1. Topic 4: Statistical Inference

2. Outline Statistical inference –confidence intervals –significance tests Statistical inference for β 1 Statistical inference for β 0 Tower of Pisa example

3. Theory for Statistical Inference X i iid Normal(μ,σ 2 ), parameters unknown

4. Theory for Statistical Inference Consider variable t is distributed as t(n-1) Use distribution in inference for m –confidence intervals –significance tests

5. Confidence Intervals where t c = t(1-α/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

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

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)

8. Significance tests

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

10. NOTE IN THIS COURSE USE α=.05 UNLESS SPECIFIED OTHERWISE

11. Theory for β 1 Inference

12. Confidence Interval for β 1 b 1 ± t c s(b 1 ) where t c = t(1-α/2,n-2), the upper (1-α/2)100 percentile of the t distribution with n-2 degrees of freedom 1-α is the confidence level

13. Significance tests for β 1

14. Theory for β 0 Inference

15. Confidence Interval for β 0 b 0 ± t c s(b 0 ) where t c = t(1-α/2,n-2), the upper (1-α/2)100 percentile of the t distribution with n-2 degrees of freedom 1-α is the confidence level

16. Significance tests for β 0

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

18. Notes Usually the CI and significance test for β 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

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.

20. SAS Proc Reg proc reg data=a1; model lean=year/clb; run; clb option generates confidence intervals

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<.00018.6365610.00080 CIs given here….CI for intercept is uninteresting

22. Review What is the default value of α that we will use in this class? What is the default confidence level that we use in this class? Suppose you could choose the X’s. How would you choose them if you wanted a precise estimate of the slope? intercept? both?

23. Background Reading Chapter 2 –2.3 : Considerations Chapter 16 –16.10 : Planning sample sizes with power Appendix A.6