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Objective: To study the effect of explanatory variables on the explained variable with the
help of linear regression model & to analyze the effects of the explanatory variables.
Theory: In regression analysis, the key idea behind is the statistical dependence of one
variable, the independent variable, on one or more other variables, explanatory
variables. The two variable regression functions is given by-
Y= 0+ 1Xi+ui
Where, Y is called as the dependent variableXi is called as the independent variableUi is called as the random variable or stochastic variable
Project: In our project, we try to analyze the Mileage (MPG), in miles per gallon, of
passenger cars upon its various components-
i) Top Speed (SP), in miles per hour
ii) Horse Power (HP)
iii) Cubic feet of cab space or Volume of the engine (VOL)
iv) Weight of the vehicle (WT), in hundreds of pound
We develop the regression equation as
Y= 0+ 1Xi+2X2+3X3+4X4+ui
Where,
Y=Mileage of the car, in miles per gallon (MPG)
Xi =Top Speed of the vehicle (SP)
X2=Horse Power of the engine (HP)
X3=Cubic feet of engine cab space (VOL)
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X4=Weigh
t of the vehicle
(WT), in
hundreds of
pound
ui =Other
factors effectingMileage
Data:
SlNo.
MPG
SP
HP
VOL
WT
1 65.4 96 49 89 17.5
2 56 97 55 92 20
3 55.9 97 55 92 204 49 105 70 92 20
5 46.5 96 53 92 20
6 46.2 105 70 89 20
7 45.4 97 55 92 20
8 59.2 98 62 50 22.5
9 53.3 98 62 50 22.5
10 43.4 107 80 94 22.5
11 41.1 103 73 89 22.5
12 40.9 113 92 50 22.5
13 40.9 113 92 99 22.514 40.4 103 73 89 22.5
15 39.6 100 66 89 22.5
16 39.3 103 73 89 22.5
17 38.9 106 78 91 22.5
18 38.8 113 92 50 22.5
19 38.2 106 78 91 22.5
20 42.2 109 90 103 25
21 40.9 110 92 99 25
22 40.7 101 74 107 25
23 40 111 95 101 2524 39.3 105 81 96 25
25 38.8 111 95 89 25
26 38.4 110 92 50 25
27 38.4 110 92 117 25
28 38.4 110 92 99 25
29 46.9 90 52 104 27.5
30 36.3 112 103 107 27.5
31 36.1 103 84 114 27.5
32 36.1 103 84 101 27.5
33 35.4 111 102 97 27.5
34 35.3 111 102 113 27.5
35 35.1 102 81 101 27.5
36 35.1 106 90 98 27.5
37 35 106 90 88 27.5
38 33.2 109 102 86 30
39 32.9 109 102 86 30
40 32.3 120 130 92 30
41 32.2 106 95 113 30
42 32.2 106 95 106 30
43 32.2 109 102 92 30
44 32.2 106 95 88 30
45 31.5 105 93 102 30
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Note:
V0L=cubic feet of cabspace
HP= engine horse power
MPG=average miles pergallon
SP=top speed, miles perhour
WT=vehicle weight, hundreds ofpound
Regression statistics in SPSS sheet:
Variables Entered/RemovedModel Variables Entered Variables Removed Method
1 WT, VOL, SP, HP . Enter
a All requested variables entered.b Dependent Variable: MPG
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Model SummaryModel R R Square Adjusted R Square Std. Error of the
Estimate
1 .909 .827 .817 4.28
a Predictors: (Constant), WT, VOL, SP, HPb Dependent Variable: MPG
ANOVAModel Sum of
SquaresDf Mean
SquareF Sig.
1 Regression 6106.493 4 1526.623 83.409 .000
Residual 1281.193 70 18.303
Total 7387.687 74
a Predictors: (Constant), WT, VOL, SP, HPb Dependent Variable: MPG
CoefficientsUnstandardized Coefficients
StandardizedCoefficients
t Sig.
Model B Std. Error Beta
1 (Constant) 70.222 4.760 14.751 .000
SP -1.452E-02 .040 -.025 -.360 .720
HP -3.031E-02 .023 -.160 -1.347 .182
VOL -3.154E-02 .028 -.067 -1.117 .268
WT -.909 .138 -.733 -6.607 .000
a Dependent Variable: MPG
Residuals Statistics
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Normal P-P Plot of Regression Standardiz
Dependent Variable: MPG
Observed Cum Prob
1.00.75.50.250.00E
xpectedCumProb
1.00
.75
.50
.25
0.00
Minimum Maximum Mean Std. Deviation N
Predicted Value 7.80 48.62 34.67 9.08 75
Residual -4.96 16.78 9.71E-15 4.16 75
Std. PredictedValue
-2.958 1.537 .000 1.000 75
Std. Residual -1.159 3.921 .000 .973 75
a Dependent Variable: MPG
Charts:
Prediction:
i) From the SPSS output we found R2, the fitness of good, to be 0.827. This
signifies that the sample regression line fits the model very well.
ii) The value of F is found to be 83.409 & sig is equal .000. It means that the
mode is highly significant.
iii) The significance value are found to be .000, .720, .182, .268, .000 for const,
SP, HP, VOL & WT respectively. It means that Constant, HP & VOL values
are significant.
Summary: From the entire regression analysis we have seen that the out of four explanatory
variables, three variables are showing negative but significant value, So the Mileage of vehicles
are negatively related to SP, HP & VOL., but positively related to WT of the vehicle We can
conclude that the model is comparatively a good model.
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A PROJECT ON REGRESSION ANALYSIS
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By Jankim Hazarika
MBE 2nd semester
Roll no. 04