Topic 4 Statistical Inference Outline Statistical inference confidence

Topic 4: Statistical Inference

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

Theory for Statistical Inference • Xi iid Normal(μ, σ2), parameters unknown

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

Confidence Intervals where tc= 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

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

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)

Significance tests

Significance tests • • • Under H 0 t* will have distribution t(n-1) P(reject H 0 | H 0 true) = a (Type I error) Under Ha, t* will have noncentral t(n-1) dists P(DNR H 0 | Ha true) = b (Type II error) Type II error related to the power of the test

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

Theory for β 1 Inference

Confidence Interval for β 1 b 1 ± tcs(b 1) where tc = 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

Significance tests for β 1

Theory for β 0 Inference

Confidence Interval for β 0 b 0 ± tcs(b 0) where tc = 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

Significance tests for β 0

Notes • The normality of b 0 and b 1 follows from the fact that each of these is a linear combination of the Yi, each of which is an independent normal • For b 1 see KNNL p 42 • For b 0 try this as an exercise

Notes • Usually the CI and significance test for β 0 is not of interest • If the ei are not normal but are relatively symmetric, then the CIs and significance tests are reasonable approximations

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.

SAS Proc Reg proc reg data=a 1; model lean=year/clb; run; clb option generates confidence intervals

Variable Intercept year Parameter Estimates Parameter Standard 95% Confidence DF Estimate Error t Value Pr > |t| Limits 1 -61. 12088 25. 12982 -2. 43 0. 0333 -116. 43124 -5. 81052 1 9. 31868 0. 30991 30. 07 <. 0001 8. 63656 10. 00080 CIs given here…. CI for intercept is uninteresting

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?

Background Reading • Chapter 2 – 2. 3 : Considerations • Chapter 16 – 16. 10 : Planning sample sizes with power • Appendix A. 6