Question

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.

Answer #1

**ANSWER::**

**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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