1. Statistical Methods Lecture 29

2. Standard Errors for Mean Values
QTM1310/ Sharpe
Standard Errors for Mean Values
Confidence Interval for the Mean Response
Last time, we said we were modeling our line to infer the “line of means”—the expected value of our response variable for each given value of the explanatory variable.
The confidence interval for the mean response, mv, at a value xv, is:
where:
SE, and the confidence interval, becomes smaller with increasing n.
SE, and the confidence interval, are larger for samples with more spread around the line (when se is larger). 2

3. Standard Errors for Mean Values
QTM1310/ Sharpe
Standard Errors for Mean Values
Confidence Interval for the Mean Response
The confidence interval for the mean response, mv, at a value xv, is:
where:
SE becomes larger the further xν gets from . That is, the confidence interval broadens as you move away from . (See figure at right.)
point out here that there are many points outside the intervals, because we aren’t trying to predict individual values, we’re doing inference about the mean(s) at the x’s. 3

4. Standard Errors for Predicted Values
QTM1310/ Sharpe
Standard Errors for Predicted Values
Prediction Interval for an Individual Response
Now, we tackle the more difficult (as far as additional variability) of predicting a single value at a value xv. When conditions are met, that interval is:
where:
Because of the extra term , the confidence interval for individual values is broader that those for the predicted mean value. 4

5. Difference Between Confidence and Prediction Intervals
QTM1310/ Sharpe
Difference Between Confidence and Prediction Intervals
Confidence interval for a mean:
The result at 95% means:
“We are 95% confident that the mean value of y is between 4.40 and 4.70 when x = 10.1.”
Emphasize: made up numbers! Just looking at interpretation for now! 5

6. Difference Between Confidence and Prediction Intervals
QTM1310/ Sharpe
Difference Between Confidence and Prediction Intervals
Prediction interval for an individual value:
The result at 95% means:
“We are 95% confident that a single
measurement of y will be between 3.95
and 5.15 when x = 10.1.” 6

7. Using Confidence and Prediction Intervals
QTM1310/ Sharpe
Using Confidence and Prediction Intervals
Example : External Hard Disks
A study of external disk drives reveals a linear relationship between the Capacity (in GB) and the Price (in $). Regression resulted in the following:
Price = Capacity
se = 17.95, and SE(b1) =
Find the predicted Price of a 1000 GB hard drive.
Find the 95% confidence interval for the mean Price of all 1000 GB hard drives.
Find the 95% prediction interval for the Price of one 1000 GB hard drive. 7

8. Using Confidence and Prediction Intervals
QTM1310/ Sharpe
Using Confidence and Prediction Intervals
Example : External Hard Disks
A study of external disk drives reveals a linear relationship between the Capacity (in GB) and the Price (in $). Regression resulted in the following:
Price = Capacity
se = 17.95, and SE(b1) =
Find the predicted Price of a 1000 GB hard drive.
Price = (1000) = 8

9. Using Confidence and Prediction Intervals
QTM1310/ Sharpe
Using Confidence and Prediction Intervals
Example : External Hard Disks
A study of external disk drives reveals a linear relationship between the Capacity (in GB) and the Price (in $). Regression resulted in the following:
Price = Capacity
se = 17.95, and SE(b1) =
Find the 95% confidence interval for the mean Price of all 1000 GB hard drives. 9

10. Using Confidence and Prediction Intervals
QTM1310/ Sharpe
Using Confidence and Prediction Intervals
Example : External Hard Disks
A study of external disk drives reveals a linear relationship between the Capacity (in GB) and the Price (in $). Regression resulted in the following:
Price = Capacity
se = 17.95, and SE(b1) =
Find the 95% prediction interval for the price of one 1000 GB hard drive. 10

11. Looking back
Construct and interpret a confidence interval for the mean value
Construct and interpret a prediction interval for an individual value
Assumptions: quantitative data from incr. normal popn as sample gets smaller, r. sample
Particular form of the test statistic
Two-tailed test: effect on the p-value (have to double tail prob) or rejection region (have to take given alpha & divide it between two tails)