Dependability of Regression Equation Drawing Inferences Optimising the independent variables in a Multiple Regression Regression Modeling Technique

1. Prof V Nallasivam

2. Regression Modeling
Technique
Dependability of
Regression Equation
Drawing Inferences
Optimising the
independent variables in a
Multiple Regression
Prof V Nallasivam

3. Year
R & D
Expenditure
(in Crs)
Annual
Profit
(in Crs)
x y
2002 2 20
2003 3 25
2004 5 34
2005 4 30
2006 11 40
2007 5 31
Total 30 180
Prof V Nallasivam

4. Prof V Nallasivam
ˆ 20 2y x= +

5. ANOVA - (Test the Model)
Scatter diagram
Standard Error of Estimate
Testing the significance of Regression
Co-efficient (b) against Zero.
Co-efficient of determination
Prof V Nallasivam
Dependability of a Regression Equation

6. Prof V Nallasivam

7. 2 Explained Variation
r
Total Variation
=
Prof V Nallasivam
2 2
[ ]r Correlation=
Coefficient of Determination

8. x y
1 4 4 -14 196 -14 196
2 8 8 -10 100 -10 100
3 12 12 -6 36 -6 36
4 16 16 -2 4 -2 4
5 20 20 2 4 2 4
6 24 24 6 36 6 36
7 28 28 10 100 10 100
8 32 32 14 196 14 196
144 144 0 672 0 672
ˆyˆy y− 2
ˆ( )y y−y y− 2
( )y y−
ˆy ˆy y− 2
ˆ( )y y− y y− 2
( )y y−
Prof V Nallasivam

9. Prof V Nallasivam
ˆ 4y x=

10. x y
1 4 4 -14 196 -14 196
2 8 8 -10 100 -10 100
3 12 12 -6 36 -6 36
4 16 16 -2 4 -2 4
5 20 20 2 4 2 4
6 24 24 6 36 6 36
7 28 28 10 100 10 100
8 32 32 14 196 14 196
144 144 0 672 0 672
ˆyˆy y− 2
ˆ( )y y−y y− 2
( )y y−
ˆy ˆy y− 2
ˆ( )y y− y y− 2
( )y y−
Prof V Nallasivam

11. x y
1 4 4 -14 196 -14 196
2 8 8 -10 100 -10 100
3 12 12 -6 36 -6 36
4 16 16 -2 4 -2 4
5 20 20 2 4 2 4
6 24 24 6 36 6 36
7 28 28 10 100 10 100
8 32 32 14 196 14 196
144 144 0 672 0 672
ˆyˆy y− 2
ˆ( )y y−y y− 2
( )y y−
ˆy ˆy y− 2
ˆ( )y y− y y− 2
( )y y−
Prof V Nallasivam

12. x y
1 4 4 -14 196 -14 196
2 8 8 -10 100 -10 100
3 12 12 -6 36 -6 36
4 16 16 -2 4 -2 4
5 20 20 2 4 2 4
6 24 24 6 36 6 36
7 28 28 10 100 10 100
8 32 32 14 196 14 196
144 144 0 672 0 672
ˆyˆy y− 2
ˆ( )y y−y y− 2
( )y y−
ˆy ˆy y− 2
ˆ( )y y− y y− 2
( )y y−
Prof V Nallasivam

13. x y
1 4 4 -14 196 -14 196
2 8 8 -10 100 -10 100
3 12 12 -6 36 -6 36
4 16 16 -2 4 -2 4
5 20 20 2 4 2 4
6 24 24 6 36 6 36
7 28 28 10 100 10 100
8 32 32 14 196 14 196
144 144 0 672 0 672
ˆyˆy y− 2
ˆ( )y y−y y− 2
( )y y−
ˆy ˆy y− 2
ˆ( )y y− y y− 2
( )y y−
Prof V Nallasivam

14. x y
1 4 4 -14 196 -14 196
2 8 8 -10 100 -10 100
3 12 12 -6 36 -6 36
4 16 16 -2 4 -2 4
5 20 20 2 4 2 4
6 24 24 6 36 6 36
7 28 28 10 100 10 100
8 32 32 14 196 14 196
144 144 0 672 0 672
ˆyˆy y− 2
ˆ( )y y−y y− 2
( )y y−
ˆy ˆy y− 2
ˆ( )y y− y y− 2
( )y y−
Prof V Nallasivam
E V T V

