Question

Problem

An automobile maker claims that its newest Raptor V5s (a sports car) have a mean fuel efficiency of 48.5 miles per gallon on highways. A consumer advocate suspects, however, that the actual mean fuel efficiency of the Raptor V5s on highways is less than 48.5 miles per gallon.

To test this hypothesis, the consumer advocate obtains a random sample of 35 Raptor V5s, measures their fuel efficiencies on highways, and calculates the sample mean fuel efficiency to be 46.2 miles per gallon, with a sample standard deviation of 5.5 miles per gallon.

Is there sufficient evidence to conclude that the mean fuel efficiency of the Raptor V5s on highways is less than 48.5 miles per gallon? Use an α=0.05α=0.05 level of significance.

Question 1

State the null and alternative hypotheses (using symbols).

Question 2

Explain what it would mean for the consumer advocate's study to make a Type II Error.

Question 3 (Optional)

Verify the requirements for this hypothesis test. Briefly show that each requirement is satisfied.

Question 4

Test the hypothesis using either the classical (test statistic) approach or the P-value approach. Show all steps to your approach. Then state whether you reject H0H0 or do not reject H0H0.

Question 5

Is there sufficient evidence to conclude that the mean fuel efficiency of the Raptor V5s on highways is less than 48.5 miles per gallon? State your conclusion in one or two sentences.

Answer 1

x̅ = 46.2, s = 5.5, n = 35

1.

Null and Alternative hypothesis:

Ho : µ = 48.5

H1 : µ < 48.5

2.

Type II error is failing to reject that fuel efficient is 48.5 miles per gallon while in reality it is less than 48.5

3. Requirements:

The sample is random and independent.

The distribution is normally distributed.

Sample size is reasonably large.

4.

Test statistic:

t = (x̅- µ)/(s/√n) = (46.2 - 48.5)/(5.5/√35) = -2.4740

df = n-1 = 34

Critical value :

Left tailed critical value, t-crit = T.INV(0.05, 34) = -1.691

Reject Ho if t < -1.691

p-value :

p-value = T.DIST(-2.474, 34, 1) = 0.0093

Decision:

p-value < α, Reject the null hypothesis

5.

Conclusion:

There is enough evidence to conclude that the mean fuel efficiency of the Raptor V5s on highways is less than 48.5 miles per gallon at 0.05 significance level.