The U.S. Department of Energy’s Fuel Economy Guide provides fuel efficiency data

Question

The U.S. Department of Energy’s Fuel Economy Guide provides fuel efficiency data for cars and trucks. A portion of the data for 311 compact, midsized, and large cars is contained in the file fueldata.xlsx which you can download from the class D2L shell. The dataset contains the following variables: Class identifies the size of the car; Compact, Midsize, or Large Displacement = engine displacement in liters FuelType shows whether the car uses premium (P) or regular (R) fuel HwyMPG = fuel efficiency rating for highway driving in miles per gallon a. Develop an estimated regression equation to predict the fuel efficiency for highway driving given the engine’s displacement. Plot the residuals from this regression to see if you see any pattern. Conduct a hypothesis test to determine whether the coefficient on Displacement is statistically significantly different from 0 using the 0.05 level of significance. How much of the variation in the sample values of HwyMPG does this estimated regression equation explain? b. Create a scatter chart with HwyMPG on the y-axis and displacement on the x-axis for which the points representing compact, midsize, and large automobiles are shown in different shapes and/or colors. What does this chart suggest about the relationship between the class of automobile (compact, midsize, and large) and HwyMPG? c. Add dummy variables ClassMidsize and ClassLarge to the simple linear regression model in part a. The value of ClassMidsize is 1 if the car is a midsize car and 0 otherwise; the value of ClassLarge is 1 if the car is a large car and 0 otherwise. Thus, for a compact car, the value of ClassMidsize and the value of ClassLarge are both 0. Estimate the new regression equation. What happened to the adjusted R2 when these variables were added to the regression? What does this tell you? d. Use significance level of 0.05 to determine whether the dummy variables added to the model in part c are significant. e. Finally add the dummy variable FuelPremium, (which is 1 if the car uses premium fuel and 0 if the car uses regular fuel) to the regression. What happened to the adjusted R2 when these variables were added to the regression? What does this tell you? f. For the estimated regression equation developed in part e, test for the significance of the overall regression relationship (using the F-statistic) and for each of the independent variables the 0.05 level of significance for each test. Please include your regression results with your answers (and submit the excel file to the D2L dropbox for the Unit 7 Homework)

Return to the ‘cars.xlsx’ dataset that was used in the Unit 6 Homework. Recall that this dataset includes data on the 0-60 time (TIME), top speed (SPEED), curb weight (WEIGHT), and horsepower (HP) of 30 automobiles. There are good reasons based in physics to believe that the relationship between horsepower and 0-60 time is non-linear. Look again at the scatter plot of HP (x-axis) and 0-60 time (y-axis) from the Unit 6 Homework. To test the goodness of fit of one possible non-linear relationship estimate the following quadratic regression: TIMEi = 0 + 1HPi + 2HPi2 + Please include your regression results with your answers (and submit the excel file to the D2L dropbox for the Unit 7 Homework). Compare the results from this regression to those from the Unit 6 Homework. Which do you prefer and why?

2. Use the "BEER" data set to answer the following question. (Also on the course D2L site). The data set includes 30 observations on beer consumption and related data. The variables included are as follows: q = quantity of beer purchased (in liters) pB = price of beer (in dollars) pL = price of other liquor (in dollars) pR = price of remaining (non-alcoholic) goods & services (in dollars) m = income (in dollars) a. What sign do you expect for each coefficient and why? b. Estimate a regression to explain the quantity of beer purchased as a function of the price, price of liquor, price of remaining goods, and income. Please include your regression results with your answers (and submit the excel file to the D2L dropbox for the Unit 7 Homework). c. Interpret the parameters. d. Are any of the regressors individually significant in explaining the quantity of beer purchased? If so, at what level? e. Suppose that beer industry representatives hypothesize that the marginal propensity to buy beer out of an additional dollar of income is $0.01. Conduct a hypothesis test to determine if there is any validity to their conjecture. Use = 1%. f. Conduct a hypothesis test to determine if the regressors are jointly significant in explaining the quantity of beer purchased. Be sure to state the null and alternative hypotheses formally.

