In: Statistics and Probability
A study found that highway drivers in one state traveled at an average speed of 59.7 miles per hour (MPH). Assume the population standard deviation is 6. 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 58 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 58 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 58 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 59.7 MPH. Therefore, the probability of observing a sample mean less than 58 MPH (Same, Increase, Decrease).
a)
Here, μ = 59.7, σ = 1.0954 and x = 58. We need to compute P(X <=
58). The corresponding z-value is calculated using Central Limit
Theorem
z = (x - μ)/σ
z = (58 - 59.7)/1.0954 = -1.55
Therefore,
P(X <= 58) = P(z <= (58 - 59.7)/1.0954)
= P(z <= -1.55)
= 0.0606
b)
Here, μ = 59.7, σ = 0.8944 and x = 58. We need to compute P(X <= 58). The corresponding z-value is calculated using Central Limit Theorem
z = (x - μ)/σ
z = (58 - 59.7)/0.8944 = -1.9
Therefore,
P(X <= 58) = P(z <= (58 - 59.7)/0.8944)
= P(z <= -1.9)
= 0.0287
c)
Here, μ = 59.7, σ = 0.7746 and x = 58. We need to compute P(X <= 58). The corresponding z-value is calculated using Central Limit Theorem
z = (x - μ)/σ
z = (58 - 59.7)/0.7746 = -2.19
Therefore,
P(X <= 58) = P(z <= (58 - 59.7)/0.7746)
= P(z <= -2.19)
= 0.0143
d)
As the sample size increases, the standard error of the mean ( Decrease) As the sample size increases, the standard error of the mean ( Move farther) the population mean of 59.7 MPH. Therefore, the probability of observing a sample mean less than 58 MPH ( Decrease).