15. Prof V Nallasivam
2
ˆ( ) 672
1
( ) 672
y y
r
y y
−
= = =
−
∑
∑

16. Y = 12
ˆ 4y x=
Y
X
ˆ 12y =
Example-1
Prof V Nallasivam

17. x y
1 6 9 0 0 -3 9
1 12 9 0 0 3 9
3 6 9 0 0 -3 9
3 12 9 0 0 3 9
5 6 9 0 0 -3 9
5 12 9 0 0 3 9
7 6 9 0 0 -3 9
7 12 9 0 0 3 9
0 72
ˆyˆy y− 2
ˆ( )y y−y y− 2
( )y y−
ˆy ˆy y− 2
ˆ( )y y− y y− 2
( )y y−
Prof V Nallasivam

18. Prof V Nallasivam
ˆ 9y =

19. x y
1 6 9 0 0 -3 9
1 12 9 0 0 3 9
3 6 9 0 0 -3 9
3 12 9 0 0 3 9
5 6 9 0 0 -3 9
5 12 9 0 0 3 9
7 6 9 0 0 -3 9
7 12 9 0 0 3 9
0 72
ˆyˆy y− 2
ˆ( )y y−y y− 2
( )y y−
ˆy ˆy y− 2
ˆ( )y y− y y− 2
( )y y−
Prof V Nallasivam
E V T V

20. Prof V Nallasivam
2
ˆ( ) 0
0
( ) 72
y y
r
y y
−
= = =
−
∑
∑

21. Example-2
Prof V Nallasivam

22. Research & Development
Expenditure - Profit
Prof V Nallasivam

23. x y
2 20 24 -6 36 -10 100
3 25 26 -4 16 -5 25
5 34 30 0 0 4 16
4 30 28 -2 4 0 0
11 40 42 12 144 10 100
5 31 30 0 0 1 1
30 180 180 200 241
ˆyˆy y−ˆy y−y y−y y−
ˆy ˆy y− 2
ˆ( )y y− y y− 2
( )y y−
Prof V Nallasivam
E V T V

24. Prof V Nallasivam
2
ˆ( ) 200
0.829
( ) 241
y y
r
y y
−
= = =
−
∑
∑

25. y
ˆy
y
Total
Variation
Unexplained
Variation
Explained
Variation
Prof V Nallasivam

26. x y
2 20 24 -4 16
3 25 26 -1 1
5 34 30 4 16
4 30 28 2 4
11 40 42 -2 4
5 31 30 1 1
30 180 180 0 42
ˆy ˆy y− 2
ˆ( )y y−
ˆy ˆy y−
2
ˆ( )y y−
Prof V Nallasivam
Standard Error of Estimate

27. Prof V Nallasivam
2
ˆ( ) 42
3.24
2 4
e
y y
SE
n
−
= = =
−
∑

28. Formulation of Hypothesis
Significance Level [
αα
Formulation of Hypothesis
Significance Level [
α
Formulation of Hypothesis
Significance Level [
α
Formation of Hypothesis
Significance Level
Probability Distribution
Find the Table Value
Find the Calculated Value
Prof V Nallasivam
Test Regression Coefficient ‘b’ against ZERO

29. -Statistic Parameter
CV
Standard Error
=
2 0
4.44
0.45r
b B
CV
SE
− −
= = =
Prof V Nallasivam

30. 0- 2.78 2.78
4.44
Probability Curve t Distribution
Acceptance Region Rejected RegionRejected Region
Table Value
Calculated Value
P Value 0.025 0.025
Prof V Nallasivam

31. Acceptance Region Rejected Region
7.71
19.048
Table Value
Calculated Value
P Value 0.05
Prof V Nallasivam
ANOVA
0.012

32. Prof V Nallasivam

33. I Hypothesis Testing
II Estimation of Population Parameters
a) Point Estimate
b) Interval Estimate
Prof V Nallasivam

34. r
b B
t
SE
−
=
2.0 2.1
0.22
0.45
t
−
= = −
Population Growth Rate of Profit = 2.1
Prof V Nallasivam

