Statistics 350 Lecture 12

Today Last Day: Exam Today: part of Chapter 4 and start Chapter 5 Read Sections 5.1-5.3

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?

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?

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.

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 4.1--4.3

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$.

Matrix Representations Read Sections 5.1-5.3 Will re-write the regression model in matrix notation and re-do estimation This is particularly important as we move to multiple linear regression

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

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

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

Apartment Example For these data:

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