A car company says that the mean gas mileage for its luxury sedan is at least 21 miles per gallon. You believe the claim is incorrect and find that a random sample of five cars has a mean gas mileage of 19 miles per gallon and a standard deviation of 4 miles per gallon. Assume the gas mileage of all of the company’s luxury sedans is normally distributed. At á = 0.05, test the company’s claim.
• What is the difference between a critical value and a test statistic?
• Decide whether you should use a normal sampling distribution or a t-sampling distribution to perform the hypothesis test.
• Why would you use a z-test rather than a t-test?
• Which do you think you will use more often? Justify your decisions.
• Then use the distribution to test the claim.
• Write a short paragraph about the results of the test and what you can conclude about the claim.
Ho: Mean >= 21 (claim)
Ha: mean < 21
t = 1.118
P-value = 0.163
Fail to reject Ho
1. The difference between a critical value and a test statistic lies in their purpose and interpretation. A critical value is a threshold value that defines the boundary for rejecting or retaining a null hypothesis in a hypothesis test. It is determined based on the significance level (α) chosen for the test and the specific test being conducted. On the other hand, a test statistic is a numerical value calculated from the sample data that helps assess the evidence against the null hypothesis. It quantifies the distance between the sample result and the expected value under the null hypothesis.
2. In this case, since the population standard deviation is unknown, it is appropriate to use a t-sampling distribution to perform the hypothesis test. This is because the sample standard deviation is used as an estimate of the population standard deviation.
3. A z-test is used when the population standard deviation is known or when the sample size is sufficiently large (generally over 30) and the population is approximately normally distributed. On the other hand, a t-test is used when the population standard deviation is unknown and needs to be estimated from the sample data, or when the sample size is small (typically under 30) and the population may not be normally distributed.
4. In this particular scenario, we should use a t-test because the population standard deviation is unknown and we have a small sample size of five cars.
5. To test the claim, we will conduct a one-sample t-test. The null hypothesis (H0) for this test would be: the mean gas mileage for the luxury sedan is at least 21 miles per gallon. The alternative hypothesis (Ha) would be: the mean gas mileage for the luxury sedan is less than 21 miles per gallon.
6. By performing the t-test using the given sample data (mean = 19, standard deviation = 4, sample size = 5, significance level α = 0.05), we can calculate the t-statistic. Then, we compare the t-statistic to the critical value from the t-distribution with (n-1) degrees of freedom. If the t-statistic falls in the rejection region (i.e., the calculated t-value is smaller than the critical value), we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
7. Based on the results of the t-test, if the calculated t-value is smaller than the critical value, we can conclude that there is sufficient evidence to reject the claim made by the car company. In this case, a lower mean gas mileage of 19 miles per gallon suggests that the luxury sedan's actual gas mileage may be less than the claimed 21 miles per gallon.