Car Class #NAME? Fuel Type Hwy MPG 1 Compact 3.1 P 25 2 Compact 3.1 P 25 3 Compact 3 P 25 4 Compact 3 P 25 5 Compact 3 P 25 6 Compact 3 P 25 7 Compact 2.4 P 25 8 Compact 3.5 P 25 9 Compact 3 P 25 10 Compact 2.4 P 24 11 Compact 2.8 P 24 12 Compact 2.5 P 24 13 Compact 3 P 24 14 Compact 2.5 P 24 15 Compact 2.4 R 24 16 Compact 5.3 P 24 17 Compact 6 P 24 18 Compact 3.5 R 24 19 Compact 3.5 R 24 20 Compact 3.5 R 24 21 Compact 3.5 R 24 22 Compact 3.5 R 24 23 Compact 3.8 R 24 24 Compact 3.5 R 24 25 Compact 3.5 P 24 26 Compact 5.3 P 24 27 Compact 4.6 P 24 28 Compact 3.5 P 24 29 Compact 4.2 P 24 30 Compact 4.6 P 24 31 Compact 4.6 P 24 32 Compact 4.6 P 24 33 Compact 3.5 P 24 34 Compact 3.6 P 24 35 Compact 3.2 P 24 36 Compact 3.2 P 24 37 Compact 3.5 R 24 38 Compact 2.4 R 24 39 Compact 3.5 R 24 40 Compact 3.5 R 24 41 Compact 3.2 P 23 42 Compact 4.2 P 23 43 Compact 4.8 P 23 44 Compact 4.8 P 23 45 Compact 3.5 P 23 46 Compact 4.6 R 23 47 Compact 4.6 R 23 48 Compact 5.7 R 23 49 Compact 5.7 R 23 50 Compact 4.6 R 23 51 Compact 4.6 R 23 52 Compact 4.2 P 23 53 Compact 4.2 P 23 54 Compact 4.8 P 23 55 Compact 3.5 P 23 56 Compact 3 P 23 57 Compact 4.4 P 23 58 Compact 3 P 22 59 Compact 3.9 R 22 60 Compact 3.5 P 22 61 Compact 4.2 P 22 62 Compact 4.2 P 22 63 Compact 4.6 R 22 64 Compact 4.6 R 22 65 Compact 3.5 R 22 66 Compact 5.7 R 22 67 Compact 5.7 R 22 68 Compact 3.5 R 22 69 Compact 4.8 P 22 70 Compact 3 P 22 71 Compact 4.2 P 22 72 Compact 5 P 22 73 Compact 3.5 P 22 74 Compact 5.5 P 22 75 Compact 4.2 P 21 76 Compact 5.5 P 21 77 Compact 5.5 P 21 78 Compact 4.5 P 21 79 Compact 5.5 P 21 80 Compact 4.6 P 21 81 Compact 4.2 P 20 82 Compact 4.2 P 20 83 Compact 6 P 20 84 Compact 4.5 P 20 85 Compact 5.5 P 20 86 Compact 6 P 19 87 Compact 5.2 P 19 88 Compact 5.2 P 19 89 Compact 4.4 P 19 90 Compact 5.5 P 19 91 Compact 6.2 P 19 92 Compact 6.2 P 18 93 Compact 6.2 P 18 94 Compact 6.1 P 18 95 Compact 6.1 P 18 96 Compact 4.6 R 18 97 Compact 4.6 R 18 98 Compact 6.1 P 18 99 Compact 5.5 P 17 100 Compact 5.5 P 17 101 Compact 6.2 P 17 102 Compact 6 P 17 103 Compact 5.5 P 17 104 Compact 5 P 17 105 Compact 5 P 17 106 Compact 5.5 P 16 107 Compact 6 P 16 108 Compact 5.5 P 16 109 Compact 6 P 16 110 Compact 5.7 P 16 111 Compact 5.7 P 15 112 Midsize 1.8 R 37 113 Midsize 2 R 35 114 Midsize 1.6 R 35 115 Midsize 1.8 R 35 116 Midsize 1.6 R 34 117 Midsize 2.4 R 34 118 Midsize 2 R 33 119 Midsize 1.6 R 33 120 Midsize 2 R 33 121 Midsize 2 R 33 122 Midsize 1.8 R 33 123 Midsize 2.5 R 33 124 Midsize 2 R 33 125 Midsize 1.6 R 32 126 Midsize 1.6 R 32 127 Midsize 1.6 R 32 128 Midsize 2 R 32 129 Midsize 2.4 R 32 130 Midsize 2 R 32 131 Midsize 1.8 R 32 132 Midsize 2.5 R 32 133 Midsize 2.4 R 32 134 Midsize 2 P 31 135 Midsize 2.4 R 31 136 Midsize 2 R 31 137 Midsize 2 R 31 138 Midsize 2.4 R 31 139 Midsize 2.4 R 31 140 Midsize 2.4 R 31 141 Midsize 2.4 R 31 142 Midsize 2.4 R 31 143 Midsize 2.2 R 31 144 Midsize 2.4 R 31 145 Midsize 2.4 R 31 146 Midsize 2.5 R 31 147 Midsize 1.8 R 31 148 Midsize 2 R 31 149 Midsize 2.4 R 31 150 Midsize 2.4 R 31 151 Midsize 2 P 30 152 Midsize 2.4 R 30 153 Midsize 2.4 R 30 154 Midsize 2 