MAT 303 Module One Problem Set Report Template

MAT 303 Module One Problem Set Report Multiple Regression Brian Tynan Brian.Tynan@snhu.edu Southern New Hampshire University

Note: Replace the bracketed text on page one (the cover page) with your personal information. 1. Introduction The statistical analyses that is being completed and reviewed in this case is being done to establish the relationship between a vehicle’s weight and the horsepower of a vehicle and the effects that it has on the fuel efficiency of said vehicle. There are 32 different vehicles that are being used for this data set that was obtained from the car market to analyze the fuel economy of these specific vehicles. The information that is obtained can be utilized to assist the car market with finding a good balance between the weight and horsepower of a vehicle while providing a vehicle that is fuel efficient. One of the things that they could do would be to shave off some of the weight from the vehicle while adding some additional horsepower to the vehicle while maintaining the same level of fuel efficiency. Or they can lower the horsepower and add some weight and or they can lower both to develop an even more fuel- efficient vehicle. The type of analysis that is being run for this problem set is being done to create a multiple regression model that is analyzing the fuel economy verses weight and fuel economy verses horsepower. This information will provide data that will allow us to find the fitted values and residuals that can be plotted against one and that can be used to generate a q-q plot that can be used to test the assumptions of normality of the residuals that can than be used to fine the confidence interval. This data can be used to determine if the weight and horsepower of a said vehicle has a statistical significant effect on the fuel economy of the vehicle. 2. Data Preparation When comparing the weight and horsepower to fuel economy the following variable are considered the important variables that are being used for the data set: weight (wt in the code), horsepower (hp in the code), quarter mile time (qsec in the code), and fuel economy in miles (US) gallon (mpg in the code). The study for this problem set will focus on the weight, horsepower, and the fuel economy or efficiency in miles. The data set consists of 12 columns which is one column for each variable and there are 32 rows which is for each vehicle being evaluated. 3. Multiple Regression Model Correlation Analysis The scatterplot below illustrates the correlation between the fuel economy of a vehicle and its weight. As you can see in the scatterplot, the weight increases the fuel economy or efficiency decreases which is considered a negative correlation. A strong negative correlation is indicated with the below scatterplot and the Pearson’s product moment correlation of -0.8677. 2

Reporting Results E(Y )=b0+b1 X1+b2 X2 E(Y )=37.22727−3.87783 X1−0.03177 X2 The R-squared value for this model is 0.8268. This model shows the total variance in miles per gallon illustrated by the multiple regression model with the weight and horsepower of the vehicles with the predictors of 82.7%. The adjusted R-squared value is adjustment of the R-squared value which allows for alternative models of the same or similar response variable to be compared. The beta estimates in this data set each represent the slope in respect to the predictor variable of the multiple regression model. This model shows that the beta estimate for weight is b1=-3.87783 and the beta estimate for the horsepower as b2=-0.03177. This shows that as the predictor variable increases the response variable decreases by the value of the beta. A model’s prediction of the mean response variable when the input values for the predictor variable are entered provides the fitted value. The estimated regression error that is based on the sample multiple regression function is the residual. 4

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H0 : Bi=0 fori=1,2 Ha :Bi ≠ 0 The P-value for the variable weight is shows as 1.12e-06 and the P-value of the variable for horsepower is 0.00145. These values show a significantly smaller value than the given level of significance of 5%. With this data we should reject the null hypothesis and accept the alternative hypothesis that analyzes the individual variables for the weight and horsepower which shows a statistical significant relationship in correlation with the fuel economy variable. Making Predictions Using the Model The equation used to predict fuel efficiency for a car that has a weight of 2.95 and a horsepower of 179 is as follows: ^ Y 1 = b 0 + b 1 X 1 i + b 2 X 2 i ^ Y 1 = 37.22727 − 3.87783 ( 2.95 ) − 0.03177 ( 179 ) ^ Y 1 = 20.1003 If the car being analyzed achieves an average of 22.7 miles per gallon, our sample observation would be denoted by Y = 22.7 To find the residual data, we would use the following equation e i = Y i − ^ Y i . So e i = 22.7 − 20.1003 = 2.5997 . The residual for this data is 2.5997. The data shows that 95% prediction interval for the individual response for the fuel economy for this car is as follows: ( 14.6450 , 25.5556 ) . The prediction interval for the individual response provides us with a 95% certainty that a car’s fuel economy (in miles per gallon) will lie between the lower and upper bounds as long as the car’s weight is 2.95 and its horsepower is 179. The data is taking into consideration the nondefinitive data related to the variations regarding the Y based on the regression error, ϵ ,along with the data sampling that is not certain related to estimating the regression parameters . Considering regression error as well as the sampling uncertainty related to estimating the regression parameters is the main reason the prediction interval for the individual response is wider than the confidence interval for this mean. The 95% Confidence interval for the fuel economy for this car is shown as ( 18.8249 , 21.3758 ) . This confidence interval for this mean lets us know that we can be 95% certain that the average fuel economy (in miles per gallon) for this group of cars should be within the lower and upper bounds; as long as the weight is 2.95 and the horsepower is 179. 4. Conclusion 7

I would recommend the use of this model based on the analysis performed as it shows accurately that weight and horsepower has a significant impact on the fuel economy of efficiency of a vehicle. The heavier that the vehicle is the less fuel efficient it will be and the same can be said for the horsepower. The more horsepower the vehicle has the lower the fuel economy will be. The data shows that there is a strong correlation between the weight and horsepower of a vehicle versus the fuel economy or efficiency. This data provides the car makers with information that would assist them with predicting the fuel economy or efficiency based on the weight and horsepower of a vehicle. This is important because this will provide them with the opportunity to find a balance between the weight and the horsepower of a vehicle to offer a more fuel-efficient vehicle. This could be beneficial as this may increase the sales of their vehicles based on the fuel efficiency offered by the cars compared to their competitors. 8