Kxu Stat Anderson Ch12 Student

8/3/2019 Kxu Stat Anderson Ch12 Student

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Chapter 12Simple Linear Regression

Simple Linear Regression Model

Least Squares Method

Coefficient of Determination

Model Assumptions

Testing for SignificanceUsing the Estimated Regression Equation

for Estimation and Prediction

Computer Solution

Residual Analysis: Validating Model Assumptions


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The equation that describes how y is related to x and

an error term is called the regression model.

The simple linear regression model is:

y =b0 +b1x +e

b0 andb1 are called parameters of the model.

e is a random variable called the error term.

Simple Linear Regression Model


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n The simple linear regression equation is:

E(y) =b0 +b1x

Graph of the regression equation is a straight line.

b0 is the y intercept of the regression line.

b1 is the slope of the regression line.

E(y) is the expected value of y for a given x value.

Simple Linear Regression Equation


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n Positive Linear Relationship

E(y)

x

Slopeb1is positive

Regression line

Interceptb0


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n Negative Linear Relationship

E(y)

x

Slopeb1is negative

Regression lineIntercept

b0


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n No Relationship

E(y)

x

Slopeb1is 0

Regression line

Interceptb0


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n The estimated simple linear regression equation is:

The graph is called the estimated regression line.

b0 is the y intercept of the line.

b1 is the slope of the line.

is the estimated value of y for a given x value.

Estimated Simple Linear Regression Equation

0 1y b b x

y


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Estimation Process

Regression Modely =b0 +b1x +e

Regression EquationE(y) =b0 +b1x

Unknown Parametersb0,b1

Sample Data:

x yx1 y1. .

. .xn yn

EstimatedRegression Equation

Sample Statisticsb0, b1

b0

and b1

provide estimates ofb0 andb1

0 1y b b x


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Least Squares Criterion

where:

yi = observed value of the dependent variable

for the ith observation

yi = estimated value of the dependent variable

for the ith observation

Least Squares Method

min (y yi i )2

^


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Slope for the Estimated Regression Equation

bx y x y n

x x n

i i i i

i i1 2 2

( ) /

( ) /

The Least Squares Method


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n y-Intercept for the Estimated Regression Equation

where:

xi = value of independent variable for ith observationyi = value of dependent variable for ith observation

x = mean value for independent variable

y = mean value for dependent variable

n = total number of observations

_

_

The Least Squares Method

0 1b y b x


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Example: Reed Auto Sales

Simple Linear Regression

Reed Auto periodically has a special week-longsale. As part of the advertising campaign Reed runsone or more television commercials during theweekend preceding the sale. Data from a sample of 5

previous sales are shown on the next slide.


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n Simple Linear Regression

Number of TV Ads Number of Cars Sold1 143 24

2 181 173 27


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Slope for the Estimated Regression Equation

b1 = 220 - (10)(100)/5 = _____

24 - (10)2/5

y-Intercept for the Estimated Regression Equationb0 = 20 - 5(2) = _____

Estimated Regression Equation

y = 10 + 5x^



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Scatter Diagram

y = 10 + 5x

0

5

10

15

20

25

30

0 1 2 3 4

TV Ads

CarsSold

^


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Relationship Among SST, SSR, SSE

SST = SSR + SSE

where:

SST = total sum of squares

SSR = sum of squares due to regressionSSE = sum of squares due to error

The Coefficient of Determination

( ) ( ) ( )y y y y y yi i i i 2 2 2^^


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Coefficient of Determination

r2 = SSR/SST = 100/114 =

The regression relationship is very strong because

88% of the variation in number of cars sold can beexplained by the linear relationship between thenumber of TV ads and the number of cars sold.



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The Correlation Coefficient

Sample Correlation Coefficient

where:

b1 = the slope of the estimated regression

equation

2

1 )of(sign rbrxy

ionDeterminatoftCoefficien)of(sign 1brxy

xbby 10


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Sample Correlation Coefficient

The sign of b1 in the equation is +.

rxy = +.9366


2

1 )of(sign rbrxy

10 5y x

=+ .8772xyr


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Model Assumptions

Assumptions About the Error Term e1. The error e is a random variable with mean of

zero.

