Questions: Calculate the range, population variance, and population standard deviation for the following data set. If necessary, round to one more decimal place than the largest number of decimal places given in the data. 8, 10, 7, 3, 7, 6, 4, 8, 4, 31

Solution Steps

To calculate the range, we identify the minimum and maximum values in the data set.

• Minimum (\(min\)) = 3
• Maximum (\(max\)) = 31 Then, we calculate the range as \(Range = max - min = 31 - 3 = 28\).

First, we compute the mean (\(\mu\)) of the data set as \(\mu = \frac{1}{n}\sum_{i=1}^{n}x_i = 8.8\). Then, for each value in the data set, we calculate the square of its difference from the mean and sum all the squared differences. Finally, we divide by the number of observations to get the population variance: \(\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \mu)^2 = 59\).

We take the square root of the population variance to get the population standard deviation: \(\sigma = \sqrt{\sigma^2} = 7.7\).

Final Answer: