A study found that highway drivers in one state traveled at

A study found that highway drivers in one state traveled at an average speed of 57.3 miles per hour (MPH). Assume the population standard deviation is 6.9 MPH. Complete parts a through d below.

a. What is the probability that a sample of 30 of the drivers will have a sample mean less than 56 MPH?

(Round to four decimal places as needed.)

b. What is the probability that a sample of 45 of the drivers will have a sample mean less than 56 MPH?

(Round to four decimal places as needed.)

c. What is the probability that a sample of 60 of the drivers will have a sample mean less than 56 MPH?

(Round to four decimal places as needed.)

d. Explain the difference in these probabilities. (Select one each)

As the sample size increases, the standard error of the mean (Same, Increase, Decrease) As the sample size increases, the standard error of the mean (Same, Move farther, Move closer) the population mean of 57.3 MPH. Therefore, the probability of observing a sample mean less than 56 MPH (Same, Increase, Decrease).

Solution

Mean ( u ) =57.3
Standard Deviation ( sd )=6.9
Number ( n ) = 30
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  
a)
P(X < 56) = (56-57.3)/6.9/ Sqrt ( 30 )
= -1.3/1.2598= -1.0319
= P ( Z <-1.0319) From Standard NOrmal Table
= 0.1511                  
b)
P(X < 56) = (56-57.3)/6.9/ Sqrt ( 45 )
= -1.3/1.0286= -1.2639
= P ( Z <-1.2639) From Standard NOrmal Table
= 0.1031                  
c)
P(X < 56) = (56-57.3)/6.9/ Sqrt ( 60 )
= -1.3/0.8908= -1.4594
= P ( Z <-1.4594) From Standard NOrmal Table
= 0.0722