A_Gonzalez_Module_One_Problem_Set
docx
keyboard_arrow_up
School
Southern New Hampshire University *
*We aren’t endorsed by this school
Course
303
Subject
Mathematics
Date
Apr 3, 2024
Type
docx
Pages
6
Uploaded by LieutenantHackerFinch8328
MAT 303 Module One Problem Set Report
Multiple Regression
Alfonso Gonzalez
Alfonso.Gonzalez@SNHU.edu
Southern New Hampshire University
1.
Introduction The focus of the current problem set revolves around conducting a statistical analysis on a data set concerning car fuel economy, specifically utilizing the mtcars.csv file. This analysis aims to uncover factors that correlate with improved fuel efficiency—which is a critical element for automobile manufacturers that want to design and market vehicles that are more fuel-economic. The mtcars data set comprises various variables potentially influencing fuel economy, including miles per gallon (mpg) as the key indicator of fuel efficiency, cylinder count (cyl) affecting power and consumption, engine displacement (disp), gross horsepower (hp), rear axle ratio (drat), vehicle weight (wt), quarter-mile time (qsec), engine cylinder configuration (vs), transmission type (am), and number of transmission gears (gear). These elements are important in determining a car's fuel economy. The statistical analysis's outcomes could guide car manufacturers in identifying crucial factors contributing to fuel efficiency, inform new model designs, drive marketing strategies, and balance performance features against fuel economy. The analysis includes descriptive statistics to summarize data features, correlation analysis to explore variable interrelations, regression analysis for modeling fuel economy against predictors, and ANOVA for examining fuel economy variations across different categories. This approach aims to help in the design and production of vehicles that better meet fuel economy objectives.
2.
Data Preparation Based on the information provided the important variables I am being asked to analyze in this data set are:
Fuel Efficiency (mpg):
As the response variable for the regression model, it's central to understanding how different factors impact a car's fuel consumption.
Rear Axle Ratio (drat):
One of the predictor variables in the regression model, indicating its potential impact on fuel efficiency.
Horsepower (hp):
The other predictor variable in the regression model, suggesting its role in determining a car's fuel efficiency.
These variables are pivotal in analyzing the trade-offs between power (as indicated by horsepower) and fuel efficiency, as well as how the rear axle ratio might influence these aspects. The dataset contains 32 rows and 12 columns, encompassing these critical variables among others.
3.
Multiple Regression Model
Correlation Analysis 2
Based on the scatter plots provided, we can see that both rear axle ratio (drat) and horsepower (hp) play significant roles in determining fuel efficiency (mpg). The scatter plot depicting fuel efficiency against rear axle ratio illustrates a positive relationship between the two variables, suggesting that as the rear axle ratio increases, so does fuel efficiency. However, the distribution of points does not exhibit a perfect linear pattern, indicating that while rear axle ratio influences fuel efficiency, it is not the only thing determining it. Variability in mpg across different rear axle ratios implies the presence of other unaccounted variables affecting fuel efficiency. Same thing with the scatter plot of fuel efficiency against horsepower which reveals a negative correlation between the variables. In this case higher horsepower seems to be associated with lower fuel efficiency. The spread of data points increases as horsepower rises, suggesting the influence of other factors alongside horsepower on fuel efficiency. The decrease in fuel efficiency with increased horsepower can be attributed to the higher fuel consumption of more powerful engines. Just looking at the scatter plots we can tell that both rear axle ratio and horsepower are important factors in determining fuel efficiency. The Pearson Correlation Coefficient between fuel efficiency (MPG) and horsepower (HP) is -0.776 and between MPG and rear axle ratio (DRAT) is 0.681. The negative value of -0.776 for fuel efficiency and horsepower indicates an inverse relationship so as horsepower increases fuel efficiency will decrease. The positive value of 0.681 for fuel efficiency and rear axle ratio suggests a direct relationship, as the rear axle ratio increases so does fuel efficiency. In this case both correlations are relatively strong indicating significant relationships.
Reporting Results 3
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
The analysis we conducted on the data to estimate how fuel efficiency (mpg) is influenced by rear axle ratio (drat) and horsepower (hp) provided us with valuable insights. Using a multiple regression model, represented by the equation , where is the intercept, and are the coefficients for the predictors, and is the error term we got some specific predictions. The specific prediction equation derived from the analysis is This equation indicates that an increase in rear axle ratio leads to an increase in fuel efficiency, whereas an increase in horsepower leads to a decrease in fuel efficiency. The model’s value is 0.7412 and the adjusted value is 0.7233. These values suggest that approximately 74.12% of the variance in mpg is explained by the model. The Adjusted considers how many things we're predicting at once. It gives us a good idea of how well our model fits the data. Even with this adjustment, it shows that we're still explaining a big chunk of the differences in fuel efficiency. The beta estimate for drat (4.6982) suggests a positive relationship with fuel efficiency, indicating that vehicles with higher rear axle ratios are generally more fuel-efficient. On the other hand, the beta estimate for hp (-0.0518) indicates an inverse relationship with fuel efficiency, suggesting that higher horsepower is associated with lower fuel efficiency.
