458 CHAPTER 9 Inferences from Two Samples Statdisk Step4: The significance level is a = 0.05. Step5: Using Technology Steps 5 and 6 can be skipped when using technology; see the accompanying Statdisk display showing the test statistic and P-value. YOUR TURN. Do Part (a) of Exercise 5 “Better Tips by Giving Candy.” Manual Calculation If not using technology, we use the t distribution with the test statistic given in the Key Elements box. Step6: The test statistic is calculated using these statistics (with extra decimal places) obtained from the listed sample data: ANSUR I 1988: n = 12, x = 1739.417 mm, s = 66.6012 mm ANSUR II 2012: n = 15, x = 1777.8 mm, s = 47.86618 mm t = 1x1 - x22 - 1m1 - m22 Bs2 1 n1 + s2 2 n2 = 11739.417 - 1777.82 - 0 B66.60122 12 + 47.866182 15 = -1.679 P-Value With test statistic t = -1.679, we refer to Table A-3 (t distribution). The number of degrees of freedom is the smaller of n1 - 1 and n2 - 1, or the smaller of 112 - 12 and 115 - 12, which is 11. With df = 11 and a left-tailed test, Table A-3 indicates that the P-value is greater than 0.05 (and less than 0.10). Technology will provide the P-value of 0.0546 when using the original data or unrounded sample statistics. Step7: Because the P-value is greater than the significance level of 0.05, we fail to reject the null hypothesis. (“If the P is low, the null must go.” Here, the P-value is not low, so it doesn’t go.) INTERPRETATION Step8: There is not sufficient evidence to support the claim that the mean height of the 1988 male population is less than the mean height of the 2012 male population. Are people getting taller? Maybe, maybe not, but the given data do not provide enough evidence to support that claim. The sample mean from 1988 is 1739.4 mm and the sample mean from 2012 is 1777.8 mm, so the 2012 sample mean is larger. Perhaps a larger sample might provide sufficient evidence to support the claim. Also, the sample data are from U.S. male Army personnel, and this sample might be from a population that is different from the general population. Critical Value Method If technology is not available, the critical value method of testing a claim about two means is generally easier than the P-value method. Example 1 can be solved using the critical value method. When finding critical values in Table A-3, we use df = smaller of n1 - 1 and n2 - 1 as a relatively easy way to avoid using the really messy calculation required with Formula 9-1. In Example 1 with sample sizes of n1 = 12 and n2 = 15, the number Expensive Diet Pill There are many past examples in which ineffective treatments were marketed for substantial profits. Capsules of “Fat Trapper” and “Exercise in a Bottle,” manufactured by the Enforma Natural Products company, were advertised as being effective treatments for weight reduction. Advertisements claimed that after taking the capsules, fat would be blocked and calories would be burned, even without exercise. Because the Federal Trade Commission identified claims that appeared to be unsubstantiated, the company was fined $10 million for deceptive advertising. The effectiveness of such treatments can be determined with experiments in which one group of randomly selected subjects is given the treatment, while another group of randomly selected subjects is given a placebo. The resulting weight losses can be compared using statistical methods, such as those described in this section. T m e w t w f
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