The Office of Student Services at a large western state univ

The Office of Student Services at a large western state university maintains information on the study habits of its full-time students. Their studies indicate that the mean amount of time undergraduate students study per week is 20 hours. The hours studied follows the normal distribution with a standard deviation of six hours. Suppose we select a random sample of 144 current students. What is the probability that the mean of this sample is between 19 hours and 20 hours?

A. 2.00

B. 0.4772

C. -2.00

D. Cannot be calculated based on the given information

Solution

Let X represents the mean amount of time undergraduate students study per week

X follows normal distribution with

mean = 20 , standard deviation = 6 , n = size of sample = 144

P(X=x) = P(Z = x-mean/std/sqrt(n))

=>

P(19 < X <20) = P(X<20) - P(X < 19)

= P(Z < 20 - 20/6/12) - P ( Z < 19-20/6/12)

= P (Z <0) - P ( Z < -2)

= 0.5 - 0.0228

= 0.4772

hence the correct choice is B)