Correlation and Multiple Regression

This document discusses the correlation and multiple regression between mileage per gallon and weight, length, and engine displacement of a car. It includes information on data collection, pairwise correlation coefficients, and the multiple regression model. The study finds that two of the independent variables are statistically insignificant, and the model fit is good.

RUNNING HEAD: CORRELATION AND MULTIPLE REGRESSION
PSY-520 Graduate Statistics
Topic 5 – Benchmark – Correlation and Regression Project

Correlation and Multiple Regression
1. Mileage per gallon (mpg), weight, length, and engine displacement of engine of a car
have been considered as the four variables in this study.
a. Mileage for a car is believed to decrease with increase in weight, height, and
displacement of engine. Mileage per gallon is taken as the dependent and weight,
length, and engine displacement are taken as the independent variables. An inverse
and linear impact of the independent variables is expected on the dependent variable
of this study.
2. The primary data about the brand, mileage per gallon, weight, length, and engine
displacement of cars has been collected from 74 car owners. The car owners were
interviewed about their brand, model of car, and fuel efficiency. The interviewed car
owners belong to relatives, neighbors, or friends of the scholar. Initially, the scholar
collected data from 30 car owners, and then extended the sample size to 74 to ensure a
large sample (n > 30). After collecting primary data, the scholar collected information
about the weight, height, and displacement of engine from the brochure of the specific
brand and model of cars. This data is secondary in nature. The primary data is important
for the validity of the correct information on fuel efficiency. The scholar was looking at
R-square = 0.5 and the sample size is found to be sufficient for the chosen value (Faul,
Erdfelder, Buchner, & Lang, 2009).
a. The collected data has been included in a SPSS file for submission. Information about
the model of the cars has not been included in the sample data file.
3. The variables are continuous in nature and pairwise Pearson’s correlation coefficients
have been calculated in SPSS. “mpg” has significant and strong negative correlation with
the independent variables. Pairwise correlations between all the three independent
variables are statistically significant and strongly positive in nature. Table 1 presents the
specific values of the correlation coefficients.
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Correlation and Multiple Regression
Table 1: Pairwise Pearson Correlation Coefficients
Pearson Correlation mpg weight length displacement
mpg 1 -.807** -.796** -.706**
weight -.807** 1 .946** .895**
length -.796** .946** 1 .835**
displacement -.706** .895** .835** 1
Note: **. Correlation is significant at the 0.01 level (2-tailed)
4. A multiple regression model has been constructed with the four variables of the study.
Table 2 presents the Multiple Linear Regression model. The regression equation is
evaluated to be mpg=48. 197−0 .004∗weight−0 . 078∗length+0. 004∗displacement .
Table 2: Multiple Regression Model
Regression Model B t Sig.
Intercept 48.197 7.816 0.000
weight -0.004 -2.204 0.031
length -0.078 -1.391 0.169
displacement 0.004 0.427 0.671
Note: Dependent Variable: mpg
5. The above model is statistically significant (F = 45.755, p < 0.05), but two of the
independent variables (length, displacement) are found to be statistically insignificant.
The coefficient of the model R-square = 0.662 implied that variation in the independent
variables are able to explain 66.2% variation in “mpg”. Hence, the model fit is found to
be good. Instead of significant negative correlations, high pairwise correlations for the
independent variables could be the reason behind the statistical insignificance of two
predictors.
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Correlation and Multiple Regression
References
Faul, F., Erdfelder, E., Buchner, A., & Lang, A. G. (2009). Statistical power analyses using
G* Power 3.1: Tests for correlation and regression analyses. Behavior research
methods, 41(4), 1149-1160.
Appendix
Figure 1: G-Power Output for power of regression with 74 sample observation
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Correlation and Multiple Regression
Regression Results of SPSS
R Square
Change F Change df1 df2
Sig. F
Change
1 .814 a 0.662 0.648 3.434 0.662 45.755 3 70 0.000
Sum of
Squares df
Mean
Square F Sig.
Regressio
n
1618.229 3 539.410 45.755 .000 b
Residual 825.231 70 11.789
Total 2443.459 73
Standardiz
ed
Coefficient
s
B Std. Error Beta
(Constant) 48.197 6.167 7.816 0.000
weight -0.004 0.002 -0.584 -2.204 0.031
length -0.078 0.056 -0.299 -1.391 0.169
displacem
ent
0.004 0.010 0.067 0.427 0.671
Minimum Maximum Mean
Std.
Deviation N
Predicted
Value
10.73 29.54 21.30 4.708 74
Residual -6.763 13.337 0.000 3.362 74
Std.
Predicted
Value
-2.244 1.751 0.000 1.000 74
Std.
Residual
-1.970 3.884 0.000 0.979 74
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a. Dependent Variable: mpg
Residuals Statisticsa
a. Dependent Variable: mpg
b. Predictors: (Constant), displacement, length, weight
Coefficientsa
Model
Unstandardized
Coefficients
t Sig.
a. Predictors: (Constant), displacement, length, weight
b. Dependent Variable: mpg
ANOVAa
Model
1
a. Dependent Variable: mpg
Model Summaryb
Model R R Square
Adjusted
R Square
Std. Error
of the
Estimate
Change Statistics
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