1. Statistics Lecture 12

2. Today Last Day: Exam Today: part of Chapter 4 and start Chapter 5
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3. Simultaneous Inference
Suppose you make a 95% confidence interval
Assuming the model and procedures are correct, what is the probability that the interval fails to cover the true parameter value…based on repeated sampled?

4. Simultaneous Inference
Now suppose you make 95% intervals for two different parameters, and suppose the statistics on which they are based are independent of each other
What is the probability that at least one interval fails to cover its parameter?

5. Simultaneous Inference
The more intervals you make, the greater the chance that at least one will fail to cover its intended parameter value
These probabilities can reach high levels with very large numbers of intervals
For computing k independent intervals,
P(at least one fails) =
For 10 intervals (95% CI’s), P(at least one fails)=.40.

6. Simultaneous Inference
You might want to make a coverage probability apply to a set of intervals simultaneously
That is, fix P(\at least one fails)=.05$ or whatever your  is
One way to do this is:
Methods for doing this are in Sections

7. Simultaneous Inference
The more intervals you make, the greater the chance that at least one will fail to cover its intended parameter value
These probabilities can reach high levels with very large numbers of intervals
For computing k independent intervals,
P(at least one fails) =
For 10 intervals (95% CI’s), P(at least one fails)=.40$.

8. Matrix Representations
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Will re-write the regression model in matrix notation and re-do estimation
This is particularly important as we move to multiple linear regression

9. Matrix Representations
Consider data arising from the multple linear regression model:

10. Matrix Representations
Can re-write the data and model in matrix notation

11. Matrix Representations
Can re-write the expected value of Y as:

12. Apartment Example
For these data:

13. Matrix Representations
Recall matrix multiplication:
Y`Y=
X`X=
X`Y=