the average estimated hours a person in the united states spent playing video games per year from 2002 to…

Explanation:

Step1: Calculate the mean

Let the data - set be (x_1 = 71,x_2 = 80,x_3 = 82,x_4 = 78,x_5 = 80,x_6 = 91,x_7 = 107,x_8 = 121,x_9 = 125,x_{10}=131,x_{11}=142). The mean (\mu=\frac{\sum_{i = 1}^{n}x_i}{n}), where (n = 11). (\sum_{i=1}^{11}x_i=71 + 80+82 + 78+80+91+107+121+125+131+142=1098). (\mu=\frac{1098}{11}\approx99.82).

Step2: Calculate the squared - differences

((x_1-\mu)^2=(71 - 99.82)^2=(-28.82)^2 = 830.5924), ((x_2-\mu)^2=(80 - 99.82)^2=(-19.82)^2 = 392.8324), ((x_3-\mu)^2=(82 - 99.82)^2=(-17.82)^2 = 317.5524), ((x_4-\mu)^2=(78 - 99.82)^2=(-21.82)^2 = 476.1124), ((x_5-\mu)^2=(80 - 99.82)^2=(-19.82)^2 = 392.8324), ((x_6-\mu)^2=(91 - 99.82)^2=(-8.82)^2 = 77.7924), ((x_7-\mu)^2=(107 - 99.82)^2=(7.18)^2 = 51.5524), ((x_8-\mu)^2=(121 - 99.82)^2=(21.18)^2 = 448.5924), ((x_9-\mu)^2=(125 - 99.82)^2=(25.18)^2 = 634.0324), ((x_{10}-\mu)^2=(131 - 99.82)^2=(31.18)^2 = 972.1924), ((x_{11}-\mu)^2=(142 - 99.82)^2=(42.18)^2 = 1779.1524).

Step3: Calculate the variance

The population variance (\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_i-\mu)^2}{n}). (\sum_{i = 1}^{11}(x_i-\mu)^2=830.5924+392.8324 + 317.5524+476.1124+392.8324+77.7924+51.5524+448.5924+634.0324+972.1924+1779.1524 = 6473.23). (\sigma^{2}=\frac{6473.23}{11}\approx588.48\approx588).

Step4: Calculate the population standard deviation

The population standard deviation (\sigma=\sqrt{\sigma^{2}}). (\sigma=\sqrt{588}\approx24.25\approx24).

Answer:

The variance of this data set is 588. The population standard deviation for this data set is 24.