statistics for business and economics chapter 10 simple linear regression
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![Page 1: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/1.jpg)
Statistics for Business and Economics
Chapter 10
Simple Linear Regression
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Learning Objectives
1. Describe the Linear Regression Model
2. State the Regression Modeling Steps
3. Explain Least Squares
4. Compute Regression Coefficients
5. Explain Correlation
6. Predict Response Variable
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Models
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Models
• Representation of some phenomenon
• Mathematical model is a mathematical expression of some phenomenon
• Often describe relationships between variables
• Types– Deterministic models– Probabilistic models
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Deterministic Models
• Hypothesize exact relationships
• Suitable when prediction error is negligible
• Example: force is exactly mass times acceleration
– F = m·a
© 1984-1994 T/Maker Co.
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Probabilistic Models
• Hypothesize two components– Deterministic– Random error
• Example: sales volume (y) is 10 times advertising spending (x) + random error
– y = 10x + – Random error may be due to factors
other than advertising
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Types of Probabilistic Models
ProbabilisticModels
Regression Models
Correlation Models
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Regression Models
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Types of Probabilistic Models
ProbabilisticModels
Regression Models
Correlation Models
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Regression Models
• Answers ‘What is the relationship between the variables?’
• Equation used– One numerical dependent (response) variable
What is to be predicted– One or more numerical or categorical
independent (explanatory) variables
• Used mainly for prediction and estimation
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Regression Modeling Steps
1. Hypothesize deterministic component
2. Estimate unknown model parameters
3. Specify probability distribution of random error term
• Estimate standard deviation of error
4. Evaluate model
5. Use model for prediction and estimation
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Model Specification
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Regression Modeling Steps
1. Hypothesize deterministic component
2. Estimate unknown model parameters
3. Specify probability distribution of random error term
• Estimate standard deviation of error
4. Evaluate model
5. Use model for prediction and estimation
![Page 14: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/14.jpg)
Specifying the Model
1. Define variables• Conceptual (e.g., Advertising, price)• Empirical (e.g., List price, regular price) • Measurement (e.g., $, Units)
2. Hypothesize nature of relationship• Expected effects (i.e., Coefficients’ signs)• Functional form (linear or non-linear)• Interactions
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Model Specification Is Based on Theory
• Theory of field (e.g., Sociology)
• Mathematical theory
• Previous research
• ‘Common sense’
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Advertising
Sales
Advertising
Sales
Advertising
Sales
Advertising
Sales
Thinking Challenge: Which Is More Logical?
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Simple
1 ExplanatoryVariable
Types of Regression Models
RegressionModels
2+ ExplanatoryVariables
Multiple
LinearNon-
LinearLinear
Non-Linear
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Linear Regression Model
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Types of Regression Models
Simple
1 ExplanatoryVariable
RegressionModels
2+ ExplanatoryVariables
Multiple
LinearNon-
LinearLinear
Non-Linear
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y x 0 1
Linear Regression Model
Relationship between variables is a linear function
Dependent (Response) Variable
Independent (Explanatory) Variable
Population Slope
Population y-intercept
Random Error
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y
E(y) = β0 + β1x (line of means)
β0 = y-intercept
x
Change in y
Change in x
β1 = Slope
Line of Means
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Population & Sample Regression Models
Population
$ $
$
$
$
Unknown Relationship
0 1y x
Random Sample
$$ $$
$$ $$
0 1ˆ ˆ ˆy x
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y
x
Population Linear Regression Model
0 1i i iy x
0 1E y x
Observedvalue
Observed value
i = Random error
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y
x
0 1ˆ ˆ ˆi i iy x
Sample Linear Regression Model
0 1ˆ ˆˆi iy x
Unsampled observation
i = Random
error
Observed value
^
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Estimating Parameters:Least Squares Method
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Regression Modeling Steps
1. Hypothesize deterministic component
2. Estimate unknown model parameters
3. Specify probability distribution of random error term
• Estimate standard deviation of error
4. Evaluate model
5. Use model for prediction and estimation
![Page 27: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/27.jpg)
Scattergram
1. Plot of all (xi, yi) pairs
2. Suggests how well model will fit
0204060
0 20 40 60
x
y
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0204060
0 20 40 60
x
y
Thinking Challenge
• How would you draw a line through the points?• How do you determine which line ‘fits best’?
