Question

In: Statistics and Probability

A study found that highway drivers in one state traveled at an average speed of 59.7...

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).

Solutions

Expert Solution

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).


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