1. Statistics for Business and Economics
Chapter 10
Simple Linear Regression

2. Learning Objectives Describe the Linear Regression Model
State the Regression Modeling Steps
Explain Least Squares
Compute Regression Coefficients
Explain Correlation
Predict Response Variable
As a result of this class, you will be able to...

3. Models

4. Models Representation of some phenomenon
Mathematical model is a mathematical expression of some phenomenon
Often describe relationships between variables
Types
Deterministic models
Probabilistic models .

5. Deterministic Models Hypothesize exact relationships
Suitable when prediction error is negligible
Example: force is exactly mass times acceleration
F = m·a
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6. 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

7. Types of Probabilistic Models
Regression Models
Correlation Models 7

8. Regression Models

9. Types of Probabilistic Models
Regression Models
Correlation Models 7

10. 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

11. Regression Modeling Steps
Hypothesize deterministic component
Estimate unknown model parameters
Specify probability distribution of random error term
Estimate standard deviation of error
Evaluate model
Use model for prediction and estimation

12. Model Specification

13. Regression Modeling Steps
Hypothesize deterministic component
Estimate unknown model parameters
Specify probability distribution of random error term
Estimate standard deviation of error
Evaluate model
Use model for prediction and estimation

14. Specifying the Model Define variables
Conceptual (e.g., Advertising, price)
Empirical (e.g., List price, regular price)
Measurement (e.g., $, Units)
Hypothesize nature of relationship
Expected effects (i.e., Coefficients’ signs)
Functional form (linear or non-linear)
Interactions

15. Model Specification Is Based on Theory
Theory of field (e.g., Sociology)
Mathematical theory
Previous research
‘Common sense’

16. Thinking Challenge: Which Is More Logical?
Sales
Sales
With positive linear relationship, sales increases infinitely.
Discuss concept of ‘relevant range’.
Advertising
Advertising
Sales
Sales
Advertising
Advertising 17

17. Types of Regression Models
Simple
1 Explanatory
Variable
2+ Explanatory
Variables
Multiple
This teleology is based on the number of explanatory variables & nature of relationship between X & Y.
Linear
Non-
Linear
Linear
Non-
Linear 19

18. Linear Regression Model

19. Types of Regression Models
Simple
1 Explanatory
Variable
Regression Models
2+ Explanatory
Variables
Multiple
Linear
Non-
This teleology is based on the number of explanatory variables & nature of relationship between X & Y. 27

20. Linear Regression Model
Relationship between variables is a linear function
Population y-intercept
Population Slope
Random Error y     x   1
Dependent (Response) Variable
Independent (Explanatory) Variable

21. Line of Means y x E(y) = β0 + β1x (line of means) Change in y
β1 = Slope
Change in x
β0 = y-intercept x 28

22. Population & Sample Regression Models
Random Sample
 $
Unknown Relationship
 $
 $
 $
 $
 $ 31

23. Population Linear Regression Modely
Observed value
i = Random error x
Observed value 35

24. Sample Linear Regression Modely
i = Random error ^
Unsampled observation x
Observed value 36

25. Estimating Parameters: Least Squares Method

26. Regression Modeling Steps
Hypothesize deterministic component
Estimate unknown model parameters
Specify probability distribution of random error term
Estimate standard deviation of error
Evaluate model
Use model for prediction and estimation

27. Scattergram Plot of all (xi, yi) pairs
Suggests how well model will fit 20 40 60 x y

28. Thinking Challenge How would you draw a line through the points?
How do you determine which line ‘fits best’? 20 40 60 x y 42

29. Least Squares
‘Best fit’ means difference between actual y values and predicted y values are a minimum
But positive differences off-set negative
Least Squares minimizes the Sum of the Squared Differences (SSE) 49

30. Least Squares Graphically^ e ^ 4 2 ^ e e ^ 1 3 x 52

31. Coefficient Equations
Prediction Equation
Slope
y-intercept 53

32. Computation Table xi yi xi yi xiyi x1 y1 x1 y1 x1y1 x2 y2 x2 y2 x2y2 :xn yn xn 2 yn
xnyn 2 2
Σxi
Σyi
Σxi
Σyi
Σxiyi 54

33. Interpretation of Coefficients^
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)
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 ^

34. Least Squares Example
You’re a marketing analyst for Hasbro Toys. You gather the following data:
Ad $ Sales (Units)
Find the least squares line relating sales and advertising.

