Question

Suppose that the waiting time for a bus is normally distributed with a mean of 5 minutes and a standard deviation of 2.5 minutes.

(a) Find the probability that the waiting time for the bus is between 3 minutes and 7 minutes.

(b) If you randomly choose 4 passengers, find the sampling distribution of their average waiting time. Please indicate shape, mean and standard deviation.

(c) For the randomly selected 4 passengers, find the probability that their average waiting time is above 6 minutes. Let X¯ be the average waiting time of randomly selected 4 passengers.

Answer 1

Solution :

Given that ,

mean =   = 5

standard deviation = = 2.5

P(3< x < 7) = P[(3-5) /2.5 < (x - ) / < (7-5) /2.5 )]

= P( -0.8< Z <0.8 )

= P(Z < 0.8) - P(Z < -0.8)

Using z table   

= 0.7881-0.2119

probability= 0.5762

(B)

n = 4

mean = = 5

standard deviation = = / n = 2.5/ 4 = 1.25

P( >6 ) = 1 - P( <6 )

= 1 - P[( - ) / < (6 -5) /1.25 ]

= 1 - P(z <0.8 )

Using z table

= 1 - 0.7881

= 0.2119

probability=0.2119