STA291Statistical Methods

Lecture 29

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).

Standard Errors for Mean Values

n

sxxbSESE e

vv

22

12ˆ

vnv SEty ̂ˆ *2

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:

Standard Errors for Mean Values

n

sxxbSESE e

vv

22

12ˆ

vnv SEty ̂ˆ *2

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.)

x

x

Standard Errors for Predicted Values

Because of the extra term , the confidence interval for individual values is broader that those for the predicted mean value.

2es

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:

Difference Between Confidence and Prediction IntervalsConfidence 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.”

ˆ 10.1 4.55 0.15

n

sxxbSEty e

vnv

22

12*

2ˆ

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.”

ˆ 10.1 4.55 0.60y

Difference Between Confidence and Prediction Intervals

22

21

2*2ˆ e

evnv s

n

sxxbSEty

Using Confidence and Prediction IntervalsExample : 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 = 18.64 + 0.104Capacity

se = 17.95, and SE(b1) = 0.0051

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.

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 = 18.64 + 0.104Capacity

se = 17.95, and SE(b1) = 0.0051

Find the predicted Price of a 1000 GB hard drive.

Using Confidence and Prediction Intervals

Price = 18.64 + 0.104(1000) = 122.64

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 = 18.64 + 0.104Capacity

se = 17.95, and SE(b1) = 0.0051

Find the 95% confidence interval for the mean Price of all 1000 GB hard drives.

Using Confidence and Prediction Intervals

14.140,$14.105$50.17$64.122$7

95.17111010000051.0571.264.122$

ˆ

222

22

12*

2

n

sxxbSEty e

vnv

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 = 18.64 + 0.104Capacity

se = 17.95, and SE(b1) = 0.0051

Find the 95% prediction interval for the price of one 1000 GB hard drive.

Using Confidence and Prediction Intervals

Looking back

oConstruct and interpret a confidence interval for the mean valueoConstruct and interpret a prediction interval for an individual value