1 Slide

Simple Linear RegressionEstimation and Residuals

Chapter 14BA 303 – Spring 2011

2 Slide

Point Estimation

0 1y b b x

If 3 TV ads are run prior to a sale, we expect the mean number of cars sold to be:

^y = 10 + 5(3) = 25 cars

3 Slide

/ y t sp yp 2

where:confidence coefficient is 1 -

andt/2 is based on a t distributionwith n - 2 degrees of freedom

Confidence Interval Estimate of E(yp)

The CI is an interval estimate of the mean value of y for a given value of x.

Confidence Interval of E(yp)

4 Slide

2

ˆ 2( )1

( )p

py

i

x xs s

n x x

Estimate of the Standard Deviation of py

Confidence Interval for E(yp)

2

ˆ 2 2 2 2 2(3 2)12.16025 5 (1 2) (3 2) (2 2) (1 2) (3 2)pys

ˆ1 12.16025 1.44915 4pys

5 Slide

The 95% confidence interval estimate of the mean number of cars sold when 3 TV ads are run is:

Confidence Interval for E(yp)

25 - 4.61

/ y t sp yp 2

25 + 3.182(1.4491)

20.39 to 29.61 cars

25 + 4.61

6 Slide

where:confidence coefficient is 1 -

andt/2 is based on a t distributionwith n - 2 degrees of freedom

Prediction Interval Estimate of yp

/ 2 indpy t s

The PI is an interval estimate of an individual value of y for a given value of x. The margin of error is larger than for a CI.

Prediction Interval

7 Slide

2

ind 2( )11 ( )

p

i

x xs s

n x x

Estimate of the Standard Deviation of an Individual Value of yp

ˆ1 12.16025 1 5 4pys

ˆ 2.16025(1.20416) 2.6013pys

Prediction Interval for yp

8 Slide

The 95% prediction interval estimate of the number of cars sold in one particular week when 3 TV ads are run is:

Prediction Interval for yp

25 - 8.28

25 + 3.1824(2.6013)

/ 2 indpy t s

16.72 to 33.28 cars

25 + 8.28

9 Slide

Comparison

16.72 to 33.28 carsPrediction Interval:

Confidence Interval: 20.39 to 29.61 cars

Point Estimate: 25

10 Slide

PRACTICEPREDICTION INTERVALS AND CONFIDENCE INTERVALS

11 Slide

Data

1 2.83 8.05 13.2

ttable 3.182 =0.05, /2=0.025d.f. = n – 2 = 3

s 2.033

3

10x

ix iy

2)( xxi

14 Slide

RESIDUAL ANALYSIS

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Residual Analysis

ˆi iy y Much of the residual analysis is based on an examination of graphical plots.

Residual for Observation i The residuals provide the best information about e .

If the assumptions about the error term e appear questionable, the hypothesis tests about the significance of the regression relationship and the interval estimation results may not be valid.

16 Slide

Residual Plot Against x

If the assumption that the variance of e is the same for all values of x is valid, and the assumed regression model is an adequate representation of the relationship between the variables, then

The residual plot should give an overall impression of a horizontal band of points

17 Slide

x

ˆy y

0

Good PatternRe

sidua

l

Residual Plot Against x

18 Slide

Residual Plot Against x

x

ˆy y

0

Resid

ual

Nonconstant Variance

19 Slide

Residual Plot Against x

x

ˆy y

0

Resid

ual

Model Form Not Adequate

20 Slide

Residuals

1 14 15 -13 24 25 -12 18 20 -21 17 15 23 27 25 2

ix iy

)ˆ( ii yy iyix iy

21 Slide

Residual Plot Against x

0 2 4

-3

-2

-1

0

1

2

3

22 Slide

Standardized Residual for Observation i

Standardized Residuals

ˆ

ˆi i

i i

y y

y ys

ˆ 1i i iy ys s h

2

2( )1( )i

ii

x xhn x x

where:

23 Slide

Standardized Residuals

1 1 0.2500 0.4500 1.60203 1 0.2500 0.4500 1.60202 0 0.0000 0.2000 1.93211 1 0.2500 0.4500 1.60203 1 0.2500 0.4500 1.6020

4

ix2)( xxi ih ii yys ˆ

2

2

)()(xxxx

i

i

s=2.1602x=2

24 Slide

Standardized Residuals

1 14 15 1.6020 -0.62423 24 25 1.6020 -0.62422 18 20 1.9321 -1.03511 17 15 1.6020 1.24843 27 25 1.6020 1.2484

ix iyiyii yy

ii

syy

ˆ

)ˆ(

ii yys ˆ

25 Slide

Standardized Residual Plot

The standardized residual plot can provide insight about the assumption that the error term e has a normal distribution.

If this assumption is satisfied, the distribution of the standardized residuals should appear to come from a standard normal probability distribution.

26 Slide

Standardized Residual Plot

0 2 4

-1.5000

-1.0000

-0.5000

0.0000

0.5000

1.0000

1.5000

27 Slide

Standardized Residual Plot

All of the standardized residuals are between –1.5 and +1.5 indicating that there is no reason to question the assumption that e has a normal distribution.

28 Slide

Outliers and Influential Observations

Detecting Outliers

• Minitab classifies an observation as an outlier if its standardized residual value is < -2 or > +2.• This standardized residual rule sometimes fails to identify an unusually large observation as being an outlier.

• This rule’s shortcoming can be circumvented by using studentized deleted residuals.• The |i th studentized deleted residual| will be larger than the |i th standardized residual|.

• An outlier is an observation that is unusual in comparison with the other data.

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PRACTICESTANDARDIZED RESIDUALS

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Standardized Residuals

1

2

3

4

5

ix2)( xxi ih ii yys ˆ

2

2

)()(xxxx

i

i

2)( xxi10

x3

2.0330s

32 Slide

COMPUTER SOLUTIONS

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Computer Solution

Performing the regression analysis computations without the help of a computer can be quite time consuming.

34 Slide

Our Solution – Calculations

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Our Solution – Calculations

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Basic MiniTab Output

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MiniTab Residuals, Prediction Intervals, and Confidence Intervals

38 Slide

Excel Output

39 Slide