Question

The Length of time required by students to complete a one hour exam is a random variable with a density function give by:

f(x) = (3/2)x^2 + x (0<=x<=1)

0 elsewhere

a. What is the probability that a randomly selected student will finish in less than 45 minutes?

b. If 40 students are chosen at random, what is the probability that the sample average will be less than 45 minutes?

c. If instead the sample size had been 10, could you have found the probability of the sample average being less than 45? Explain

Answer 1

a)

45 min = 0.75 hour

b)

E(T) =

E(T^2) =

Var(T) =E(T^2- (E(T))^2

= 11/20 - (17/24)^2

= 0.0482638

sd(T) = sqrt( 0.0482638 ) = 0.219690

E(Tbar) = E(T) = 17/24

sd(Tbar) = sd(T)/sqrt(n)

= 0.0347360

Z = (Tbar - 17/24 )/0.034736    {by central limit theorem , as n = 40 > 30 }

P(Tbar < 0.75)

= P(Z <(0.75 - 17/24)/ 0.0347360 )

= P(Z < 1.19952 )

= 0.8848

c)

No, because then central limit would not be applicable