1. Confidence and Prediction Intervals
Lesson MR - A
Confidence and Prediction Intervals

2. Objectives Construct confidence intervals for a mean response
Construct prediction intervals for an individual response

3. Vocabulary
Confidence intervals for a mean response – intervals constructed about the predicted value of y, at a given level of x, that are used to measure the accuracy of the mean response of all individuals in the population
Prediction intervals for an individual response – intervals constructed about the predicted value of y that are used to measure the accuracy of a single individual’s predicted value

4. Key Concept
Confidence intervals about an individual response will have more variability than mean responses

5. Example 1 Assume that we have a linear model with
The response variable y = the change in cholesterol level
The explanatory variable x = the amount of medication given, in mg
Assume that our least squares regression line is
The intercept 20 can be interpreted as a placebo effect
y = 6x + 20

6. Prediction Intervals
For a dose of x = 15 mg, the predicted value of the response is
This can be interpreted in two ways
We predict that the mean change, for all patients receiving 15 mg, is 110
We predict that the mean change, for one specific patient receiving 15 mg, is 110
y = 6x + 20 = 6(15) + 20 = 110

7. Interpretation 1 Confidence Interval for a Mean Response
The mean change, for all patients receiving 15 mg, is 110
What is the margin of error for this prediction?
What is a confidence interval for this prediction?
An answer to this question could be
“A 95% confidence interval for the mean change is (102, 118)”

8. Interpretation 2 Prediction Interval for an Individual Response
The mean change, for a specific individual receiving 15 mg, is 110
What is the margin of error for this prediction?
What is a confidence interval for this prediction?
An answer to this question could be
“A 95% confidence interval for this individual’s change is (92, 128)”

9. Prediction Interval Types
These are two types of intervals
Confidence intervals for a mean response
Prediction intervals for an individual response
The main difference is
Confidence intervals address the precision of a mean response
Prediction intervals address the precision of an individual’s response

10. Confidence Intervals for the Mean Response of y, y^
(x* - x)2
Lower bound = y – tα/2 · se
n (xi – x)2 Σ
(x* - x)2
Upper bound = y + tα/2 · se
n (xi – x)2 Σ
note: x* is the given value of the explanatory variable, n is the number of observations, and tα/2 critical value with degrees of freedom = n – 2

11. Confidence Intervals for an Individual Response of y^
(x* - x)2
Lower bound = y – tα/2 · se
n (xi – x)2 Σ
(x* - x)2
Upper bound = y + tα/2 · se
n (xi – x)2 Σ
note: x* is the given value of the explanatory variable, n is the number of observations, and tα/2 critical value with degrees of freedom = n – 2

12. Example
The computation of these prediction intervals is quite involved
It is strongly suggested to use technology to compute these prediction intervals

13. Summary and Homework Summary Homework
In a regression analysis, we can compute the sampling distribution of the sample slope
Using this, we can test the significance of the slope of the least squares regression line using a test statistic that has a t-distribution
The null hypotheses can be structured in the usual three ways (two-tailed, left-tailed, and right-tailed)
Confidence intervals can be computed in the usual ways
Homework
pg 757 – 758: 1, 2, 3, 7, 12