MAT 303 Module Two Problem Set Report Template (1)
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Southern New Hampshire University *
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Feb 20, 2024
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MAT 303 Module Two Problem Set Report
Interaction Terms and Qualitative Predictors
Tara Culpepper
tara.culpepper@snhu.edu
Southern New Hampshire University
Note: Replace the bracketed text on page one (the cover page) with your personal information.
1. Introduction The data set that we are exploring in this problem set is the relationship between fuel efficiency
and correlating variables. These results may be used in designing and manufacturing future vehicles to fully understand how, for example, a car's weight could affect the fuel efficiency. The type of analysis that is appropriate for this analysis is multiple regression models. Each regression model will have two different variables to compare to fuel efficiency.
2. Data Preparation The important variables in this problem set our fuel economy, the weight, rear axle ratio, and quarter second time. We will primarily be focusing on the relationship between the fuel economy, and the following variables.
qsec : quarter second
drat: rear axle ratio
hp: horse power
mpg: fuel efficiency 2
There are six rows and 12 columns in my data set
3. Model with Interaction Term
Correlation Analysis The correlation matrix is what provides us with values that show the relationship between two of our values. For example: The relationship between fuel economy and weight is -0.8677 meaning there is a strong negative correlation between both fuel economy and weight. As weight increases the fuel economy decreases.
The relationship between fuel economy and horsepower is -0.7762, which shows a very moderate negative correlation. As horsepower increases, the fuel economy decreases. The relationship between the rear axle ratio and fuel economy has a positive correlation. That correlation coefficient reflects 0.6812. The correlation is positive, leading us to think that as the rear axle ratio increases the fuel economy will also increase. 3
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Please see the table below for reflection. Reporting Results The general form and the prediction equation of the regression model for fuel economy, using horsepower, quarter-mile time, and rear axle ratio as predictors:
The multiple regression model is:
E
(
y
)=
12.53794
−
0.0693
x
❑
1
+
5.89123
x
❑
2
−
0.30296
x
❑
3
−
0.01123
x
❑
2
x
❑
3
y
=
fuelefficiency
4
x
❑
1
=
horsepower
x
❑
2
=
rear axleratio
x
❑
2
x
❑
3
=
hp
:
drat
x
❑
3
=
quarter of asecond
The value of R
❑
2
or R-squared is 0.7472 and the value of R
❑
a
2
or adjusted R-squared is 0.7097. That means that 75% of the variance is explained by the model using predictors. We can use this model to predict the change in fuel economy of a car that has a 160 horsepower for each unit increase in quarter mile time. To do so we use the above regression model but only need to look at the following:
E(Y) = -0.693 (1) - 0.01123 (160)(1) = ? -18.661
This suggests that the fuel economy of a car will decrease by 18.661 with a horsepower of 160 with each unit increase in quarter mile time. 5
We can continue to use this model to predict the fuel economy of a car with 160 but instead it is for each unit increase in rear axle ratio by using the following regression model:
E(Y) = -0.693 (1) + 1.70650 (160) (1) = ? 272.347 This means that the fuel economy will increase by 272.347 with a horsepower of 160 for each unit increase in rear axle ratio. This proves a positive correlation that as the rear axle ratio increases the fuel economy increases.