Fitted values are the model's predictions for the response variable, while residuals are the differences between observed and fitted values, representing the error in predictions. The analysis of fitted values and residuals through a Residuals against Fitted Values plot and a Normal Q-Q plot provides insight into the assumptions of homoscedasticity and normality. The Residuals vs. Fitted Values plot did not reveal any clear patterns that violate the assumption of homoscedasticity, suggesting that the error variance is fairly constant across the range of predictions. The Normal Q-Q Plot showed that residuals are approximately normally distributed, with slight deviations at the extremes, indicating a minor departure from normality.
In conclusion, the regression model provides a good fit for predicting fuel efficiency based on rear axle ratio and horsepower, with both predictors significantly influencing mpg. The assumptions of homoscedasticity and normality of residuals are not severely violated, suggesting that the linear regression model is reasonably appropriate for this data. 4
Evaluating Model Significance When we looked at how much rear axle ratio('drat') and horsepower ('hp') affect fuel efficiency, we discovered that both really matter. The p-values, which are like the odds that their effects are just random,
were low – around 0.05 for 'drat' and even lower for 'hp'. This means there's strong evidence suggesting that both 'drat' and 'hp' seriously impact fuel efficiency. It seems like 'drat' might boost fuel efficiency, while 'hp' tends to drag it down. The overall model we used to predict fuel efficiency is also good, with an
F-statistic of about 41.52 and a p-value less than 0.05. This indicates that our model isn't just a lucky guess and that the factors we chose really do explain the differences in fuel efficiency. Plus, when we looked at the range of values for the coefficients of 'drat' and 'hp' – from roughly 2.261 to 7.135 for 'drat' and from about -0.0708 to -0.0328 for 'hp' – we found that they're reliable. This means we can be about 95% confident that 'drat' and 'hp' have the effects we observed. Overall, this thorough analysis shows that 'drat' and 'hp' are both crucial factors affecting fuel efficiency, and our model gives us a solid understanding of how they work together.
Making Predictions Using the Model
When we looked at how important the predictors rear axle ratio (drat) and horsepower (hp) are for fuel efficiency in our regression model, we found they both really matter. Both were statistically significant at the 5% level, which means they're not just by chance. Drat seems to boost fuel efficiency, while hp seems
to bring it down. The overall importance of our model was confirmed by a test called an F-test, which showed that our model does a good job of explaining the differences in fuel efficiency.
For example, if we use our model to predict the fuel efficiency for a car with a drat of 3.15 and hp of 120, we estimate it would get around 19.37 mpg. But the actual mpg was 20.5, leaving us with a difference, or residual, of about 1.13. To understand how much our predictions might vary, we calculated a 95% prediction interval, which suggests that for similar cars, the mpg could be anywhere from 12.64 to 26.10. This shows the range of possible outcomes.
Comparing this to the 95% confidence interval for the average fuel efficiency of similar cars, which is narrower, ranging from 17.57 to 21.18 mpg, we see the difference between predicting individual outcomes and estimating the average for a group.
This difference between prediction and confidence intervals tells us something important. Prediction intervals are wider because they consider not only the variability in individual observations but also the uncertainty in estimating the average. Confidence intervals, on the other hand, only deal with the uncertainty around the average estimate. So, prediction intervals give us a broader range of potential outcomes, reflecting both the natural variation in the data and the accuracy of our model's predictions. This thorough analysis emphasizes how much drat and hp affect fuel efficiency and reminds us of the uncertainties we face when making predictions, whether for a single case or for a whole group.
4.
Conclusion 5
Based on the analysis conducted and assuming we have a large enough sample size, I would suggest using this model with some caution. The model shows statistical significance, which means it's not likely due to chance, indicated by low p-values for both the overall F-test and individual predictors like rear axle
ratio and horsepower. It also has good predictive power, with an value suggesting that a big chunk of the differences in fuel efficiency can be explained by these predictors. The coefficients for rear axle ratio (drat) and horsepower (hp) are both statistically significant, showing that they have a meaningful relationship with fuel efficiency. A positive coefficient for drat means cars with higher rear axle ratios tend to be more fuel-efficient, while a negative coefficient for hp means that more horsepower is linked to
lower fuel efficiency. These results show that the model fits the data well. The and adjusted values tell us how much of the variation in fuel efficiency our model can explain, and they're pretty high, suggesting our model is doing a good job. The significance of the model, indicated by the F-test, and the significance of individual predictors suggest that the relationships we found are reliable. The confidence and prediction intervals show us how uncertain our predictions are, with wider prediction intervals indicating more variability in individual observations around the predicted average.
The practical importance of these analyses is in helping with decisions about vehicle design and consumer
choices. By understanding how rear axle ratio, horsepower, and fuel efficiency are related, car makers can
design vehicles that use these factors to make cars more fuel-efficient. Consumers can also use this information when choosing a car, looking for features that make a car more fuel-efficient. The model's ability to predict fuel efficiency based on specific car characteristics can be really useful in the automotive industry for making cars that use less fuel and for marketing to consumers who care about fuel economy.
But there are some things to keep in mind. Other variables that we didn't include in the model might also affect fuel efficiency. And even though our model seems to fit well, there are still some differences between our predictions and the actual fuel efficiency, which suggests there might be other factors we're not considering. So while this model gives us some good insights, it's important to look at other factors too to get a complete picture of what affects fuel efficiency.
6
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help