35. 0- 2.78 2.78
- 0.22
Probability Curve t Distribution
Acceptance Region Rejected RegionRejected Region
Table Value
Calculated Value
P Value 0.025 0.025
Prof V Nallasivam

36. From
Sample
statistic estimate, population Parameter
From
Sample
y estimate, population Y
From
Sample
estimate, population
From
Sample
b estimate, population B
From
Sample
a estimate, population A
ˆyˆY
ˆy ˆY
Prof V Nallasivam

37. Parameter = statistic ± [Standard Error × Critical Value]
Parameter = statistic + [Standard Error × Critical Value]
Parameter = statistic - [Standard Error × Critical Value]
General Formula to Calculate Interval Estimate
Upper Limit
Lower Limit
Prof V Nallasivam

38. Confidence Level Significance Level
90%
(0.9)
10%
(0.1)
95%
(0.95)
5%
(0.05)
99%
(0.99)
1%
(0.01)
Prof V Nallasivam

39. y
ˆy
y
Upper Limit 3.25
5
Prof V Nallasivam
Lower Limit 0.75
Point Estimate 2.00
34
31
y
From Sample b confidence Interval of B
b

40. y
ˆy
y
Upper Limit 38.39
5
Prof V Nallasivam
Lower Limit 21.11
Point Estimate 30
34
31
y
From Sample confidence Interval ofˆy ˆY

41. y
ˆy
y
Upper Limit 27.23
5
Prof V Nallasivam
Lower Limit 12.77
Point Estimate 20
34
31
y
From Sample a confidence Interval of A
a

42. y
ˆy
y
Upper Limit 39.45
5
Prof V Nallasivam
Lower Limit 20.55
Point Estimate 30
34
31
y
From Sample y confidence Interval of Y

43. Prof V Nallasivam

44. Prof V Nallasivam
Dependent Variable
Sales
Independent Variables
Market Potential
Number of dealers
Number of Sales People
Competitors Activities
Number of Service People
Number of Existing Customers

45. Prof V Nallasivam
Reg
n
SALES POTENTI DEALERS PEOPLE COMPT SERVICE CUSTOM
1 5.00 25.00 1.00 6.00 5.00 2.00 20.00
2 60.00 150.00 12.00 30.00 4.00 5.00 50.00
3 20.00 45.00 5.00 15.00 3.00 2.00 25.00
4 11.00 30.00 2.00 10.00 3.00 2.00 20.00
5 45.00 75.00 12.00 20.00 2.00 4.00 30.00
6 6.00 10.00 3.00 8.00 2.00 3.00 16.00
7 15.00 29.00 5.00 18.00 4.00 5.00 30.00
8 22.00 43.00 7.00 16.00 3.00 6.00 40.00
9 29.00 70.00 4.00 15.00 2.00 5.00 39.00
10 3.00 40.00 1.00 6.00 5.00 2.00 5.00
11 16.00 40.00 4.00 11.00 4.00 2.00 17.00
12 8.00 25.00 2.00 9.00 3.00 3.00 10.00
13 18.00 32.00 7.00 14.00 3.00 4.00 31.00
14 23.00 73.00 10.00 10.00 4.00 3.00 43.00
15 81.00 150.00 15.00 35.00 4.00 7.00 70.00

46. Prof V Nallasivam
1 1 2 2 3 3 4 4 5 5 6 6y a b x b x b x b x b x b x= + + + + + +
Sales = -3.17 + 0.227Pot + 0.819Dealers + 1.091People
-1.893Compet – 0.549Service + 0.66Cust.