R 30 155 Midsize 2.4 R 30 156 Midsize 2.4 R 30 157 Midsize 2.4 R 30 158 Midsize 2.4 R 30 159 Midsize 2 R 30 160 Midsize 2.5 R 30 161 Midsize 2.4 R 30 162 Midsize 2 P 29 163 Midsize 2 P 29 164 Midsize 2 P 29 165 Midsize 2 P 29 166 Midsize 2 P 29 167 Midsize 2.3 R 29 168 Midsize 2.3 R 29 169 Midsize 2 R 29 170 Midsize 2 R 29 171 Midsize 3.5 R 29 172 Midsize 2.5 R 29 173 Midsize 2.5 R 29 174 Midsize 2.5 R 29 175 Midsize 2.5 R 29 176 Midsize 3.5 R 29 177 Midsize 3.5 R 29 178 Midsize 3.5 R 29 179 Midsize 2.5 P 29 180 Midsize 2 P 29 181 Midsize 3.5 R 29 182 Midsize 1.8 R 29 183 Midsize 2.4 R 29 184 Midsize 2.3 R 29 185 Midsize 2.3 R 29 186 Midsize 2.3 R 29 187 Midsize 3.5 R 29 188 Midsize 2.4 P 28 189 Midsize 2.4 P 28 190 Midsize 2 P 28 191 Midsize 3 P 28 192 Midsize 3 P 28 193 Midsize 2.5 P 28 194 Midsize 2.5 P 28 195 Midsize 2.4 P 28 196 Midsize 2.4 P 28 197 Midsize 2.4 P 28 198 Midsize 2.4 P 28 199 Midsize 3.5 R 28 200 Midsize 2 R 28 201 Midsize 2 R 28 202 Midsize 2 R 28 203 Midsize 2 R 28 204 Midsize 3.9 R 28 205 Midsize 3.5 R 28 206 Midsize 3.3 R 28 207 Midsize 3.5 R 28 208 Midsize 3.5 R 28 209 Midsize 3 P 28 210 Midsize 2.3 P 28 211 Midsize 2 P 28 212 Midsize 3.8 R 28 213 Midsize 3.5 R 28 214 Midsize 2.3 R 28 215 Midsize 2.7 R 28 216 Midsize 2.3 R 28 217 Midsize 3.5 R 28 218 Midsize 2.3 R 28 219 Midsize 3.8 R 28 220 Midsize 3.5 R 28 221 Midsize 2 P 27 222 Midsize 3.1 P 27 223 Midsize 2.5 P 27 224 Midsize 2.5 P 27 225 Midsize 2.5 P 27 226 Midsize 2.5 P 27 227 Midsize 2.5 R 27 228 Midsize 2.5 R 27 229 Midsize 2.5 R 27 230 Midsize 2.5 R 27 231 Midsize 3.3 R 27 232 Midsize 3.5 P 27 233 Midsize 3.1 P 27 234 Midsize 3 P 27 235 Midsize 3.5 P 27 236 Midsize 3.5 P 27 237 Midsize 3.5 P 27 238 Midsize 2.7 R 27 239 Midsize 2.7 R 27 240 Midsize 2.7 R 27 241 Midsize 2 R 27 242 Midsize 2.7 R 27 243 Midsize 2.4 R 27 244 Midsize 3 R 26 245 Midsize 3 R 26 246 Midsize 3.6 R 26 247 Large 3 P 26 248 Large 3 P 26 249 Large 3 P 26 250 Large 2 P 26 251 Large 2.8 P 26 252 Large 2.5 P 26 253 Large 2.5 P 26 254 Large 2.5 P 26 255 Large 2.4 P 26 256 Large 2.4 P 26 257 Large 2.4 R 26 258 Large 3.5 R 26 259 Large 3.6 R 26 260 Large 3.3 R 26 261 Large 2.7 R 26 262 Large 2.7 R 26 263 Large 3.8 R 26 264 Large 3.3 R 26 265 Large 3.2 P 26 266 Large 3.5 P 26 267 Large 3 P 26 268 Large 3 P 26 269 Large 3 P 26 270 Large 2.3 P 26 271 Large 3.5 P 26 272 Large 2.3 P 26 273 Large 3.6 P 26 274 Large 3.6 R 26 275 Large 3.6 R 26 276 Large 3.6 R 26 277 Large 3.6 R 26 278 Large 3.6 R 26 279 Large 3.6 R 26 280 Large 3.5 R 26 281 Large 3.5 R 26 282 Large 3 P 25 283 Large 3.5 P 25 284 Large 2.5 P 25 285 Large 2.5 P 25 286 Large 2.5 P 25 287 Large 2.5 P 25 288 Large 3.5 R 25 289 Large 4.2 P 25 290 Large 4.2 P 25 291 Large 4.2 P 25 292 Large 3.8 R 25 293 Large 3.1 P 25 294 Large 3 P 25 295 Large 3 P 25 296 Large 3 P 25 297 Large 3 P 25 298 Large 3.5 P 25 299 Large 3.5 P 25 300 Large 3.8 P 25 301 Large 3.5 P 25 302 Large 5.3 P 25 303 Large 3.6 R 25 304 Large 3.6 R 25 305 Large 3.6 R 25 306 Large 3.6 R 25 307 Large 2.4 R 25 308 Large 3 R 25 309 Large 3 R 25 310 Large 3 R 25 311 Large 3 R 25

