It is claimed by a consumer advocacy group that a specific car model obtains a m
ID: 3358457 • Letter: I
Question
It is claimed by a consumer advocacy group that a specific car model obtains a mean of 35 miles per gallon on the highway with a standard deviation of 2 miles per gallon. The company that manufactures the car claims that it exceeds this estimate in highway driving. To support its assertion, the company randomly selects 81 cars. The company conducts an appropriate hypothesis test at level of significance .025.
a. What is the probability of making a Type I error? If they would use 64 cars instead, what would be the probability of making a Type I error?
b. If the p-value is equal to .09, what was the value of the sample mean of the 81 cars?
c. If the CEO claims that the mean is different than 35 miles per gallon, what would be the p-value assuming the same value of xbar and n?
d. A consumer advocacy group wants to construct a 95% confidence interval for the true mean fuel efficiency. She takes a random sample of 64 cars and computes the sample mean to be equal to 35.45. (Assume that the population standard deviation is still 2 miles per gallon.) FIRST, Construct the 95% confidence interval. The advocacy group believes that there is a 95% chance that the true mean is between the two numbers you just computed. SECOND, Please respond to this claim.
Explanation / Answer
a.
The probability of making a Type I error is level of signicance. So, the probability of making a Type I error is 0.025.
The probability of Type I error is independent of the size of the sample. So, probability of making a Type I error will remain same equal to 0.025 even if they would use 64 cars.
b.
p-value is equal to .09 => P[Z > z] = 0.09
=> z = 1.34 (Using Z table)
where z = (Observed value - Hypothesized value)/ Standard error
Hypothesized value = 35
Standard error = SD / sqrt(n) = 2 / sqrt(81) = 0.2222
So,
z = (Observed value - 35)/ 0.2222 = 1.34
Observed value = 35 + 1.34 * 0.2222 = 35.29775
So, the value of sample mean is 35.29775
c.
If the CEO claims that the mean is different than 35 miles per gallon, then the mean is less or greater than 35 miles per gallon. So, this is a two-tail test.
p-value = 2 * 0.09 = 0.18
d.
Standard error of sample mean = 2 / sqrt(64) = 0.25
Z value of 95% confidence interval is 1.96
Margin of error = 1.96 * 0.25 = 0.49
So, 95% confidence interval is,
(35.45 - 0.49, 35.45 + 0.49)
(34.96, 35.94)
The claim is not correct. The claim is that there is a 95% chance that the true mean is between the two numbers.
Correct Interpretation - If we take multiple samples, and when we calculate a confidence interval in this way, 95% of the time, the true mean will be between these two numbers.
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