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, 10, 15, 3, 14, 6, 13, 6, 5, 7. the range is. (simplify your answer.) the population mean is. (simplify your answer. round to the nearest tenth as needed.) the population variance is. (simplify your answer. round to the nearest tenth as needed.) the population standard deviation is. (simplify your answer. round to the nearest tenth as needed.)

Explanation:

Step1: Calculate the range

The range is the difference between the maximum and minimum values.
The data set is \(2, 10, 15, 3, 14, 6, 13, 6, 5, 7\).
The maximum value \(x_{max}=15\), the minimum value \(x_{min} = 2\).
\(Range=x_{max}-x_{min}=15 - 2=13\)

Step2: Calculate the population mean \(\mu\)

The formula for the population mean is \(\mu=\frac{\sum_{i = 1}^{N}x_{i}}{N}\), where \(N = 10\) (the number of data points) and \(x_{i}\) are the data values.
\(\sum_{i=1}^{10}x_{i}=2 + 10+15+3+14+6+13+6+5+7=81\)
\(\mu=\frac{81}{10}=8.1\)

Step3: Calculate the population variance \(\sigma^{2}\)

The formula for the population variance is \(\sigma^{2}=\frac{\sum_{i = 1}^{N}(x_{i}-\mu)^{2}}{N}\)
\((2 - 8.1)^{2}=(-6.1)^{2}=37.21\)
\((10 - 8.1)^{2}=(1.9)^{2}=3.61\)
\((15 - 8.1)^{2}=(6.9)^{2}=47.61\)
\((3 - 8.1)^{2}=(-5.1)^{2}=26.01\)
\((14 - 8.1)^{2}=(5.9)^{2}=34.81\)
\((6 - 8.1)^{2}=(-2.1)^{2}=4.41\)
\((13 - 8.1)^{2}=(4.9)^{2}=24.01\)
\((6 - 8.1)^{2}=(-2.1)^{2}=4.41\)
\((5 - 8.1)^{2}=(-3.1)^{2}=9.61\)
\((7 - 8.1)^{2}=(-1.1)^{2}=1.21\)
\(\sum_{i = 1}^{10}(x_{i}-\mu)^{2}=37.21+3.61 + 47.61+26.01+34.81+4.41+24.01+4.41+9.61+1.21=193.9\)
\(\sigma^{2}=\frac{193.9}{10}=19.4\)

Step4: Calculate the population standard deviation \(\sigma\)

The formula for the population standard deviation is \(\sigma=\sqrt{\sigma^{2}}\)
\(\sigma=\sqrt{19.4}\approx4.4\)

Answer:

The range is \(13\).
The population mean is \(8.1\).
The population variance is \(19.4\).
The population standard deviation is \(4.4\).