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Least Squares
• ‘Best fit’ means difference between actual y values and predicted y values are a minimum
– But positive differences off-set negative
2 2
1 1
ˆ ˆn n
ii ii i
y y
• Least Squares minimizes the Sum of the Squared Differences (SSE)
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Least Squares Graphically
2
y
x
1
3
4
^^
^^
2 0 1 2 2ˆ ˆ ˆy x
0 1ˆ ˆˆi iy x
2 2 2 2 21 2 3 4
1
ˆ ˆ ˆ ˆ ˆLS minimizes n
ii
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Coefficient Equations
1 1
11 2
12
1
ˆ
n n
i ini i
i ixy i
nxx
ini
ii
x y
x ySS nSS
x
xn
Slope
y-intercept
0 1ˆ ˆy x Prediction Equation
0 1ˆ ˆy x
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Computation Table
xi2
x12
x2
2
: :
xn2
Σxi2
yi
y1
y2
:
yn
Σyi
xi
x1
x2
:
xn
Σxi
2
2
2
2
2
yi
y1
y2
:
yn
Σyi
xiyi
Σxiyi
x1y1
x2y2
xnyn
![Page 33: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/33.jpg)
Interpretation of Coefficients
^
^
^1. Slope (1)
• Estimated y changes by 1 for each 1unit increase
in x— If 1 = 2, then Sales (y) is expected to increase by 2
for each 1 unit increase in Advertising (x)
2. Y-Intercept (0)
• Average value of y when x = 0— If 0 = 4, then Average Sales (y) is expected to be
4 when Advertising (x) is 0
^
^
![Page 34: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/34.jpg)
Least Squares Example
You’re a marketing analyst for Hasbro Toys. You gather the following data:
Ad $ Sales (Units)1 12 13 24 25 4
Find the least squares line relatingsales and advertising.
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0
1
2
3
4
0 1 2 3 4 5
Scattergram Sales vs. Advertising
Sales
Advertising
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Parameter Estimation Solution Table
2 2
1 1 1 1 1
2 1 4 1 2
3 2 9 4 6
4 2 16 4 8
5 4 25 16 20
15 10 55 26 37
xi yi xi yi xiyi
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Parameter Estimation Solution
1 1
11 2 2
12
1
15 1037
5ˆ .7015
555
n n
i ini i
i ii
n
ini
ii
x y
x yn
x
xn
ˆ .1 .7y x
0 1ˆ ˆ 2 .70 3 .10y x
![Page 38: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/38.jpg)
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Param=0 Prob>|T|
INTERCEP 1 -0.1000 0.6350 -0.157 0.8849
ADVERT 1 0.7000 0.1914 3.656 0.0354
Parameter Estimation Computer Output
0^
1^
ˆ .1 .7y x
![Page 39: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/39.jpg)
Coefficient Interpretation Solution
1. Slope (1)
• Sales Volume (y) is expected to increase by .7 units for each $1 increase in Advertising (x)
^
2. Y-Intercept (0)
• Average value of Sales Volume (y) is -.10 units when Advertising (x) is 0— Difficult to explain to marketing manager
— Expect some sales without advertising
^
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0
1
2
3
4
0 1 2 3 4 5
Regression Line Fitted to the Data
Sales
Advertising
ˆ .1 .7y x
![Page 41: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/41.jpg)
Least Squares Thinking Challenge
You’re an economist for the county cooperative. You gather the following data:
Fertilizer (lb.) Yield (lb.) 4 3.0 6 5.510 6.512 9.0
Find the least squares line relatingcrop yield and fertilizer.
© 1984-1994 T/Maker Co.
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02468
10
0 5 10 15
Scattergram Crop Yield vs. Fertilizer*
Yield (lb.)
Fertilizer (lb.)
![Page 43: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/43.jpg)
Parameter Estimation Solution Table*
4 3.0 16 9.00 12
6 5.5 36 30.25 33
10 6.5 100 42.25 65
12 9.0 144 81.00 108
32 24.0 296 162.50 218
xi2yi xi yi xiyi
2
![Page 44: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/44.jpg)
Parameter Estimation Solution*
1 1
11 2 2
12
1
32 24218
4ˆ .6532
2964
n n
i ini i
i ii
n
ini
ii
x y
x yn
x
xn
0 1ˆ ˆ 6 .65 8 .80y x
ˆ .8 .65y x
![Page 45: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/45.jpg)
Coefficient Interpretation Solution*
2. Y-Intercept (0)
• Average Crop Yield (y) is expected to be 0.8 lb. when no Fertilizer (x) is used
^
^1. Slope (1)
• Crop Yield (y) is expected to increase by .65 lb. for each 1 lb. increase in Fertilizer (x)
![Page 46: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/46.jpg)
Regression Line Fitted to the Data*
02468
10
0 5 10 15
Yield (lb.)
