A consumer research group is interested in testing an automobile manufacturer’s claim
that a new economy model will travel at least 25 miles per gallon of gasoline (H0: μ � 25).
a. With a .02 level of significance and a sample of 30 cars, what is the rejection rule based
on the value of for the test to determine whether the manufacturer’s claim should be
rejected? Assume that σ is 3 miles per gallon.
b. What is the probability of committing a Type II error if the actual mileage is 23 miles
per gallon?
c. What is the probability of committing a Type II error if the actual mileage is 24 miles
per gallon?
d. What is the probability of committing a Type II error if the actual mileage is 25.5 miles
per gallon?
To answer these questions, we need to perform a hypothesis test and calculate the rejection rule and the probabilities of Type II errors.
a. The rejection rule is based on the significance level (α) and the sample size (n). In this case, the significance level is 0.02 (α = 0.02) and the sample size is 30 cars (n = 30).
Since we know σ, the population standard deviation (σ = 3 miles per gallon), we can calculate the standard error of the mean (SEM) using the formula:
SEM = σ / sqrt(n)
SEM = 3 / sqrt(30) ≈ 0.5477
To calculate the rejection rule, we determine the critical value (z) that corresponds to the desired significance level (α = 0.02) using a standard normal distribution table or a statistical software. In this case, since it's a two-tailed test, we need to split the significance level into two equal parts (0.02 / 2 = 0.01) and find the z-value that corresponds to the two tails of the distribution.
The rejection rule is:
Reject H0 if the test statistic (z) is less than -zα/2 or greater than zα/2.
Using a standard normal distribution table or a statistical software, we find that zα/2 is approximately -2.33.
Therefore, the rejection rule is:
Reject H0 if z < -2.33 or z > 2.33.
b. To calculate the probability of committing a Type II error for an actual mileage of 23 miles per gallon, we need to first determine the cutoff value for rejecting the null hypothesis (H0). In this case, the null hypothesis is μ = 25 (the claimed mileage) and the alternative hypothesis is μ < 25 (the actual mileage is less than the claimed mileage).
The cutoff value for rejecting H0 depends on the desired significance level (α) and the sample size (n). Since the significance level is 0.02, we can use the rejection rule from part (a) as a cutoff value.
Now, we need to calculate the test statistic (z) for the actual mileage of 23 miles per gallon using the formula:
z = (x̄ - μ) / (SEM)
where x̄ is the sample mean.
For an actual mileage of 23 miles per gallon, the test statistic is:
z = (23 - 25) / 0.5477 ≈ -3.65
Using a standard normal distribution table or a statistical software, we find the probability of z < -3.65 (the test statistic) is approximately 0.0001.
Therefore, the probability of committing a Type II error if the actual mileage is 23 miles per gallon is approximately 0.0001.
c. To calculate the probability of committing a Type II error for an actual mileage of 24 miles per gallon, we follow the same steps as in part (b).
For an actual mileage of 24 miles per gallon, the test statistic is:
z = (24 - 25) / 0.5477 ≈ -1.82
Using a standard normal distribution table or a statistical software, we find the probability of z < -1.82 is approximately 0.0344.
Therefore, the probability of committing a Type II error if the actual mileage is 24 miles per gallon is approximately 0.0344.
d. For an actual mileage of 25.5 miles per gallon, we calculate the test statistic:
z = (25.5 - 25) / 0.5477 ≈ 0.9145
Using a standard normal distribution table or a statistical software, we find the probability of z > 0.9145 is approximately 0.1790.
Therefore, the probability of committing a Type II error if the actual mileage is 25.5 miles per gallon is approximately 0.1790.