1. Statistics Lecture 4

2. 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 and
Homework #2:
Chapter 2 Problems (page 89-99): 2, 3, 4, 7, 51, 54
Due: January 26
Read Sections
Mid-Term Exam Dates:
Mid-term #1 – Friday, February 9, 2007
Mid-term #2 – Friday, March 9, 2007

3. 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

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

5. 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

6. 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

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

8. Simple Linear Regression: Inference
Distribution of b1:

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

10. Simple Linear Regression: Inference
Test statistic:
Hypothesis Tests:

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

12. Simple Linear Regression: Inference
Test statistic:
Hypothesis Tests:

13. 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