Beteta - Module One Problem Set Report
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Southern New Hampshire University *
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303
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Mathematics
Date
Apr 3, 2024
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docx
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MAT 303 Module One Problem Set Report
Multiple Regression
Diego Beteta
diego.beteta@snhu.edu
Southern New Hampshire University
1.
Introduction The dataset we are exploring consists of specifications and performance metrics for various car models, named mtcars. It includes measures like miles per gallon (fuel efficiency), horsepower, weight, and several other characteristics. Analyzing this data can offer insights into how different specifications influence a car's fuel efficiency or performance. Automobile manufacturers or enthusiasts could utilize the results of our analysis to identify influential factors that impact vehicle performance. In this problem set, we will run multiple regression analyses to understand the relationships between various car specifications and performance metrics and predict specific outcomes based on multiple input variables.
2.
Data Preparation The mtcars dataset provides comprehensive specifications and performance metrics for a selection of cars. Central to our analysis are variables such as mpg, which quantifies fuel efficiency; hp, a measure of the car's power; wt, reflecting the vehicle's weight; and am, distinguishing between automatic and manual transmissions. Beyond these, the dataset includes nuanced specifications like the number of cylinders and engine displacement. With 32 rows, the dataset encompasses information on 32 distinct car models. Also, 12 columns capture a dozen different attributes for each vehicle, offering a holistic view of each car's makeup and performance.
3.
Multiple Regression Model
Correlation Analysis
Mpg against drat:
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The scatterplot showcases a positive correlation between the rear axle ratio and fuel efficiency. As the rear axle ratio increases, cars tend to be more fuel-efficient. While the data points are somewhat spread, the upward trend is noticeable, suggesting that cars with higher rear axle ratios might be designed or performed in a manner that results in better fuel efficiency.
Mpg against hp:
The scatterplot reveals an inverse relationship between horsepower and fuel efficiency. As the horsepower of a car increases, its fuel efficiency tends to decrease, suggesting that cars with greater power might not be as fuel-efficient. It's a trade-off between power and economy, where vehicles designed for higher performance might not prioritize fuel conservation.
The correlation between fuel efficiency and rear axle ratio is positive and moderately strong at 0.6812. This indicates that fuel efficiency also tends to increase as the rear axle ratio increases.
The correlation between fuel efficiency and horsepower is negative and relatively strong at −0.7762. This suggests an inverse relationship: whereas horsepower increases, fuel efficiency tends to decrease.
In essence, while higher rear axle ratios are associated with better fuel efficiency, increased horsepower is linked to reduced fuel efficiency. The strength of these correlations suggests that both variables (rear axle ratio and horsepower) significantly influence a car's fuel efficiency.
Reporting Results
General Form: E
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Prediction Model: ^
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Based on the R-script results, the prediction model equation is:
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The R-squared value is 0.7412, and the adjusted R-squared value is 0.7233.
In our model, the R-squared value of 0.741 means that the car’s rear axle ratio and horsepower can explain 74.1% of the changes in fuel efficiency. The Adjusted R-squared value of 0.723, which is lower, gives a more accurate picture by considering the number of factors we're using. Both values tell us that our chosen factors (rear axle ratio and horsepower) predict a car's fuel efficiency well, explaining over 70% of the variation we see in the data.
The beta estimate for the rear axle ratio is 4.6982. This means that, on average, for every one-
unit increase in the rear axle ratio, we can expect the fuel efficiency (mpg) to increase by approximately 4.6982 units, keeping everything else constant. On the other hand, the beta estimate for horsepower (hp) is -0.0518, indicating that for every one-unit rise in horsepower, the fuel efficiency is expected to drop by roughly 0.0518 units, given that other factors remain unchanged. In essence, while cars with a higher rear axle ratio tend to be more fuel-efficient, increased power (as denoted by higher horsepower) decreases fuel efficiency.
A fitted value, often called predicted value, is the value our model expects based on the information given. For instance, in our car dataset, the fitted value of mpg for each car is the model's fuel efficiency estimate, given that car's rear axle ratio and horsepower.
A residual is the discrepancy between an actual observed value and its corresponding fitted value as predicted by the regression model. It measures the error of our prediction for a particular data point. When analyzing the fit of a regression model, the residuals help understand where the model's predictions deviate from the true values. If the model is accurate, these residuals should be randomly dispersed around zero without any noticeable trend.
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This plot checks the assumption of homoscedasticity, implying that the variance of the residuals should remain consistent across all levels of the independent variables.
Ideally, the residuals should display a random scatter around the horizontal zero line. However, our plot shows a slight funnel shape, suggesting potential heteroscedasticity. The funnel shape in our plot means that our model might be less reliable across the entire range of data.
For some cars (likely those with higher predicted fuel efficiency), our model's predictions might be more off-target than for others. This uneven spread can make it hard to trust the model's predictions uniformly across different cars.
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The Q-Q plot assesses the normal distribution of residuals. Points closely following the 45-
degree reference line suggest a normal distribution.
In our plot, deviations are observed at the tails while most points align with this line, particularly
in the center. This suggests slight non-normality in the residuals, particularly for extreme values.
The deviations in the Q-Q plot indicate that while the residuals are largely normally distributed, there are minor deviations, especially at the tails. Both observations suggest that while the model is a reasonable fit, refinements or transformations could
enhance its accuracy and adherence to regression assumptions.
