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Unformatted text preview: Lecture 29 Nancy Pfenning Stats 1000 Reviewing Confidence Intervals and Tests for Ordinary OneSample, Matched Pairs, and TwoSample 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 onesample 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 Pvalue 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 Pvalue 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 calciumtakers same as mean difference for placebotakers] vs....
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 Fall '06
 taeyoungpark
 Statistics, Normal Distribution, Standard Deviation, Statistical hypothesis testing

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