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, 5, 15, 6, 11, 7, 2, 9

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:

Calculate the range and population mean

$$ \text{Data set: } \{2, 8, 15, 5, 15, 6, 11, 7, 2, 9\} \quad (N = 10) $$

$$ \text{Range} = \text{Maximum} - \text{Minimum} = 15 - 2 = 13 $$

$$ \mu = \frac{\sum x_i}{N} = \frac{2 + 8 + 15 + 5 + 15 + 6 + 11 + 7 + 2 + 9}{10} = \frac{80}{10} = 8.0 $$

Calculate the population variance

$$ \sum (x_i - \mu)^2 = (2-8)^2 + (8-8)^2 + (15-8)^2 + (5-8)^2 + (15-8)^2 + (6-8)^2 + (11-8)^2 + (7-8)^2 + (2-8)^2 + (9-8)^2 $$

$$ \sum (x_i - \mu)^2 = (-6)^2 + 0^2 + 7^2 + (-3)^2 + 7^2 + (-2)^2 + 3^2 + (-1)^2 + (-6)^2 + 1^2 $$

$$ \sum (x_i - \mu)^2 = 36 + 0 + 49 + 9 + 49 + 4 + 9 + 1 + 36 + 1 = 194 $$

$$ \sigma^2 = \frac{\sum (x_i - \mu)^2}{N} = \frac{194}{10} = 19.4 $$

Calculate the population standard deviation

$$ \sigma = \sqrt{\sigma^2} = \sqrt{19.4} \approx 4.4045 $$

$$ \sigma \approx 4.4 \quad (\text{rounded to the nearest tenth}) $$

Answer:

The range is <blank>13</blank>.
(Simplify your answer.)

The population mean is <blank>8.0</blank>.
(Simplify your answer. Round to the nearest tenth as needed.)

The population variance is <blank>19.4</blank>.
(Simplify your answer. Round to the nearest tenth as needed.)

The population standard deviation is <blank>4.4</blank>.
(Simplify your answer. Round to the nearest tenth as needed.)