Fertilizer (lb.)
ˆ .8 .65y x
![Page 47: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/47.jpg)
Probability Distribution of Random Error
![Page 48: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/48.jpg)
Regression Modeling Steps
1. Hypothesize deterministic component
2. Estimate unknown model parameters
3. Specify probability distribution of random error term
• Estimate standard deviation of error
4. Evaluate model
5. Use model for prediction and estimation
![Page 49: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/49.jpg)
Linear Regression Assumptions
1. Mean of probability distribution of error, ε, is 0
2. Probability distribution of error has constant variance
3. Probability distribution of error, ε, is normal
4. Errors are independent
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Error Probability Distribution
x1 x2 x3
yE(y) = β0 + β1x
x
![Page 51: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/51.jpg)
Random Error Variation
• Measured by standard error of regression model– Sample standard deviation of : s^
• Affects several factors– Parameter significance– Prediction accuracy
^• Variation of actual y from predicted y, y
![Page 52: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/52.jpg)
y
xxi
Variation Measures
0 1ˆ ˆˆi iy x
yi 2ˆ( )i iy y
Unexplained sum of squares
2( )iy yTotal sum of squares
2ˆ( )iy yExplained sum of squares
y
![Page 53: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/53.jpg)
Estimation of σ2
22 ˆ2 i i
SSEs where SSE y y
n
2
2
SSEs s
n
![Page 54: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/54.jpg)
Calculating SSE, s2, s Example
You’re a marketing analyst for Hasbro Toys. You gather the following data:
Ad $ Sales (Units)1 12 13 24 25 4
Find SSE, s2, and s.
![Page 55: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/55.jpg)
Calculating SSE Solution
1 1
2 1
3 2
4 2
5 4
xi yi
.6
1.3
2
2.7
3.4
ˆy y
.4
-.3
0
-.7
.6
2ˆ( )y yˆ .1 .7y x
.16
.09
0
.49
.36
SSE=1.1
![Page 56: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/56.jpg)
Calculating s2 and s Solution
2 1.1.36667
2 5 2
SSEs
n
.36667 .6055s
![Page 57: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/57.jpg)
Evaluating the Model
Testing for Significance
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Regression Modeling Steps
1. Hypothesize deterministic component
2. Estimate unknown model parameters
3. Specify probability distribution of random error term
• Estimate standard deviation of error
4. Evaluate model
5. Use model for prediction and estimation
![Page 59: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/59.jpg)
Test of Slope Coefficient
• Shows if there is a linear relationship between x and y
• Involves population slope 1
• Hypotheses – H0: 1 = 0 (No Linear Relationship)
– Ha: 1 0 (Linear Relationship)
• Theoretical basis is sampling distribution of slope
![Page 60: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/60.jpg)
y
Population Linex
Sample 1 Line
Sample 2 Line
Sampling Distribution of Sample Slopes
1
Sampling Distribution
11
11SS
^
^
All Possible Sample Slopes
Sample 1: 2.5
Sample 2: 1.6
Sample 3: 1.8
Sample 4: 2.1 : :
Very large number of sample slopes
![Page 61: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/61.jpg)
Slope Coefficient Test Statistic
1
1 1
ˆ
2
12
1
ˆ ˆ2
wherexx
n
ini
xx ii
t df nsS
SS
x
SS xn
![Page 62: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/62.jpg)
Test of Slope Coefficient Example
You’re a marketing analyst for Hasbro Toys.
You find β0 = –.1, β1 = .7 and s = .6055.Ad $ Sales (Units)
1 12 13 24 25 4
Is the relationship significant at the .05 level of significance?
