1. Measures of Regression and Prediction Intervals
Section 9.3
Measures of Regression and Prediction Intervals

2. Section 9.3 Objectives
Interpret the three types of variation about a regression line
Find and interpret the coefficient of determination
Find and interpret the standard error of the estimate for a regression line
Construct and interpret a prediction interval for y

3. Variation About a Regression Line
Three types of variation about a regression line
Total variation
Explained variation
Unexplained variation
To find the total variation, you must first calculate
The total deviation
The explained deviation
The unexplained deviation

4. Variation About a Regression Line
Total Deviation =
Explained Deviation =
Unexplained Deviation = y
(xi, yi)
Unexplained deviation
Total deviation
Explained deviation
(xi, ŷi)
(xi, yi) x

5. Variation About a Regression Line
Total variation
The sum of the squares of the differences between the y-value of each ordered pair and the mean of y.
Explained variation
The sum of the squares of the differences between each predicted y-value and the mean of y.
Total variation =
Explained variation =

6. Variation About a Regression Line
Unexplained variation
The sum of the squares of the differences between the y-value of each ordered pair and each corresponding predicted y-value.
Unexplained variation =
The sum of the explained and unexplained variation is equal to the total variation.
Total variation = Explained variation + Unexplained variation

7. Coefficient of Determination
The ratio of the explained variation to the total variation.
Denoted by r2

8. Example: Coefficient of Determination
The correlation coefficient for the advertising expenses and company sales data as calculated in Section 9.1 is r ≈ Find the coefficient of determination. What does this tell you about the explained variation of the data about the regression line? About the unexplained variation?
Solution:
About 83.4% of the variation in the company sales can be explained by the variation in the advertising expenditures. About 16.9% of the variation is unexplained.

9. The Standard Error of Estimate
The standard deviation of the observed yi -values about the predicted ŷ-value for a given xi -value.
Denoted by se.
The closer the observed y-values are to the predicted y- values, the smaller the standard error of estimate will be.
n is the number of ordered pairs in the data set

10. The Standard Error of Estimate
In Words In Symbols
Make a table that includes the column heading shown.
Use the regression equation to calculate the predicted y-values.
Calculate the sum of the squares of the differences between each observed y-value and the corresponding predicted y-value.
Find the standard error of estimate.

11. Example: Standard Error of Estimate
The regression equation for the advertising expenses and company sales data as calculated in section 9.2 is ŷ = x Find the standard error of estimate.
Solution:
Use a table to calculate the sum of the squared differences of each observed y-value and the corresponding predicted y-value.

12. Solution: Standard Error of Estimatex y
ŷ i
(yi – ŷ i)2
2.4
225
225.81
(225 – )2 =
1.6
184
185.23
(184 – )2 =
2.0
220
205.52
(220 – )2 =
2.6
240
235.96
(240 – )2 =
1.4
180
175.08
(180 – )2 =
186
(186 – )2 =
2.2
215
215.66
(215 – )2 =
Σ =
unexplained variation

13. Solution: Standard Error of Estimate
n = 8, Σ(yi – ŷ i)2 =
The standard error of estimate of the company sales for a specific advertising expense is about $10.29.

14. Prediction Intervals
Two variables have a bivariate normal distribution if for any fixed value of x, the corresponding values of y are normally distributed and for any fixed values of y, the corresponding x-values are normally distributed.

15. Prediction Intervals
A prediction interval can be constructed for the true value of y.
Given a linear regression equation ŷ = mx + b and x0, a specific value of x, a c-prediction interval for y is
ŷ – E < y < ŷ + E where
The point estimate is ŷ and the margin of error is E. The probability that the prediction interval contains y is c.

16. Constructing a Prediction Interval for y for a Specific Value of x
In Words In Symbols
Identify the number of ordered pairs in the data set n and the degrees of freedom.
Use the regression equation and the given x-value to find the point estimate ŷ.
Find the critical value tc that corresponds to the given level of confidence c.
d.f. = n – 2
Use Table 5 in Appendix B.

17. Constructing a Prediction Interval for y for a Specific Value of x
In Words In Symbols
Find the standard error of estimate se.
Find the margin of error E.
Find the left and right endpoints and form the prediction interval.
Left endpoint: ŷ – E Right endpoint: ŷ + E
Interval: ŷ – E < y < ŷ + E

18. Example: Constructing a Prediction Interval
Construct a 95% prediction interval for the company sales when the advertising expenses are $2100. What can you conclude? Recall, n = 8, ŷ = x , se =
Solution:
Point estimate:
ŷ = (2.1) ≈
Critical value:
d.f. = n –2 = 8 – 2 = tc = 2.447

19. Solution: Constructing a Prediction Interval
Left Endpoint: ŷ – E
Right Endpoint: ŷ + E – ≈ ≈
< y <
You can be 95% confident that when advertising expenses are $2100, the company sales will be between $183,735 and $237,449.

20. Section 9.3 Summary
Interpreted the three types of variation about a regression line
Found and interpreted the coefficient of determination
Found and interpreted the standard error of the estimate for a regression line
Constructed and interpreted a prediction interval for y