A data scientist wants to determine the likelihood of people shopping in a mall based on their age. To make the determination, 20 randomly selected shoppers at a mall were asked to report their ages. The resulting age counts are:

| Age | 15 | 16 | 17 | 20 | 21 | 24 | 35 | 36 | 42 | 51 | 73 |
|-----|----|----|----|----|----|----|----|----|----|----|----|
| # of People | 2 | 1 | 3 | 2 | 4 | 2 | 1 | 2 | 1 | 1 | 1 |

What was the standard deviation of the age of mall shoppers? There are two options to receive full credit for this problem:

1. Create a sum of squares table to calculate the standard deviation.
2. Fully write out each step of a single equation that includes all data and solve step by step.

Is it reasonable to assume that the data from any sample of 20 shoppers in the mall will have a normal distribution? Why or why not?

The standard deviation of the age of mall shoppers is approximately 5.41. It is reasonable to assume a normal distribution, but further analysis is required.

Explanation:

The first step in calculating the standard deviation is to find the mean of the data set. In this case, we add up all the ages and divide by the total number of shoppers (20). The sum of the ages is: 15(2) + 16(1) + 17(3) + 20(2) + 21(4) + 24(2) + 35(1) + 36(2) + 42(1) + 51(1) + 73(1) = 693. So the mean age is 693/20 = 34.65.

Next, we subtract the mean from each individual age and square the result. Then we sum up all these squared differences. The sum of squares is: (15-34.65)^2(2) + (16-34.65)^2(1) + (17-34.65)^2(3) + (20-34.65)^2(2) + (21-34.65)^2(4) + (24-34.65)^2(2) + (35-34.65)^2(1) + (36-34.65)^2(2) + (42-34.65)^2(1) + (51-34.65)^2(1) + (73-34.65)^2(1) = 1000.50.

To calculate the standard deviation, we divide the sum of squares by the total number of shoppers minus one (n-1), then take the square root of the result. In this case, since we have 20 shoppers, the standard deviation is:

sqrt(1000.50/19) ≈ 5.41.

Therefore, the standard deviation of the age of mall shoppers in this sample is approximately 5.41.

It is reasonable to assume that the data from any sample of 20 shoppers in the mall will have a normal distribution if the population of shoppers in the mall exhibits a normal distribution of ages. However, without more information about the population, we cannot definitively conclude that the data from any sample of 20 shoppers will always have a normal distribution. Additional data and analysis would be needed to make a more accurate assessment.

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