47. CORRELATION

49. COMPUTER OUTPUT [SPSS]

50. Model R R Square Adjusted R Square Std. Error of the Estimate
1 .989 .977 .960 4.39102
Model Summary
a Predictors: (Constant), CUSTOMER, COMPT, SERVICE, POTENTIA, DEALERS, PEOPLE

51. Model Sum of Squares df Mean Square F Sig.
1 Regression 6609.485 6 1101.581 57.133 .000
Residual 154.249 8 19.281
Total 6763.733 14
ANOVA
a Predictors: (Constant), CUSTOMER, COMPT, SERVICE, POTENTIA, DEALERS, PEOPLE
b Dependent Variable: SALES

52. Unstandardized
Coefficients
Standardized
Coefficients
t Sig.
Model B Std. Error Beta
1 (Constant) -3.173 5.813 -.546 .600
POTENTIA .227 .075 .439 3.040 .016
DEALERS .819 .631 .164 1.298 .230
PEOPLE 1.091 .418 .414 2.609 .031
COMPT -1.893 1.340 -.085 -1.413 .195
SERVICE -.549 1.568 -.041 -.350 .735
CUSTOMER 6.594E-02 .195 .050 .338 .744
Coefficients
a Dependent Variable: SALES

53. Prof V Nallasivam
1 1 2 2y a b x b x= + +
Sales = - 10.616 + 0.234 Pot + 1.424People

54. Prof V Nallasivam

55. Men Women
Months
Employed
Base
Salary
Months
Employed
Base
Salary
6 7.50 5 6.2
10 8.60 13 8.7
12 9.10 15 9.4
18 10.30 21 9.8
30 13.00
Prof V Nallasivam

56. Ho: There is no difference in the base
Salary between Male and Female
1 2:oH x x=
Prof V Nallasivam

57. 1
1
2
1
5
9.7
4.415
n
x
s
=
=
=
2
2
2
2
4
8.525
2.609
n
x
s
=
=
=
Men Women
( =0.01; =7)
Calculated t Value = 0.92
Table Value t 2.998α γ =
Prof V Nallasivam

58. Rejected Region
0- 2.365 2.356
0.92
Acceptance Region Rejected Region
0.0250.025 P - Value
Prof V Nallasivam

59. Prof V Nallasivam

60. Months
Employed
Base
Salary
6 7.50
10 8.60
12 9.10
18 10.30
30 13.00
5 6.2
13 8.7
15 9.4
21 9.8
Prof V Nallasivam

61. OBS ACTUAL PREDICTED
VALUE
RESIDUAL
1 7.5000 7.2085 0.2915
2 8.6000 8.1413 0.4587
3 9.1000 8.6077 0.4923
4 10.3000 10.0069 0.2913
5 13.0000 12.8054 0.1946
6 6.2000 6.9753 -0.7753
7 8.7000 8.8409 -0.1407
8 9.4000 9.3073 0.0927
9 9.8000 10.7066 -0.9066
Prof V Nallasivam

62. Months
Employed
Sex
Base
Salary
M 6 0 7.50
M 10 0 8.60
M 12 0 9.10
M 18 0 10.30
M 30 0 13.00
F 5 1 6.2
F 13 1 8.7
F 15 1 9.4
F 21 1 9.8Prof V Nallasivam

63. Ho: There is no difference in the base
Salary between Male and Female
Prof V Nallasivam

64. Prof V Nallasivam
Unstandardized
Coefficients
Standardized
Coefficients
t Sig.
Model B Std. Error Beta
1 (Constant) 6.248 .291 21.439 .000
MONEM
P
.227 .016 .937 14.089 .000
SEX -.789 .238 -.220 -3.309 .016

65. Prof V Nallasivam
0.789
3.31
0.238r
b B
CV
SE
− −
= = =

66. Prof V Nallasivam
Unstandardized
Coefficients
Standardized
Coefficients
t Sig.
Model B Std. Error Beta
1 (Constant) 6.248 .291 21.439 .000
MONEM
P
.227 .016 .937 14.089 .000
SEX -.789 .238 -.220 -3.309 .016
0.025

67. Rejected Region
0- 2.45 2.45
- 3.309
Acceptance Region Rejected Region
0.016 0.0250.025 P - Value
Prof V Nallasivam

68. 7.5000 7.6109 -0.1109
8.6000 8.5192 0.0808
9.1000 8.9734 0.1266
10.3000 10.3358 -0.0358
13.0000 13.0607 -0.0607
6.2000 6.5949 -0.3949
8.7000 8.4115 0.2885
9.4000 8.8656 0.5344
9.8000 10.2281 -0.4281
Prof V Nallasivam
y ˆy ˆy y−

69. Prof V Nallasivam