What hypothesis testing procedure should you use to answer these questions? Assume all samples are simple random samples, and assume α = 0.05 if it isn’t specified.
A car company says that the mean gas mileage for its luxury sedan is at least 21 mpg. You believe the claim is incorrect and find that a random sample of 5 cars has a mean gas mileage of 19 mpg and a sample standard deviation of 4 mpg. Assume the gas mileage of all of the company’s luxury sedans is normally distributed. At α = 0.05, test the company’s claim.
To test the company's claim that the mean gas mileage for its luxury sedan is at least 21 mpg, you can use a one-sample t-test.
The null hypothesis, denoted as H0, would be that the mean gas mileage for the luxury sedan is 21 mpg.
The alternative hypothesis, denoted as Ha, would be that the mean gas mileage for the luxury sedan is less than 21 mpg.
Since you are given a sample mean (19 mpg), sample standard deviation (4 mpg), and a sample size of 5, you can calculate the test statistic. The formula for the t-test statistic is:
t = (x̄ - μ) / (s / √n)
where x̄ is the sample mean, μ is the population mean (in this case, the claimed mean of 21 mpg), s is the sample standard deviation, and n is the sample size.
Substituting in the values you have:
t = (19 - 21) / (4 / √5)
After calculating the t-test statistic, you can compare it to the critical value from the t-distribution table at α = 0.05 for a one-tailed test. The degrees of freedom for this test is (n-1) = (5-1) = 4.
If the t-test statistic is smaller than the critical value, you would reject the null hypothesis and conclude that there is evidence to support the belief that the mean gas mileage is less than 21 mpg. Otherwise, if the t-test statistic is larger than the critical value, you would fail to reject the null hypothesis and conclude that there is not enough evidence to suggest that the mean gas mileage is less than 21 mpg.