35. Scattergram Sales vs. Advertising4 3 2 1 1 2 3 4 5
Advertising 57

36. Parameter Estimation Solution Tablexi yi xi 2 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 58

37. Parameter Estimation Solution59

38. Parameter Estimation Computer Output
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Param=0 Prob>|T|
INTERCEP
ADVERT ^ 0 ^ 1

39. Coefficient Interpretation Solution
Slope (1)
Sales Volume (y) is expected to increase by .7 units for each $1 increase in Advertising (x) ^
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 ^

40. Regression Line Fitted to the Data
Sales 4 3 2 1 1 2 3 4 5
Advertising 57

41. Least Squares Thinking Challenge
You’re an economist for the county cooperative. You gather the following data:
Fertilizer (lb.) Yield (lb.)
Find the least squares line relating crop yield and fertilizer.
© T/Maker Co. 62

42. Scattergram Crop Yield vs. Fertilizer*
Yield (lb.) 10 8 6 4 2 5 10 15
Fertilizer (lb.) 65

43. Parameter Estimation Solution Table*2 2 xi yi xi yi
xiyi 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 66

44. Parameter Estimation Solution*67

45. Coefficient Interpretation Solution*^
Slope (1)
Crop Yield (y) is expected to increase by .65 lb. for each 1 lb. increase in Fertilizer (x)
Y-Intercept (0)
Average Crop Yield (y) is expected to be 0.8 lb. when no Fertilizer (x) is used ^

46. Regression Line Fitted to the Data*
Yield (lb.) 10 8 6 4 2 5 10 15
Fertilizer (lb.) 65

47. Probability Distribution of Random Error

48. Regression Modeling Steps
Hypothesize deterministic component
Estimate unknown model parameters
Specify probability distribution of random error term
Estimate standard deviation of error
Evaluate model
Use model for prediction and estimation

49. Linear Regression Assumptions
Mean of probability distribution of error, ε, is 0
Probability distribution of error has constant variance
Probability distribution of error, ε, is normal
Errors are independent

50. Error Probability Distribution
E(y) = β0 + β1x x x1 x2 x3 91

51. Random Error Variation^
Variation of actual y from predicted y, y
Measured by standard error of regression model
Sample standard deviation of  : s ^
Affects several factors
Parameter significance
Prediction accuracy

52. Variation Measures y x xi yi y Unexplained sum of squares
Total sum of squares
Explained sum of squares y x xi 78

53. Estimation of σ2

54. Calculating SSE, s2, s Example
You’re a marketing analyst for Hasbro Toys. You gather the following data:
Ad $ Sales (Units)
Find SSE, s2, and s.

55. Calculating SSE Solutionxi yi 1 1 .6 .4
.16 2 1
1.3
-.3
.09 3 2 2 4 2
2.7
-.7
.49 5 4
3.4 .6
.36
SSE=1.1

56. Calculating s2 and s Solution

57. Testing for Significance
Evaluating the Model
Testing for Significance

58. Regression Modeling Steps
Hypothesize deterministic component
Estimate unknown model parameters
Specify probability distribution of random error term
Estimate standard deviation of error
Evaluate model
Use model for prediction and estimation

59. 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

60. Sampling Distribution of Sample Slopesy
Sample 1 Line
All Possible Sample Slopes
Sample 1: 2.5
Sample 2: 1.6
Sample 3: 1.8
Sample 4: : : Very large number of sample slopes
Sample 2 Line
Population Line x b 1
Sampling Distribution 1 S ^
105

61. Slope Coefficient Test Statistic
106

62. Test of Slope Coefficient Example
You’re a marketing analyst for Hasbro Toys.
You find β0 = –.1, β1 = .7 and s =
Ad $ Sales (Units)
Is the relationship significant at the .05 level of significance? ^ ^

63. Test of Slope Coefficient Solution
H0:
Ha:
 
df 
Critical Value(s):
1 = 0
1  0
Test Statistic:
Decision:
Conclusion:
.05
5 - 2 = 3 t
3.182
-3.182
.025
Reject H0
109

64. Solution Table xi yi xi yi xiyi 1 1 1 1 1 2 1 4 1 2 3 2 9 4 6 4 2 16 48 5 4 25 16 20 15 10 55 26 37
108

65. Test Statistic Solution

66. Test of Slope Coefficient Solution
H0: 1 = 0
Ha: 1  0
  .05
df  = 3
Critical Value(s):
Test Statistic:
Decision:
Conclusion: t
3.182
-3.182
.025
Reject H0
Reject at  = .05
There is evidence of a relationship
109

67. Test of Slope Coefficient Computer Output
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Param=0 Prob>|T|
INTERCEP
ADVERT
‘Standard Error’ is the estimated standard deviation of the sampling distribution, sbP. ^ ^ S ^
t = 1 / S 1 ^ 1 1
P-Value

