Hypothesis testing is the basis of inferential statistics. Statisticians are always coming up with new tests and testing new characteristics of population parameters. One of the simplest tests that currently exist is the one-sample test for means.

A random sample is drawn. If the population variance is known, then we use the Z test; if the population variance is unknown, we use the T test. In addition, there are some additional assumptions that can be made. For example, if a sample size is greater than 30, but the variance is unknown, we can still use the Z test.

Let us consider the following problem. We have a huge amount of data showing that the mean acceleration time (0 to 60 miles per hour) is 10.2 seconds when using regular unleaded gasoline. Now suppose that we make 41 such tests using premium unleaded gasoline because we want to find out whether premium unleaded gasoline gives us a reduction (improvement) in mean acceleration time. The sample mean acceleration for this group turns out to be 9.7 seconds with sample standard deviation of 2.1 seconds. Go to http://davidmlane.com/hyperstat/, an online statistics text, to answer the following questions related to Mr. James's problem. Read Chapter 9: The Logic of Hypothesis Testing and Chapter 10: Testing Hypothesis using Standard Errors. After reading these two Chapters, use your knowledge to answer the following questions:

• What is the hypothesis of interest in this question?
• Would you recommend a Z-test or t-test? Give a reason for your answer.
• What is the value of the test statistic for this test?
• What is the distribution of the test statistic you mentioned in the last question?
• Perform the test by first computing the p value for the test.
• Why is a one-sided test better than a two-sided test in this situation
• Can you change the situation so that a two-sided test would be appropriate and a one-sided test would not be appropriate?