Question

the ages (in years) of a random sample of shoppers at a gaming store are shown. mean, variance, and standard deviation of the sample data set.
12, 19, 23, 14, 19, 18, 21, 18, 15, 19
the range is 11.
(simplify your answer.)
the mean is 17.8.
(simplify your answer. round to the nearest tenth as needed.)
the variance is \boxed{}.
(simplify your answer. round to the nearest tenth as needed.)

Explanation:

Step1: Recall the formula for sample variance

The formula for the sample variance \( s^2 \) is \( s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1} \), where \( x_i \) are the data points, \( \bar{x} \) is the sample mean, and \( n \) is the number of data points. We know that \( \bar{x}=17.8 \) and \( n = 10 \) (since there are 10 data points: 12, 19, 23, 14, 19, 18, 21, 18, 15, 19).

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

• For \( x_1 = 12 \): \( (12 - 17.8)^2=(- 5.8)^2 = 33.64 \)
• For \( x_2 = 19 \): \( (19 - 17.8)^2=(1.2)^2=1.44 \)
• For \( x_3 = 23 \): \( (23 - 17.8)^2=(5.2)^2 = 27.04 \)
• For \( x_4 = 14 \): \( (14 - 17.8)^2=(-3.8)^2 = 14.44 \)
• For \( x_5 = 19 \): \( (19 - 17.8)^2=(1.2)^2 = 1.44 \)
• For \( x_6 = 18 \): \( (18 - 17.8)^2=(0.2)^2=0.04 \)
• For \( x_7 = 21 \): \( (21 - 17.8)^2=(3.2)^2 = 10.24 \)
• For \( x_8 = 18 \): \( (18 - 17.8)^2=(0.2)^2 = 0.04 \)
• For \( x_9 = 15 \): \( (15 - 17.8)^2=(-2.8)^2 = 7.84 \)
• For \( x_{10}=19 \): \( (19 - 17.8)^2=(1.2)^2 = 1.44 \)

Step3: Sum up the squared deviations

\( \sum_{i = 1}^{10}(x_i-\bar{x})^2=33.64 + 1.44+27.04+14.44+1.44+0.04+10.24+0.04+7.84+1.44 \)
\( =33.64+1.44 = 35.08 \); \( 35.08+27.04 = 62.12 \); \( 62.12+14.44 = 76.56 \); \( 76.56+1.44 = 78 \); \( 78+0.04 = 78.04 \); \( 78.04+10.24 = 88.28 \); \( 88.28+0.04 = 88.32 \); \( 88.32+7.84 = 96.16 \); \( 96.16+1.44 = 97.6 \)

Step4: Calculate the variance

Using the formula \( s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1} \), with \( \sum_{i = 1}^{n}(x_i-\bar{x})^2 = 97.6 \) and \( n-1=9 \)
\( s^2=\frac{97.6}{9}\approx10.844\approx10.8 \) (rounded to the nearest tenth)

Answer:

\( 10.8 \)