2. The variance of e, denoted by 2, is the same for

all values of the independent variable.3. The values of e are independent.

4. The error e is a normally distributed randomvariable.


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Testing for Significance

To test for a significant regression relationship, wemust conduct a hypothesis test to determine whetherthe value ofb1 is zero.

Two tests are commonly used

t Test F Test

Both tests require an estimate of 2, the variance of ein the regression model.


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An Estimate of 2

The mean square error (MSE) provides the estimate

of 2, and the notation s2 is also used.

s2

= MSE = SSE/(n-2)

where:


2

10

2 )()(SSE iiii xbbyyy


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An Estimate of To estimate we take the square root of 2.

The resulting s is called the standard error of theestimate.

2SSEMSE

ns


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Hypotheses

H0:b1 = 0

Ha:b1 = 0

Test Statistic

where

Testing for Significance: t Test

tb

sb 1

1

2)(

1

xx

ss

i

b


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t Test

Hypotheses

H0:b1 = 0

Ha:b1 = 0

Rejection RuleFor = .05 and d.f. = 3, t.025 = _____

Reject H0 if t > t.025 = _____



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n t Test

Test Statistics

t = _____/_____ = 4.63

Conclusions

t = 4.63 > 3.182, so reject H0



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Confidence Interval forb1

We can use a 95% confidence interval forb1 to test

the hypotheses just used in the t test.H0 is rejected if the hypothesized value of b1 is notincluded in the confidence interval for b1.


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The form of a confidence interval forb1 is:

where b1 is the point estimate

is the margin of erroris the t value providing an area

of /2 in the upper tail of a

t distribution with n - 2 degrees

of freedom

Confidence Interval forb1

12/1 bstb

12/ bst

2/t


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n Hypotheses

H0:b1 = 0

Ha:b1 = 0

n Test Statistic

F= MSR/MSE

Testing for Significance: F Test


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n Rejection Rule

Reject H0 if F> F

where: F is based on an Fdistribution

with 1 d.f. in the numerator and

n - 2 d.f. in the denominator

Testing for Significance: F Test


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n F Test

Hypotheses

H0:b1 = 0

Ha:b1 = 0

Rejection RuleFor = .05 and d.f. = 1, 3: F.05 = ______

Reject H0 if F > F.05 = ______.



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n F Test

Test Statistic

F= MSR/MSE = ____ / ______ = 21.43

Conclusion

F= 21.43 > 10.13, so we reject H0.


Some Cautions about the


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Some Cautions about theInterpretation of Significance Tests

Rejecting H0:b1 = 0 and concluding that therelationship between x and y is significant does notenable us to conclude that a cause-and-effectrelationship is present between x and y.

Just because we are able to reject H0:b

1 = 0 anddemonstrate statistical significance does not enable usto conclude that there is a linear relationship betweenx and y.

Using the Estimated Regression Equation


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n Confidence Interval Estimate of E(yp)

n Prediction Interval Estimate of yp

yp+ t/2 sind

where: confidence coefficient is 1 - and

t/2

is based on a t distribution

with n - 2 degrees of freedom

Using the Estimated Regression Equationfor Estimation and Prediction

/ y t sp yp 2


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Point Estimation

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

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



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n Confidence Interval for E(yp)

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

25 + 4.61 = ______ to _______ cars


l d l


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n Prediction Interval for yp

95% prediction interval estimate of the number ofcars sold in one particular week when 3 TV ads arerun is:

25 + 8.28 = _____ to ______ cars



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Residual for Observation i

yiyi

Standardized Residual for Observation i

where:

and

Residual Analysis

^

y ys

i i

y yi i

^

^

s s hy y ii i 1^

2

2

)(

)(1

xx

xx

nh

i

i

i


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Residuals

Observation Predicted Cars Sold Residuals

1 15 -1

2 25 -1

3 20 -2

4 15 2

5 25 2


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Residual Analysis

Residual Plot

x

y y

0

Good Pattern

Residual

R id l A l i


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Residual Analysis

n Residual Plot

x

y y

0

Nonconstant Variance

Residual

R id l A l i


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Residual Analysis

n Residual Plot

x

y y

0

Model Form Not Adequate

Residual


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End of Chapter 12

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