Lesson 14 - 2

1. Lesson 14 - 2 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 intervalfor 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 intervalfor 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. 1 (x* - x)2 Lower bound = y– tα/2· se --- + ------------- n (xi – x)2 1 (x* - x)2 Upper bound = y+ tα/2· se --- + ------------- n (xi – x)2 Σ Σ Confidence Intervals for the Mean Response of y, y ^ note:x* is the given value of the explanatory variable,n is the number of observations, andtα/2 critical value with degrees of freedom = n – 2

11. 1 (x* - x)2 Lower bound = y– tα/2· se 1 + --- + ------------- n (xi – x)2 1 (x* - x)2 Upper bound = y+ tα/2· se 1 + --- + ------------- n (xi – x)2 Σ Σ Confidence Intervals for an Individual Response of y ^ note:x* is the given value of the explanatory variable,n is the number of observations, andtα/2 critical value with degrees of freedom = n – 2

12. Example

13. Summary and Homework • Summary • 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