^^
![Page 63: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/63.jpg)
Test of Slope Coefficient Solution
• H0:• Ha:• • df • Critical Value(s):
Test Statistic:
Decision:
Conclusion:
t0 3.182-3.182
.025
Reject H0 Reject H0
.025
1 = 0
1 0
.05
5 - 2 = 3
![Page 64: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/64.jpg)
Solution Table
xi yi xi2 yi
2 xiyi
1 1 1 1 1
2 1 4 1 2
3 2 9 4 6
4 2 16 4 8
5 4 25 16 20
15 10 55 26 37
![Page 65: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/65.jpg)
Test StatisticSolution
1
1
ˆ 2
1
ˆ
.6055.1914
1555
5
ˆ .703.657
.1914
xx
SS
SS
tS
![Page 66: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/66.jpg)
Test of Slope Coefficient Solution
• H0: 1 = 0
• Ha: 1 0
• .05• df 5 - 2 = 3• Critical Value(s):
Test Statistic:
Decision:
Conclusion:
1
1
ˆ
ˆ .703.657
.1914t
S
Reject at = .05
There is evidence of a relationshipt0 3.182-3.182
.025
Reject H0 Reject H0
.025
![Page 67: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/67.jpg)
Test of Slope CoefficientComputer Output
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Param=0 Prob>|T|
INTERCEP 1 -0.1000 0.6350 -0.157 0.8849
ADVERT 1 0.7000 0.1914 3.656 0.0354
t = 1 / S
P-Value
S11 1
^^^^
![Page 68: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/68.jpg)
Correlation Models
![Page 69: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/69.jpg)
Types of Probabilistic Models
ProbabilisticModels
Regression Models
Correlation Models
![Page 70: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/70.jpg)
Correlation Models
• Answers ‘How strong is the linear relationship between two variables?’
• Coefficient of correlation– Sample correlation coefficient denoted r– Values range from –1 to +1– Measures degree of association– Does not indicate cause–effect relationship
![Page 71: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/71.jpg)
Coefficient of Correlation
xy
xx yy
SSr
SS SS
2
2
2
2
xx
yy
xy
xSS x
n
ySS y
n
x ySS xy
n
where
![Page 72: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/72.jpg)
Coefficient of Correlation Values
–1.0 +1.00–.5 +.5
No Linear Correlation
Perfect Negative
Correlation
Perfect Positive
Correlation
Increasing degree of positive correlation
Increasing degree of negative correlation
![Page 73: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/73.jpg)
Coefficient of Correlation Example
You’re a marketing analyst for Hasbro Toys. Ad $ Sales (Units)
1 12 13 24 25 4
Calculate the coefficient ofcorrelation.
![Page 74: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/74.jpg)
Solution Table
xi yi xi2 yi
2 xiyi
1 1 1 1 1
2 1 4 1 2
3 2 9 4 6
4 2 16 4 8
5 4 25 16 20
15 10 55 26 37
![Page 75: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/75.jpg)
Coefficient of Correlation Solution
22
2
22
2
(15)55 10
5
(10)26 6
5
(15)(10)37 7
5
xx
yy
xy
xSS x
n
ySS y
n
x ySS xy
n
7.904
10 6xy
xx yy
SSr
SS SS
![Page 76: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/76.jpg)
Coefficient of Correlation Thinking Challenge
You’re an economist for the county cooperative. You gather the following data:
Fertilizer (lb.) Yield (lb.) 4 3.0 6 5.510 6.512 9.0
Find the coefficient of correlation.© 1984-1994 T/Maker Co.
![Page 77: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/77.jpg)
Solution Table*
4 3.0 16 9.00 12
6 5.5 36 30.25 33
10 6.5 100 42.25 65
12 9.0 144 81.00 108
32 24.0 296 162.50 218
xi2yi xi yi xiyi
2
![Page 78: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/78.jpg)
Coefficient of Correlation Solution*
22
2
22
2
(32)296 40
4
(24)162.5 18.5
4
(32)(24)218 26
4
xx
yy
xy
xSS x
n
ySS y
n
x ySS xy
n
26.956
40 18.5xy
xx yy
SSr
SS SS
![Page 79: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/79.jpg)
Proportion of variation ‘explained’ by relationship between x and y
Coefficient of Determination
2 Explained Variation
Total Variationyy
yy
SS SSEr
SS
0 r2 1
r2 = (coefficient of correlation)2
![Page 80: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/80.jpg)
Coefficient of Determination Example
You’re a marketing analyst for Hasbro Toys.
You know r = .904. Ad $ Sales (Units)
1 12 13 24 25 4
Calculate and interpret thecoefficient of determination.
![Page 81: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/81.jpg)
Coefficient of Determination Solution
r2 = (coefficient of correlation)2
r2 = (.904)2
r2 = .817
Interpretation: About 81.7% of the sample variation in Sales (y) can be explained by using Ad $ (x) to predict Sales (y) in the linear model.
