438 CHAPTER 8 Hypothesis Testing with Two Samples When you use a t@distribution to approximate the sampling distribution for d, the mean of the differences between paired data entries, you can use a t@test to test a claim about the mean of the differences for a population of paired data. A t@test can be used to test the difference of two population means when these conditions are met. 1. The samples are random. 2. The samples are dependent (paired). 3. The populations are normally distributed or n Ú 30. The test statistic is d = Σd n and the standardized test statistic is t = d - md sd 1n . The degrees of freedom are d.f. = n - 1. t-Test for the Difference Between Means Using the t-Test for the Difference Between Means (Dependent Samples) In Words In Symbols 1. Verify that the samples are random and dependent, and either the populations are normally distributed or n Ú 30. 2. State the claim mathematically State H0 and Ha. and verbally. Identify the null and alternative hypotheses. 3. Specify the level of significance. Identify a. 4. Identify the degrees of freedom. d.f. = n - 1 5. Determine the critical value(s). Use Table 5 in Appendix B. 6. Determine the rejection region(s). 7. Calculate d and sd. d = Σd n sd = BΣ1d - d22 n - 1 8. Find the standardized test statistic t = d - md sd 1n and sketch the sampling distribution. 9. Make a decision to reject or fail to If t is in the rejection region, reject the null hypothesis. then reject H0. Otherwise, fail to reject H0. 10. Interpret the decision in the context of the original claim. GUIDELINES Picturing the World The manufacturer of an appetite suppressant claims that when its product is taken while following a low-fat diet with regular exercise for 4 months, the average weight loss is 20 pounds. To test this claim, you studied 12 randomly selected people taking an appetite suppressant for 4 months. Each person followed a low-fat diet with regular exercise for all 4 months. The results are shown in the table. (Adapted from NetHealth, Inc.) Weights (in pounds) of 12 People Taking an Appetite Suppressant Original weight Weight after 4th month 1 185 168 2 194 177 3 213 196 4 198 180 5 244 229 6 162 144 7 211 197 8 273 252 9 178 161 10 192 178 11 181 161 12 209 193 At A = 0.10, does your study provide enough evidence to reject the manufacturer’s claim? Assume the weights are normally distributed.
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