.9)^{2}=34.81+4.41+50.41+34.81+26.01+3.61+26.01+0.01+15.21+3.61 = 198.9). So, (\sigma^{2}=\frac{198.9}{10}=19.89).

Step5: Calculate the standard deviation (\sigma)

The formula for the population standard deviation (\sigma=\sqrt{\sigma^{2}}). Since (\sigma^{2}=19.89), (\sigma=\sqrt{19.89}\approx4.46).

Answer:

Range: (13) Mean: (7.9) Variance: (19.89) Standard Deviation: (\approx4.46)