![Page 82: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/82.jpg)
r2 Computer Output
Root MSE 0.60553 R-square 0.8167
Dep Mean 2.00000 Adj R-sq 0.7556
C.V. 30.27650
r2 adjusted for number of explanatory variables & sample size
r2
![Page 83: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/83.jpg)
Using the Model for Prediction & Estimation
![Page 84: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/84.jpg)
Regression Modeling Steps
1. Hypothesize deterministic component
2. Estimate unknown model parameters
3. Specify probability distribution of random error term
• Estimate standard deviation of error
4. Evaluate model
5. Use model for prediction and estimation
![Page 85: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/85.jpg)
Prediction With Regression Models
• Types of predictions– Point estimates– Interval estimates
• What is predicted– Population mean response E(y) for given x
Point on population regression line
– Individual response (yi) for given x
![Page 86: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/86.jpg)
What Is Predicted
Mean y, E(y)
y
^yIndividual
Prediction, y
E(y) = x
^x
xP
^ ^y i =
x
![Page 87: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/87.jpg)
Confidence Interval Estimate for Mean Value of y at x = xp
xx
p
SS
xx
nSty
2
2/
1ˆ
df = n – 2
![Page 88: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/88.jpg)
Factors Affecting Interval Width
1. Level of confidence (1 – )• Width increases as confidence increases
2. Data dispersion (s)• Width increases as variation increases
3. Sample size• Width decreases as sample size increases
4. Distance of xp from meanx• Width increases as distance increases
![Page 89: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/89.jpg)
Why Distance from Mean?
Sample 2 Line
y
xx1 x2
y
Sample 1 LineGreater dispersion than x1
x
![Page 90: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/90.jpg)
Confidence Interval Estimate Example
You’re a marketing analyst for Hasbro Toys.
You find β0 = -.1, β 1 = .7 and s = .6055.
Ad $ Sales (Units)1 12 13 24 25 4
Find a 95% confidence interval forthe mean sales when advertising is $4.
^^
![Page 91: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/91.jpg)
Solution Table
x i y i x i2 y i
2 x iy i
1 1 1 1 1
2 1 4 1 2
3 2 9 4 6
4 2 16 4 8
5 4 25 16 20
15 10 55 26 37
![Page 92: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/92.jpg)
Confidence Interval Estimate Solution
2
/ 2
2
1ˆ
ˆ .1 .7 4 2.7
4 312.7 3.182 .6055
5 10
1.645 ( ) 3.755
p
xx
x xy t s
n SS
y
E Y
x to be predicted
![Page 93: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/93.jpg)
Prediction Interval of Individual Value of y at x = xp
Note!
2
/ 2
1ˆ 1
p
xx
x xy t S
n SS
df = n – 2
![Page 94: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/94.jpg)
Why the Extra ‘S’?
Expected(Mean) y
yy we're trying topredict
Prediction, y
xxp
E(y) = x
^^
^y i =
x i
![Page 95: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/95.jpg)
Prediction Interval Example
You’re a marketing analyst for Hasbro Toys.
You find β0 = -.1, β 1 = .7 and s = .6055.
Ad $ Sales (Units)1 12 13 24 25 4
Predict the sales when advertising is $4. Use a 95% prediction interval.
^^
![Page 96: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/96.jpg)
Solution Table
x i y i x i2 y i
2 x iy i
1 1 1 1 1
2 1 4 1 2
3 2 9 4 6
4 2 16 4 8
5 4 25 16 20
15 10 55 26 37
![Page 97: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/97.jpg)
Prediction Interval Solution
2
/ 2
2
4
1ˆ 1
ˆ .1 .7 4 2.7
4 312.7 3.182 .6055 1
5 10
.503 4.897
p
xx
x xy t s
n SS
y
y
x to be predicted
![Page 98: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/98.jpg)
Dep Var Pred Std Err Low95% Upp95% Low95% Upp95%
Obs SALES Value Predict Mean Mean Predict Predict
1 1.000 0.600 0.469 -0.892 2.092 -1.837 3.037
2 1.000 1.300 0.332 0.244 2.355 -0.897 3.497
3 2.000 2.000 0.271 1.138 2.861 -0.111 4.111
4 2.000 2.700 0.332 1.644 3.755 0.502 4.897
5 4.000 3.400 0.469 1.907 4.892 0.962 5.837
Interval Estimate Computer Output
Predicted y when x = 4
Confidence Interval
SYPredictionInterval
![Page 99: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/99.jpg)
Confidence Intervals v. Prediction Intervals
x
y
x
^^ ^
y i =
x i
![Page 100: Statistics for Business and Economics Chapter 10 Simple Linear Regression](https://reader036.vdocuments.mx/reader036/viewer/2022081514/56649d445503460f94a2195b/html5/thumbnails/100.jpg)
Conclusion
1. Described the Linear Regression Model
2. Stated the Regression Modeling Steps
3. Explained Least Squares
4. Computed Regression Coefficients
5. Explained Correlation
6. Predicted Response Variable