the average weekly unemployment benefits (in dollars) for a random selection of states are listed below. 272…

Explanation:

Step1: Calculate the mean

Let the data set be (x_1 = 272,x_2=239,\cdots,x_{15}=416). The mean (\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}), where (n = 15). (\sum_{i=1}^{15}x_i=272 + 239+289+214+276+252+190+209+251+272+310+321+327+356+416=4004) (\bar{x}=\frac{4004}{15}\approx266.93)

Step2: Calculate the squared - differences

For each (x_i), calculate ((x_i-\bar{x})^2). For example, when (x_1 = 272), ((x_1-\bar{x})^2=(272 - 266.93)^2=(5.07)^2 = 25.7049). Do this for all (x_i) values and sum them up. (\sum_{i = 1}^{15}(x_i-\bar{x})^2=(272 - 266.93)^2+(239 - 266.93)^2+\cdots+(416 - 266.93)^2) (=25.7049+780.0849+48.0249+2877.9449+1.1449+210.6449+5914.7849+3377.1449+286.7849+25.7049+1755.7049+2809.9449+3644.4049+7804.5449+22144.7849=49686.6675)

Step3: Calculate the variance

The variance (s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}) (since this is a sample). Here (n = 15), so (s^2=\frac{49686.6675}{14}\approx3549.0477)

Step4: Calculate the standard deviation

The standard deviation (s=\sqrt{s^2}), so (s=\sqrt{3549.0477}\approx59.6)

Answer:

59.6