Expected value question

Question

Dr. Smith asked ten of his students how much cash they were carrying. The responses in dollars were: 0, 25, 10, 40, 20, 20, 40, 35, 0, 100.

In an uncharacteristic act of generosity, Smith then gives each of the students 1 dollar out of his pocket. Find the mean, median, and sample standard deviation of the students' new cash totals after Smith gives them each a dollar. Round answers to the penny.

Mean = $____

Median = $ ____

Standard Deviation = $___

Answer 1

Our new dataset is 1, 26, 11, 41, 21, 21, 41, 36, 1, and 101. Summing these values we get ∑x = 1+26+11+41+21+21+41+36+1+101 = 300. The formula for the mean x̄ = (∑x)/n, and as our sample size is n = 10, the mean = 300/10 = $30.

Sorting our list from smallest to largest we get 1, 1, 11, 21, 21, 26, 36, 41, 41, 101. Thus, our median is the average of 21 and 26 (i.e median = (21+26)/2 = $23.50).

Note that the standard deviation (s) formula is s= sqrt ((∑(x_i - x̄)² / (n-1)). Note that ∑(x_i - x̄)² = (1-30)² + (26-30)² + (11-30)² + (41-30)² + (21-30)² + (21-30)² + (41-30)² + (36-30)² + (1-30)² + (101-30)² = 7540. As n-1 = 10-1 = 9, s = sqrt(7540/9) = 28.944. Rounded to the nearest penny, the standard deviation is $28.94.