Predicted price y i 31785 1000s confidence interval

Predicted Price Y
i
= 317.85 ($1,000s)

Confidence Interval Estimate for E(y)|x
p
37.12 317.85
) x (x
) x (x
n
1
s t y
ˆ
2
2
p
ε α/2
 


 

The confidence interval endpoints are 280.66 -- 354.90,
or from $280,660 -- $354,900

Chap 14-70
Estimation of Individual Values:
Example
Find the 95% confidence interval for an individual
house with 2,000 square feet
Predicted Price Y
i
= 317.85 ($1,000s)

Prediction Interval Estimate for y|x
p
102.28 317.85
) x (x
) x (x
n
1
1 s t y
ˆ
2
2
p
ε α/2
 


  

The prediction interval endpoints are 215.50 -- 420.07,
or from $215,500 -- $420,070

Chap 14-71
Finding Confidence and Prediction
Intervals PHStat

In Excel, use
PHStat | regression | simple linear regression …

Check the
“confidence and prediction interval for X=”
box and enter the x-value and confidence level
desired

Chap 14-72
Input values
Finding Confidence and Prediction
Intervals PHStat
(continued)
Confidence Interval Estimate for E(y)|x
p
Prediction Interval Estimate for y|x
p

Chap 14-73
Residual Analysis

Purposes

Examine for linearity assumption

Examine for constant variance for all
levels of x

Evaluate normal distribution assumption

Graphical Analysis of Residuals

Can plot residuals vs. x

Can create histogram of residuals to
check for normality

Chap 14-74
Residual Analysis for Linearity
Not Linear
Linear

x
residuals
x
y
x
y
x
residuals

Chap 14-75
Residual Analysis for
Constant Variance
Non-constant variance

Constant variance
x x
y
x
x
y
residuals residuals

Chap 14-76
House Price Model Residual Plot
-60
-40
-20
0
20
40
60
80
0 1000 2000 3000
Square Feet
Residuals
Excel Output
RESIDUAL OUTPUT
Predicted
House Price Residuals
1 251.92316 -6.923162
2 273.87671 38.12329
3 284.85348 -5.853484
4 304.06284 3.937162
5 218.99284 -19.99284
6 268.38832 -49.38832
7 356.20251 48.79749
8 367.17929 -43.17929
9 254.6674 64.33264
10 284.85348 -29.85348

Chap 14-77
Leverage

The leverage of a case measures its ability to
move the regression model all by itself by just
moving in the y direction.

For example, one high leverage case hid a
strong relationship between the number of long
guns found and the number of times false
information was discovered in airport screening.

Chap 14-78
Leverage

Chap 14-79
Leverage

Chap 14-80
Leverage (cont.)

Notice how the point marked with the x
changes the regression slope quite a bit:

Chap 14-81
Leverage (cont.)

The leverage of a particular case can take on
any value between 0.0 and 1.0.

The closer the leverage of a case is to 1.0, the
more impact that case has on the regression
line.

Note that a point with zero leverage has no
effect on the regression model other than being
used in calculations for the model.

Chap 14-82
Leverage (cont.)

A case can have leverage in two ways:

It might be extraordinary in one or more individual
variables.

It might be extraordinary in a combination of
variables.

There are no tests to decide if the leverage of a
case is too large.

You can, however, make a histogram of the
leverages and give special attention to those cases
with leverage values that stand out.

Chap 14-83
Influential Points

Chap 14-84
Influential Points

Chap 14-85
Influential Cases

An influential case is one that has both high leverage