Question

# In a recent publication, it was reported that the average highway gas mileage of tested models...

In a recent publication, it was reported that the average highway gas mileage of tested models of a new car was 34.2
mpg with a standard deviation of 1.5 mpg, with the mileages approximately normally distributed.

H0 : μ = 34.2
Ha : μ < 34.2
Part a: Describe a Type II error in the context of the hypothesis test.
Part b: If a simple random sample of 100 cars is selected, what values of the sample mean ̄x would represent sufficient
evidence to reject the null hypothesis at significance level α = 0.05?
Part c: If the actual population average gas mileage is 33.8 mpg, determine the probability that the null hypothesis will be
rejected if a simple random sample of 100 cars is selected.
Part d: If the sample size were greater than 100 cars, what would happen to the probability in Part c? Explain your
reasoning.

a)
type ii error - failing to reject the null hypothesis when null is false
hence we fail to conclude that mileage is less than 34.2 when it is less 34.2

b)
Z= (Xbar -mu)/(sigma/sqrt(n))

we reject if Xbar < mu - z * sigma/sqrt(n)
< 34.2 - 1.645 * 1.5/sqrt(100)    {z= 1.645 for 0.05 one-sided }
< 33.95325

c)
type ii error
= P(xbar > 33.95325 | mu = 33.8)
= P(Z > (33.95325- 33.8)/(1.5/sqrt(100)))
= P(Z > 1.02166)
= 0.1535

d)
if n increases, type ii error decreases

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