68. Correlation Models

69. Types of Probabilistic Models
Regression Models
Correlation Models
130

70. 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

71. Coefficient of Correlation
where

72. Coefficient of Correlation Values
Perfect Negative Correlation
Perfect Positive Correlation
No Linear Correlation
–1.0
–.5
+.5
+1.0
Increasing degree of negative correlation
Increasing degree of positive correlation
134

73. Coefficient of Correlation Example
You’re a marketing analyst for Hasbro Toys. Ad $ Sales (Units)
Calculate the coefficient of correlation. 83

74. Solution Table xi yi xi yi xiyi 1 1 1 1 1 2 1 4 1 2 3 2 9 4 6 4 2 16 48 5 4 25 16 20 15 10 55 26 37
108

75. Coefficient of Correlation Solution

76. Coefficient of Correlation Thinking Challenge
You’re an economist for the county cooperative. You gather the following data:
Fertilizer (lb.) Yield (lb.)
Find the coefficient of correlation.
© T/Maker Co. 62

77. Solution Table* xi yi xi yi xiyi 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 66

78. Coefficient of Correlation Solution*

79. Coefficient of Determination
Proportion of variation ‘explained’ by relationship between x and y
0  r2  1
r2 = (coefficient of correlation)2 79

80. Coefficient of Determination Example
You’re a marketing analyst for Hasbro Toys. You know r = .904.
Ad $ Sales (Units)
Calculate and interpret the coefficient of determination. 83

81. 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. 83

82. r2 Computer Output
Root MSE R-square
Dep Mean Adj R-sq
C.V r2
r2 adjusted for number of explanatory variables & sample size

83. Using the Model for Prediction & Estimation

84. Regression Modeling Steps
Hypothesize deterministic component
Estimate unknown model parameters
Specify probability distribution of random error term
Estimate standard deviation of error
Evaluate model
Use model for prediction and estimation

85. 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

86. What Is Predicted y ^ y ^ ^ ^ x xP yi = b0 + b1x Mean y, E(y)
Individual
yi = b0 + b1x ^
Mean y, E(y)
E(y) = b0 + b1x ^
Prediction, y x xP
115

87. Confidence Interval Estimate for Mean Value of y at x = xp
df = n – 2

88. Factors Affecting Interval Width
Level of confidence (1 – )
Width increases as confidence increases
Data dispersion (s)
Width increases as variation increases
Sample size
Width decreases as sample size increases
Distance of xp from meanx
Width increases as distance increases

89. Why Distance from Mean? y y x x1 x x2 Sample 1 Line Sample 2 Line
Greater dispersion than x1
The closer to the mean, the less variability.
This is due to the variability in estimated slope parameters. y
Sample 2 Line x x1 x x2
118

90. Confidence Interval Estimate Example
You’re a marketing analyst for Hasbro Toys.
You find β0 = -.1, β 1 = .7 and s =
Ad $ Sales (Units)
Find a 95% confidence interval for the mean sales when advertising is $4. ^ ^

91. Solution Table 2 2 x y x y x y i i i i i 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
120

92. Confidence Interval Estimate Solution
x to be predicted
121

93. Prediction Interval of Individual Value of y at x = xp
Note the 1 under the radical in the standard error formula.
The effect of the extra Syx is to increase the width of the interval.
This will be seen in the interval bands.
Note!
df = n – 2
122

94. e Why the Extra ‘S’? y ^ x xp yi = b0 + b1xi E(y) = b0 + b1x ^
y we're trying to predict e
Expected
The error in predicting some future value of Y is the sum of 2 errors:
1. the error of estimating the mean Y, E(Y|X)
2. the random error that is a component of the value of Y to be predicted.
Even if we knew the population regression line exactly, we would still make  error.
(Mean) y
E(y) = b0 + b1x ^
Prediction, y x xp
123

95. Prediction Interval Example
You’re a marketing analyst for Hasbro Toys.
You find β0 = -.1, β 1 = .7 and s =
Ad $ Sales (Units)
Predict the sales when advertising is $4. Use a 95% prediction interval. ^ ^

96. Solution Table 2 2 x y x y x y i i i i i 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
120

97. Prediction Interval Solution
x to be predicted
121

98. Interval Estimate Computer Output
Dep Var Pred Std Err Low95% Upp95% Low95% Upp95%
Obs SALES Value Predict Mean Mean Predict Predict
Predicted y when x = 4
Confidence Interval
Prediction Interval SY ^

99. Confidence Intervals v. Prediction Intervalsy ^
yi = b0 + b1xi
Note:
1. As we move farther from the mean, the bands get wider.
2. The prediction interval bands are wider. Why? (extra Syx) x x
124

100. Conclusion Described the Linear Regression Model
Stated the Regression Modeling Steps
Explained Least Squares
Computed Regression Coefficients
Explained Correlation
Predicted Response Variable