Below is the plot for the fitted values and residuals from the model for the data set
6
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Now the Q-Q plot:
7
Based on the above plots we can assume, because there is no discernible pattern, we can have an assumption of homoscedasticity. The Q-Q plot does show an abnormal distribution of residuals as some points deviate from the line. Evaluating Model Significance Evaluate model significance for the regression model. Address the following questions in your analysis:
Performing an overall F-test will allow us to see if it is significant at a 5% level of significance. The null hypothesis states that there is no linear relation between our predictor and response 8
variable. To counter the null the alternative hypothesis would suggest that the relationship does exist however in at least one predictor variable. H
❑
0
:
B
❑
1
=
B
❑
2
=
B
❑
3
=
B
❑
4
=
B
❑
5
=
0
H
❑
a
=
B
❑
1
≠
0
Our P-value reflects 9.611e-08 which is below our alpha value of 0.05 making the null hypothesis rejected There is significant evidence to suggest that there is a linear equation between the MPG and one predictor. This, however, does not elaborate on the variables that are correlated, so additional testing would need to be performed. Making Predictions Using the Model 9
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To make a prediction, we have to plug in 175 horsepower, 14.2 quarter miles, and 3.91 to our equation to estimate the fuel economy for a car. E(Y) = 12.53794-0.693(175)+5.89123(3.91)-0.30296(14.2)-0.01123(3.91)(14.2)
90.62 MPG
The 95% prediction interval for the fuel economy: The 95% confidence interval for the fuel economy of this car
The prediction interval tells us that there is a 95% chance that the fuel economy falls between a
lower bound of 13.1652 and an upper bound of 28.0849, falling under a continuous test of the fuel economy There is a 95% chance that the actual value falls between 17.2695 and 23.9805, which makes up our confidence interval. 10
4. Model with Interaction Term and Qualitative Predictor
Reporting Results The general form of the regression model for fuel economy using horsepower, quarter mile time, interaction term for horsepower and quarter mile time, and number of cylinders. X1 = quarter mile time
X2 = horsepower X3 = interaction term for horsepower and quarter mile time
x3&x4 = referring to 6 & 8 cylinders
The specific regression model will be E(Y) = 24.505565 -+ 0.531630 (X1) + 0.141850(X2) - 0.012526 (X1X2) - 4.408372 (X3) - 1.454339 (4)
11
The R squared value is 0.8327 and the adjusted R squared is 0.8005. This information tells us that about 83.6% of the variance in MPG is explained by the predictors presented. About 80% of the adjusted value takes into account the number of predictors. 12
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○
Residuals against Fitted Values
○
Normal Q-Q plot 13
Based on the scatter plots, just like previously, the plots do not seem to have a specific pattern which meets the requirements of homoscedasticity. The Q-Q plot shows that the data points are right over the line and do not seem to roam from the regression line and this allows us to assume normality. Evaluating Model Significance We proceed to conduct an overall F-test that will be tested at a 5% significance level. Like previously stated the Null hypothesis represents the nonexistence of a linear relation whereas 14
the alternative hypothesis claims that there is a linear relationship that exists between MPG and at least one predictor variable. H
❑
0
:
B
❑
1
=
B
❑
2
=
B
❑
3
=
B
❑
4
=
B
❑
5
=
0
H
❑
a
=
B
❑
1
≠
0
Our P-value is 2.526e-09, which represents 0. Because our P-value is less than the alpha value of 0.05, the null hypothesis is rejected. 5. Conclusion From what has been gathered from the analysis that we have performed it seems that a sample
size that is sufficiently large, would be more preferred to use in this scenario. I believe that the data didn’t show a true significance between the relationships with fuel economy. based on the
data quarter of a second, and rear axle ratio don’t seem to have a significant relationship to the fuel economy. As for weight in horsepower, they do hold a significant relationship to the effects
of the fuel economy.
To fully understand, the correct out of that does have a significant relationship with your academy. There would have to be multiple tests on each variable that is presented to us. This 15
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would just give us the opportunity to see if there is a significant statistical relationship with the fuel economy. All of this analysis that has been performed could help a car manufacturer. I think this data holds a significant value to a car manufacturer and testing and associating specific variables to the vehicle to the field economy, and whether or not they share a significance with each other.
Looking to lower the weight of a vehicle? How does that significantly affect the fuel economy? If the lower the weight in the vehicle and the higher the fuel economy what happens to the horsepower? Does it have a negative effect if the weight isn’t as heavy? All important standards
that a car manufacturer could follow if they were paying attention to this specific analysis.
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