MAT 303 Module One Problem Set Report Template
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School
Southern New Hampshire University *
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Course
303
Subject
Mathematics
Date
Apr 3, 2024
Type
Pages
6
Uploaded by mrjwallace
MAT 303 Module One Problem Set Report
Multiple Regression
Joshuah Wallace
joshuah.wallace@snhu.edu
Southern New Hampshire University
Note: Replace the bracketed text on page one (the cover page) with your personal information.
1.
Introduction
Discuss the statement of the problem with regard to the statistical analyses that are being performed.
Address the following questions in your analysis:
The data set consists of elements of vehicles that are likely to make an impact on fuel economy on
vehicles.
The results of data analysis of the data set should result in answers as to how to create a more fuel
efficient vehicle based on the elements being analyzed.
The data analysis that will be conducted on the problem set is a multiple regression analysis testing
different predictive variables against a response variable.
2.
Data Preparation
There are some important variables that you have been asked to analyze in this problem set. Identify and
explain these variables. Address the following questions in your analysis:
The data that is important are the 11 elements, or columns, of the vehicle, the 12th one being the name
of the vehicle does not matter with regards to the data project, only to identify the vehicle. The 11
elements that are being examined in conjunction with fuel economy are mile-per-gallon listing, how
many cylinders the engine has, displacement of the cylinders in the engine, horsepower, weight, rear
axle ratio, quarter mile time, engine configuration, transmission type, and the number of gears in the
vehicle.
Looking at the
mtcars.csv
file, we can see that there are 32 unique entries, or rows, not including the key
guide labeling each column. There are 12 elements per entry, or 12 columns altogether, that make up
one entry regarding variables for fuel economy on a motor vehichle.
3.
Multiple Regression Model
Correlation Analysis
Visualize and describe the relationships between variables in the data set. Address the following
questions in your analysis:
2
It appears that fuel efficiency and Rear Axle Ratio are largely unrelated since every value of rear axle
ratio has varying degrees of fuel efficiency. Fuel efficiency is maximized at around 4 DRAT observing the
absolute highest values of fuel economy, but also some of the worst values of fuel economy.
3
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As horsepower increases, fuel economy decreases on trend. The highest values of fuel economy are
observed at the lowest values of horsepower while the inverse is also true. These two variables are
inversely related to one another.
I have utilized excel to run the Pearson Correlation Coefficients and have concluded that the correlation
between fuel economy and rear axle ratio (DRAT) results in a coefficient of -0.77617 and fuel economy
and horsepower have a Pearson Correlation Coefficient of 0.68117. The coefficient for DRAT indicates
that the data is trending towards having no correlation with fuel economy while the horsepower variable
indicates that there is a correlation between horsepower and fuel economy.
Reporting Results
Report the results of the regression model. Address the following questions in your analysis:
MPG =
β
0 +
β
1drat +
β
2hp + e. This is the generic formula for a multiple regression model in which fuel
economy is the response variable, and drat and horsepower are the predictor variables
Call:
lm(formula = mpg ~ drat + hp, data = mtcars)
Residuals:
Min
1Q
Median
3Q
Max
-5.0369 -2.3487 -0.6034
1.1897
7.7500
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 10.789861
5.077752
2.125 0.042238 *
drat
4.698158
1.191633
3.943 0.000467 ***
hp
-0.051787
0.009293
-5.573 5.17e-06 ***
---
Signif. codes:
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 3.17 on 29 degrees of freedom
Multiple R-squared:
0.7412, Adjusted R-squared:
0.7233
F-statistic: 41.52 on 2 and 29 DF,
p-value: 3.081e-09
The R^2 value is .7412 and the adjusted R squared value is .7233. The R squared value being closer to 1
indicates that the data points are more accurately, or tightly centered around the regression line while
lower values closer to 0 indicate that the values are very spread out in relation to a regression line. If a
term is added that improves model fitness, the adjusted r squared value increases. If a term is added
that hinders model fitness, the adjusted r squared value will decrease.
The beta estimates formula is Y = 10.78961 + 4.69816X1 - 0.05179X2. This means that for every 1 value
of rear axle ratio increasing, a 4.69816 increase is expected in fuel economy, while a 1 increase in
horsepower on average yields a loss of .05179 mpg on fuel economy.
A fitted value is the expected value of a response variable based on a regression model while a residual
value is the difference the actual value has from the expected regression value, observing the same
predictor variables of course.
4
The residuals vs fitted values plot chart shows a random dispersion of points across the mean of 0 that
suggests no strong trends could be identified between residuals and fitted values.
Evaluating Model Significance
Evaluate model significance for the regression model. Address the following questions in your analysis:
With a p value of 3.081e-09, it is reasonable to suggest that there is a significant relationship between
both horsepower and drat since the value is well below the .05 confidence level that is expected.
Therefore the null hypothesis is to be rejected suggesting a linear relationship between drat and
horsepower to fuel economy.
Making Predictions Using the Model
Make predictions using the regression model. Address the following questions in your analysis:
Based on the beta estimation formula, we should expect the vehicle to have a fuel efficiency of
19.37mpg. The residual is the difference between the actual value of 20.5 - 19.37, the theoretical value
resulting in a residual of 1.13.
We are certain that 95% of true population coefficient will be between the values of [2.2610, 7.1353]
We are certain that 95% of the true population coefficient will be between the values of [-.0708, -.0328]
Since the value of 0 is not contained in either ranges of values, at a 95% confidence value, the
coefficients are significant.
4.
Conclusion
Describe the results of the statistical analyses and address the following questions:
Using the drat and horsepower variable as predictor variables for fuel economy is a very reliable test at
this sample size. The projected values of fuel economy are fairly true to actual values such that a 95%
confidence interval is easily achieved.
5
The models conducted have indicated that the variables of horsepower and rear axle ratio (drat) are
significant to a satisfactory degree at determining fuel economy of a motor vehicle. The F test concludes
that there is a linear relationship between fuel economy and the drat rating and horsepower better than
randomized points, and the T tests concluded that each predictor variable has a significant relationship
to linearity to fuel economy.
The analysis performed is able to provide real tangible research and results without having to create real
physical models of vehicles with different variables and test them for fuel economy. Data research
projects like this save money, time, and resources over traditional research methods from testing real
objects. Predictive analysis models like this can determine what value of predictor variables will achieve
the response variable desired, optimizing production and minimizing R&D resources.
5.
Citations
You are not required to use external resources for this report. If none were used, remove this entire
section. However, if you used any resources to help you with your interpretation, you must cite them. Use
proper APA format for citations.
Insert references here in the following format:
Author's Last Name, First Initial. Middle Initial. (Year of Publication). Title of book: Subtitle of book,
edition. Place of Publication: Publisher.
6
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