1. Statistics Lecture 3

2. Today Last Day: Section 1.6 Today: Example and Section 1.7
Homework #1:
Chapter 1 Problems (page 33-38): 2, 5, 6, 7, 22, 26, 33, 34, 35
Due: January 19
Read Sections and 1.6

3. Simple Linear Regression
Last day, went through the estimation details for the simple linear regression model
Today, will work through and example, identifying the terms introduced last day
Also, will look at an alternative approach for estimating the model parameters

4. Apartment Example (Again)
So you want to find a new apartment
You look up the apartment listings in the classified ads and two pieces of information (among others) are provided
The size of the apartment in square feet
The monthly rent

5. Apartment Example
Data:

6. Apartment Example

7. Apartment Example
Does a straight line appear to provide a reasonable approximation of the relationship between rent and apartment size?
Unlikely to fit a straight line model if the data did not display a linear pattern

8. Apartment Example
Estimating the slope 1:

9. Apartment Example
Estimating the intercept 0:

10. Apartment Example

11. Apartment Example When X=0, the predicted rent is:
So, this means that to rent an apartment with 0 square feet it costs:
Note: not all values of X make physical sense. Prediction at these points may be meaningless

12. Apartment Example
Residuals
MSE:

13. Apartment Example
Predict the average monthly rent for a 650 square foot apartment
Does this mean that the next 650 square foot apartment will have this rent?

14. Estimation Using Maximum Likelihood
View this as choosing estimates of the parameters that are most consistent with the data
For a given Xi, the distribution of Yi is:
The pdf is:

15. Estimation Using Maximum Likelihood
The observations ore independent, so the joint distribution of the reponses is:
The pdf is:

16. Estimation Using Maximum Likelihood
The likelihood function is:
Want to find the value of the parameters that maximize the likelihood function

17. Estimation Using Maximum Likelihood
Partial derivatives:

18. Estimation Using Maximum Likelihood
MLE’s:

19. Estimation Using Maximum Likelihood
Summary