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# Statistics testing for mean acceleration

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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?

##### Solution Summary

This solution provides steps to perform hypothesis testing for the data provided.

Solution provided by:
###### Education
• BSc , Wuhan Univ. China
• MA, Shandong Univ.
###### Recent Feedback
• "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
• "excellent work"
• "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
• "Thank you"
• "Thank you very much for your valuable time and assistance!"

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