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11.1 T-test Example problem using the one-sample t-test in the process to test statistical hypotheses for a research problem Research Problem: We have a random sample of 25 fifth grade pupils who can do 15 pushups on the average, with a standard deviation of 9, after completing a special physical education program. Does this value of 15 differ significantly from the population value of 12? The research question for this experiment is - Does having students complete a special physical education program result in a signficantly different number of pushups they can do, as compared with the population average? State null hypothesis and alternative hypothesis: In this problem the research question does not indicate the desired direction of the result, so the alternative hypothesis will use the not equal choice. This means that a significant result could either be significantly greater than the population value or signficantly less than the population value. 1. Set the alpha level: 2. Calculate the value of the proper statistic: Since this problem involves comparing a single group's mean with the population mean and the standard deviation for the population is not known, the proper statistical test to use is the one-sample t-test. t = 1.667 df = 24 3. State the rule for rejecting the null hypothesis: We need to find the value of t that will only be exceeded 5% of the time in either direction if we set the alpha level at .05. So that means we will be looking for the upper 2.5% of the distribution and the lower 2.5% of the distribution (2.5 + 2.5 = 5%). In other words we will be using a two-tailed test. To find the signficant values of t we use the table in Appendix C of the text book (page 318 - Distribution of t). If we enter the table under the heading "Level of significance for two-tailed test" and .05 we read down the column until we come to the row that has 24 degrees of freedom (See df column to the left of the table). We see that the table value of t is 2.064, so this is the value we will use for our rejection rule. Our rejection rule then would be: Reject H0 if t or if t -2.064 Look at this rejection rule carefully as this is the general way to state the rejection rule for a two-tailed test. 4. Decision: Fail to reject H0 Since the calculated value of t (1.667) is not greater than 2.064 nor less than -2.064, we can not reject the null hypothesis. Since our decision was to fail to reject H0, we do not have to add the alpha level or the tailedness of the test as we did when we rejected H0. 5. Statement of results: The number of pushups done by a group of fifth grade pupils who have participated in a special physical education program, does not, differ significantly from the population average.