A professor records the time (in minutes) that it takes 16 students to complete

Question

A professor records the time (in minutes) that it takes 16 students to complete an exam. Compute the S S, the variance, and the standard deviation assuming the 16 students constitute a population and assuming the 16 students constitute a sample. (Round your answers for variance and standard deviation to two decimal places.)

22 38 21 42 44 24 33 49 40 30 47 15 41 30 51 45

(a) the 16 students constitute a population SS variance standard deviation (b) the 16 students constitute a sample SS variance standard deviation

Explanation / Answer

22 38 21 42 44 24 33 49 40 30 47 15 41 30 51 45

The mean is (22 + 38 + 21 + 42 + 44 + 24 + 33 + 49 + 40 + 30 + 47 + 15 + 41 + 30 + 51 + 45) / 16 = 35.75

The differences are 22-35.75 38-35.75 21-35.75 42-35.75 44-35.75 24-35.75 33-35.75 49-35.75 40-35.75 30-35.75 47-35.75 15-35.75 41-35.75 30-35.75 51-35.75 45-35.75

= -13.75 2.25 -14.75 6.25 8.25 -11.75 -2.75 13.25 4.25 -5.75 11.25 -20.75 5.25 -5.75 15.25 9.25

Squares of the differences are -13.752 2.252 -14.752 6.252 8.252 -11.752 -2.752 13.252 4.252 -5.752 11.252 -20.752 5.252 -5.752 15.252 9.252

= 189.0625 5.0625 217.5625 39.0625 68.0625 138.0625 7.5625 175.5625 18.0625 33.0625 126.5625 430.5625 27.5625 33.0625 232.5625 85.5625

Sum of the squares of the differences SS = (189.0625 + 5.0625 + 217.5625 + 39.0625 + 68.0625 + 138.0625 + 7.5625 + 175.5625 + 18.0625 + 33.0625 + 126.5625 + 430.5625 + 27.5625 + 33.0625 + 232.5625 + 85.5625) = 1827

(a) Variance when the students constitute a population = SS / n

= 1827 / 16

= 114.1875

Standard deviation = Variance = 114.1875 = 10.69

(b) Variance when the students constitute a sample = SS / (n-1)

= 1827 / 15

= 121.8

Standard deviation = Variance = 121.8 = 11.04

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