Question

the ages (in years) of a random sample of shoppers at a gaming store are shown. determine the range, mean, variance, and standard deviation of the sample data set. 12, 15, 23, 14, 14, 17, 21, 18, 13, 17 the range is 11. (simplify your answer.) the mean is 16.4 (simplify your answer. round to the nearest tenth as needed.) the variance is □. (simplify your answer. round to the nearest tenth as needed.)

Explanation:

Step1: Recall the formula for sample variance

The formula for sample variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\), where \(n\) is the sample size, \(x_{i}\) are the data points, and \(\bar{x}\) is the sample mean. Here \(n = 10\) and \(\bar{x}=16.4\).

Step2: Calculate \((x_{i}-\bar{x})^{2}\) for each data point

• For \(x = 12\): \((12 - 16.4)^{2}=(- 4.4)^{2}=19.36\)
• For \(x = 15\): \((15 - 16.4)^{2}=(-1.4)^{2}=1.96\)
• For \(x = 23\): \((23 - 16.4)^{2}=(6.6)^{2}=43.56\)
• For \(x = 14\): \((14 - 16.4)^{2}=(-2.4)^{2}=5.76\) (do this twice as there are two 14s)
• For \(x = 17\): \((17 - 16.4)^{2}=(0.6)^{2}=0.36\) (do this twice as there are two 17s)
• For \(x = 21\): \((21 - 16.4)^{2}=(4.6)^{2}=21.16\)
• For \(x = 18\): \((18 - 16.4)^{2}=(1.6)^{2}=2.56\)
• For \(x = 13\): \((13 - 16.4)^{2}=(-3.4)^{2}=11.56\)

Step3: Sum up \((x_{i}-\bar{x})^{2}\)

\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=19.36+1.96 + 43.56+5.76+5.76+0.36+0.36+21.16+2.56+11.56=112.4\)

Step4: Calculate the sample variance

\(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}=\frac{112.4}{10 - 1}=\frac{112.4}{9}\approx12.5\)

Answer:

The variance is \(12.5\)