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, 13, 7, 14, 6, 8, 6

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: Find the range

The range is the difference between the maximum and minimum values. First, identify the maximum and minimum in the data set: 2, 8, 15, 2, 15, 7, 14, 6, 8, 6.
Minimum value = 2, Maximum value = 15.
Range = Maximum - Minimum = \( 15 - 2 = 13 \).

Step2: Calculate the population mean

The population mean (\(\mu\)) is the sum of all values divided by the number of values (\(N\)).
Sum of values: \( 2 + 8 + 15 + 2 + 15 + 7 + 14 + 6 + 8 + 6 \)
Calculate the sum: \( 2+8=10; 10+15=25; 25+2=27; 27+15=42; 42+7=49; 49+14=63; 63+6=69; 69+8=77; 77+6=83 \)
\( N = 10 \)
Mean \(\mu = \frac{83}{10} = 8.3 \).

Step3: Calculate the population variance

Population variance (\(\sigma^2\)) is given by \(\frac{\sum (x_i - \mu)^2}{N}\).
First, find each \((x_i - \mu)^2\):

• For \(x=2\): \((2 - 8.3)^2 = (-6.3)^2 = 39.69\)
• For \(x=8\): \((8 - 8.3)^2 = (-0.3)^2 = 0.09\)
• For \(x=15\): \((15 - 8.3)^2 = (6.7)^2 = 44.89\)
• For \(x=2\): \((2 - 8.3)^2 = 39.69\)
• For \(x=15\): \((15 - 8.3)^2 = 44.89\)
• For \(x=7\): \((7 - 8.3)^2 = (-1.3)^2 = 1.69\)
• For \(x=14\): \((14 - 8.3)^2 = (5.7)^2 = 32.49\)
• For \(x=6\): \((6 - 8.3)^2 = (-2.3)^2 = 5.29\)
• For \(x=8\): \((8 - 8.3)^2 = 0.09\)
• For \(x=6\): \((6 - 8.3)^2 = 5.29\)

Sum of these squared differences:
\( 39.69 + 0.09 + 44.89 + 39.69 + 44.89 + 1.69 + 32.49 + 5.29 + 0.09 + 5.29 \)
Calculate step by step:
\( 39.69+0.09=39.78; 39.78+44.89=84.67; 84.67+39.69=124.36; 124.36+44.89=169.25; 169.25+1.69=170.94; 170.94+32.49=203.43; 203.43+5.29=208.72; 208.72+0.09=208.81; 208.81+5.29=214.1 \)

Variance \(\sigma^2 = \frac{214.1}{10} = 21.41\).

Step4: Calculate the population standard deviation

Population standard deviation (\(\sigma\)) is the square root of the variance.
\(\sigma = \sqrt{21.41} \approx 4.6\) (rounded to the nearest tenth).

Answer:

The range is \(\boldsymbol{13}\)
The population mean is \(\boldsymbol{8.3}\)
The population variance is \(\boldsymbol{21.41}\)
The population standard deviation is \(\boldsymbol{4.6}\)