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Unformatted text preview: Lecture 29 Nancy Pfenning Stats 1000 Reviewing Confidence Intervals and Tests for Ordinary One-Sample, Matched- Pairs, and Two-Sample Studies About Means Example Blood pressure X was measured for a sample of 10 black men. It was found that ¯ x = 114 . 9, s = 10 . 84. Give a 90% confidence interval for mean blood pressure μ of all black men. [Note: we can assume that blood pressure tends to differ for different races or genders, and that is why a separate study is made of black men—the confounding variables of race and gender are being controlled.] This is an ordinary one-sample t procedure. A level .90 confidence interval for μ is ¯ x ± t * s √ n , where t * has 10- 1 = 9 df. Consulting the df = 9 row and .90 confidence column of Table A.2, we find t * = 1 . 83. Our confidence interval is 114 . 9 ± 1 . 83 10 . 84 √ 10 = (108 . 6 , 121 . 2). Here is what the MINITAB output looks like: N MEAN STDEV SE MEAN 90.0 PERCENT C.I. calcbeg 10 114.90 10.84 3.43 ( 108.62, 121.18) Example Blood pressure for a sample of 10 black men was measured at the beginning and end of a period of treatment with calcium supplements. To test at the 5% level if calcium was effective in lowering blood pressure, let the R.V. X denote decrease in blood pressure, beginning minus end, and μ D would be the population mean decrease. This is a matched pairs procedure. To test H : μ D = 0 vs. H a : μ D > 0, we find differences X to have sample mean ¯ d = 5 . 0, sample standard deviation s = 8 . 74. The t statistic is t = ¯ d- μ s √ n = 5- 8 . 74 √ 10 = 1 . 81, and the P-value is P ( T ≥ 1 . 81). We refer to Table A.2 for the t (9) distribution, and see that 1.81 is just under 1.83, which puts our P-value just over .05. Our test has not quite succeeded in finding the difference to be significantly greater than zero, in a statistical sense. Populations of black men treated with calcium may experience no decrease in blood pressure. MINITAB output appears below. TEST OF MU = 0.00 VS MU G.T. 0.00 N MEAN STDEV SE MEAN T P VALUE calcdiff 10 5.00 8.74 2.76 1.81 0.052 It is possible that our sample size was too small to generate statistically significant results. An- other concern is the possibility of confounding variables influencing their blood pressure change. The placebo effect may tend to bias results towards a larger decrease. Or, time may play a role: if the beginning or end measurement date happened to be in the middle of a harsh winter or a politically stressful time, results could be affected. Example Data for a control group (taking placebos) of 11 black men at the beginning and end of the same time period produced control sample mean difference ¯ d 2 =- . 64, and s 2 = 5 . 87. Now we test H : μ 1- μ 2 = 0 [same as H : μ 1 = μ 2 , or mean difference for calcium-takers same as mean difference for placebo-takers] vs....
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- Fall '06