Explanation / Answer

The regression equation to estimate fuel efficiency given an engine's displacement is given as follows:

y = 9.541331 - 0.240955*engine's displacement.

After plotting the residuals from the regression model, no such pattern is observed. The residuals are evenly scattered along the x axis.

A t test has been conducted to see if the regression coefficient of Hw.mpg is significantly different from 0. The p value of the test obtained is less than 0.001. Therefore, the regression co efficient is significant at 0.05 level of significance.

The adjusted R-squared value of the test is 0.6935. Therefore, 69 percent of the variation has been explained by the regression line.

A scatter plot has been created using MPG along y axis and engine displacement along x axis for three class of cars. The graph shows a negetive sloping line. Therefore, the relationship between the MPG and engine displacement is negative. The large sized cars has a high value of engine displacement and MPG followed by medium sized caes and small sized cars.

The regression equation obtained after adding dummy variables to the class mid size and class large is given below:

y = 9.31673 -0.22683*Hwy.MPG -0.33472*ClassLarge -0.16258*ClassMidSize

The adjusted R-squared for this regression equation is 0.7062. The adjusted R-squared value has been increased which indicates that the model has been well fitted after inclusion of dummy variables.

The p value of regression co efficient obtained for class mid sized is 0.28075. This indicates that the regression co efficient for the class mid sized is not significantly different from 0. The regression co efficient obtained for the class Large is significantly different from 0 as the p-value obtained for this class is 0.0469.

The dummy variable Fuel Type has been included in the model. The regression equation obtained after the inclusion of dummy variable is given as follows:

y = 9.51769 -0.23852*Hwy.MPG -0.32811*ClassLarge -0.18260-ClassMidsize +0.21901*Fuel.Type.

The adjusted R-squared for the regression model is 0.7062. The adjusted R squared in the previous model is 0.7005. Therefore, the adjusted R-squared value has been increased but only by a small extent. Therefore, there is a change on the inclusion of a new variable.

2. The scatter plot shows that the relationship between horsepower and time is not linear. There has been an increase in HP at time 7.5. Therefore, there is outlier present in the dataset.

The following quadratic equation has been fitted to the dataset.

Timei = 9.537395 -0.01547*HPi +.0000872*HPi^2

3. The sign of the regression co efficient for income, price of beer, price of other liquor and price of other alcoholic goods can be shown from the scatter plot.

The price of beer will have a negetive sign of regression coefficent. The price of liquor will have a positive sign. The price of other alcoholic goods will have a positive sign. The income will also have a positive sign.

The fitted regression equation is given below:

y = 82.15871 - 23.7426*pB -4.07741*pL +12.92434*pR+0.001995*m

All the regression coefficent are significant at 1% level of significance.

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