Evaluating Model Significance In determining if our model is significant, we use an F-test. This test helps us decide whether our model predicts fuel efficiency better with variables such as rear axle ratio and horsepower than a model without these variables. The null hypothesis states that our variables don't make a difference, while the alternative hypothesis says they do. For our model, the F-test gives us a P-value very close to zero (3.081e-09). Since this P-value is much smaller than 0.05 (our 5% significance level), we conclude that our model is significant. This means that the rear axle ratio and horsepower variables are useful in predicting a car's fuel efficiency; they're not just random or meaningless additions.
For the variables "rear axle ratio" and "horsepower," we checked if they significantly affect fuel efficiency. The null hypothesis for each variable is that it doesn't affect fuel efficiency, while the alternative is that it does. Looking at the P-values, we found that for rear axle ratio, the P-value is about 6
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0.000467, and for horsepower, it's around 0.000005. Since both these P-values are way smaller than 0.05, the threshold for a 5% significance level, we can confidently say that both rear axle ratio and horsepower significantly affect a car's fuel efficiency. In other words, these two factors are important: rear axle ratio positively influences fuel efficiency, while more horsepower generally means lower fuel efficiency.
95% Confidence Interval of Rear Axle Ratio: [2.26, 7.14]
We are 95% sure that the actual impact of the rear axle ratio on fuel efficiency falls between increasing it by 2.26 to 7.14 miles per gallon for each one-unit increase in the ratio. If the rear axle ratio increases by one, we can expect the car's fuel efficiency to increase by this range, keeping everything else constant.
95% Confidence Interval of Horsepower: [−0.0708, −0.0328]
We are 95% confident that the car's fuel efficiency decreases between 0.0328 and 0.0708 miles per gallon for each additional horsepower. This interval tells us that more horsepower typically reduces fuel efficiency, and we are quite sure of this range of impact.
These intervals give us a pretty good idea of how much the rear axle ratio helps and how horsepower hinders a car's fuel efficiency. The broader range for the rear axle ratio suggests more uncertainty about its precise impact compared to horsepower.
Making Predictions Using the Model
For a car with a rear axle ratio of 3.15 and 120 horsepower, our model predicts a fuel efficiency of about 19.37 miles per gallon. However, if this car achieves an average of 20.5 miles per gallon, the residual is 1.13. This means our model underestimated the car's fuel efficiency by 1.13 miles per gallon. The car did
better in real life than our model expected based on its rear axle ratio and horsepower.
The 95% prediction interval, according to the r-script, is [8.5553, 21.9081] This interval means that we are 95% confident that the actual fuel efficiency of a car with these specifications will fall between 8.5553 and 21.9081 miles per gallon. Essentially, it's saying that while our
best guess for such a car's mpg is around 19.37, depending on factors like make, model, maintenance, and driving conditions, the mpg could realistically vary quite a bit, from as low as about 8.56 mpg to as high as approximately 21.91 mpg. The prediction interval is wider than the confidence interval because it covers more uncertainty. While the confidence interval focuses on where we expect the average (mean) value for a group of similar cars to fall, the prediction interval deals with the expected range of an individual car's fuel efficiency. Each car is unique and can vary due to factors like specific make, model, driving habits, and maintenance. This variation adds more uncertainty, so the prediction interval needs to be wider to capture that. In short, the prediction interval is broader because it accounts not only for the average trend among similar cars but also for the differences from one car to another.
The confidence interval, according to the r-script, is [13.6395, 16.8239]
This means that we are 95% confident the average fuel efficiency for all cars sharing these characteristics is between 13.6395 and 16.8239 miles per gallon. It has a narrower range than a prediction interval because it focuses solely on the average for a group of similar cars rather than the potential variation in an individual car. This suggests that, on average, such cars are expected to be less 7
fuel efficient than predicted for a particular car. This difference is because confidence intervals estimate the mean response for a population with those specific characteristics, whereas prediction intervals cover the additional variation found in individual data points.
4.
Conclusion The statistical analysis conducted with the provided dataset has shown that both rear axle ratio and horsepower are significant predictors of a car's fuel efficiency (mpg). The multiple regression model we created is statistically significant, as indicated by the low P-value from the F-test, suggesting that these predictors have a tangible impact on mpg. The positive coefficient for the rear axle ratio implies that higher ratios are generally associated with better fuel efficiency. In contrast, the negative coefficient for horsepower suggests that more horsepower decreases fuel efficiency. The model demonstrates a good fit with a high R-squared value, indicating that these two variables explain a substantial portion of the variability in mpg.
However, while the model is statistically robust and provides insightful interpretations, it is important to consider its practical application cautiously. The prediction and confidence intervals generated from the model offer valuable guidance on the expected fuel efficiency range for cars with specific rear axle ratios
and horsepower values. These insights are crucial for manufacturers in designing vehicles and for consumers in making informed choices. The model's predictions should be understood within the context of its limitations — it's based on the sample at hand, and other unconsidered factors could also influence fuel efficiency. Also, the assumption of linear relationships might not capture all the complexities of how different features impact fuel efficiency. Lastly, while the model offers a valuable framework for understanding key factors influencing mpg, it should be applied with an awareness of its bounds and an eye toward potential variables not included in the analysis.
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