Statistics 350 Lecture 4

Today Last Day: Section 1.7 Today: 2.1-2.2 Homework #1: Chapter 1 Problems (page 33-38): 2, 5, 6, 7, 22, 26, 33, 34, 35 Due: January 19 Read Sections 1.1-1.3 and 1.6-1.7 Homework #2: Chapter 2 Problems (page 89-99): 2, 3, 4, 7, 51, 54 Due: January 26 Read Sections 2.1-2.5 Mid-Term Exam Dates: Mid-term #1 – Friday, February 9, 2007 Mid-term #2 – Friday, March 9, 2007

Simple Linear Regression Model: Yi=0+1Xi+i for i=1,2,…,n Recall model for errors: Chapter 1 introduce the model and also two methods for estimating the model parameters

Simple Linear Regression: Inference Typically would also like to make decisions about the model parameters These are done by conducting:

Simple Linear Regression: Inference Inference for the slope, 1: Tests for 1 = 0 are frequently done (and are usually the default null hypothesis for stat packages)…Why? Tests for 1 = b* for some specified b* are sometimes of interest (e.g. when X and Y are two different types of measurements of the same thing, then we might want to test to see if the slope is…….) A confidence interval for 1 can help to give an idea of the precision of the estimation

Simple Linear Regression: Inference To conduct hypothesis tests, need to know sampling distribution of the estimators This way can decide whether or not the observed parameter estimate is typical or atypical under the null hypothesis Under our model, the distribution of the errors are Under our model, the distribution of the Yi’s are

Simple Linear Regression: Inference Consider b1 Expected value: Variance:

Simple Linear Regression: Inference Distribution of b1:

Simple Linear Regression: Inference This is only partly useful since we still need to estimate 2

Simple Linear Regression: Inference Test statistic: Hypothesis Tests:

Simple Linear Regression: Inference Inference for 0 Sampling Distribution of b0

Simple Linear Regression: Inference Test statistic: Hypothesis Tests:

Comments When the sample size (n) is large enough, the distributions of b0 and b1 are both approximately normal by the , even when the ’s are not normally distributed This means the these estimators are robust against the normality assumption