kxu stat anderson ch12 student
TRANSCRIPT
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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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Simple Linear Regression Equation
n Positive Linear Relationship
E(y)
x
Slopeb1is positive
Regression line
Interceptb0
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Simple Linear Regression Equation
n Negative Linear Relationship
E(y)
x
Slopeb1is negative
Regression lineIntercept
b0
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Simple Linear Regression Equation
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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Example: Reed Auto Sales
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^
Example: Reed Auto Sales
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Example: Reed Auto Sales
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.
Example: Reed Auto Sales
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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
Example: Reed Auto Sales
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:
Testing for Significance
2
10
2 )()(SSE iiii xbbyyy
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Testing for Significance
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 = _____
Example: Reed Auto Sales
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n t Test
Test Statistics
t = _____/_____ = 4.63
Conclusions
t = 4.63 > 3.182, so reject H0
Example: Reed Auto Sales
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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 = ______.
Example: Reed Auto Sales
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n F Test
Test Statistic
F= MSR/MSE = ____ / ______ = 21.43
Conclusion
F= 21.43 > 10.13, so we reject H0.
Example: Reed Auto Sales
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^
Example: Reed Auto Sales
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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
Example: Reed Auto Sales
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
Example: Reed Auto Sales
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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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Example: Reed Auto Sales
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