Topic 4: Statistical Inference. Outline Statistical inference –confidence intervals –significance tests Statistical inference for β 1 Statistical inference

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Topic 4: Statistical Inference Slide 2 Outline Statistical inference confidence intervals significance tests Statistical inference for 1 Statistical inference for 0 Tower of Pisa example Slide 3 Theory for Statistical Inference X i iid Normal(, 2 ), parameters unknown Slide 4 Theory for Statistical Inference Consider variable t is distributed as t(n-1) Use distribution in inference for m confidence intervals significance tests Slide 5 Confidence Intervals where t c = t(1-/2,n-1), the upper (1-a/2)100 percentile of the t distribution with n-1 degrees of freedom 1-a is the confidence level Slide 6 Confidence Intervals is the sample mean (center of interval) s( ) is the estimated standard deviation of, sometimes called the standard error of the mean is the margin of error and describes the precision of the estimate Slide 7 Confidence Intervals Procedure such that (1- a )100% of the time, the true mean will be contained in interval Do not know whether a single interval is one that contains the mean or not Confidence describes long-run behavior of procedure If data non-Normal, procedure only approximate (central limit theorem) Slide 8 Significance tests Slide 9 Under H 0 t * will have distribution t(n-1) P(reject H 0 | H 0 true) = a (Type I error) Under H a, t * will have noncentral t(n-1) dists P(DNR H 0 | H a true) = b (Type II error) Type II error related to the power of the test Slide 10 NOTE IN THIS COURSE USE =.05 UNLESS SPECIFIED OTHERWISE Slide 11 Theory for 1 Inference Slide 12 Confidence Interval for 1 b 1 t c s(b 1 ) where t c = t(1-/2,n-2), the upper (1-/2)100 percentile of the t distribution with n-2 degrees of freedom 1- is the confidence level Slide 13 Significance tests for 1 Slide 14 Theory for 0 Inference Slide 15 Confidence Interval for 0 b 0 t c s(b 0 ) where t c = t(1-/2,n-2), the upper (1-/2)100 percentile of the t distribution with n-2 degrees of freedom 1- is the confidence level Slide 16 Significance tests for 0 Slide 17 Notes The normality of b 0 and b 1 follows from the fact that each of these is a linear combination of the Y i, each of which is an independent normal For b 1 see KNNL p42 For b 0 try this as an exercise Slide 18 Notes Usually the CI and significance test for 0 is not of interest If the e i are not normal but are relatively symmetric, then the CIs and significance tests are reasonable approximations Slide 19 Notes These procedures can easily be modified to produce one-sided confidence intervals and significance tests Because we can make this quantity small by making large. Slide 20 SAS Proc Reg proc reg data=a1; model lean=year/clb; run; clb option generates confidence intervals Slide 21 Parameter Estimates VariableDF Parameter Estimate Standard Errort ValuePr > |t| 95% Confidence Limits Intercept1-61.1208825.12982-2.430.0333-116.43124-5.81052 year19.318680.3099130.07