MATH 239 HW #4 and #5
Claire Verstrate
10/11/2019
Problem 1: Complete the tables below. Use the relationships between the output tables that we discussed in class. There were 50 observations in the dataset. There are eight missing values in the summary output and four missing values in the anova table.
# Intercept t-value: -17.5791/6.7584 = -2.60107421875
# Speed t-value: 3.9324/0.4155 = 9.46426
# Speed p-value: <2e-16
# Residual standard error: 15.38 on 49 degrees of freedom
# Multiple R squared: 0.6742
# F-statistic: 49 on 1 and 48DF;
# p-value: <2e-16
#I would like to review this more because a lot of this is still confusing.Problem 2 (textbook 3.8 ) This question involves the use of simple linear regression on the Auto data set from the textbook website.
2(a) Use the lm() function to perform a simple linear regression with mpg as the response and horsepower as the predictor. Use the summary() function to print the results. Comment on the output. Answer the following:
auto<-read.csv("Auto.csv",
header=TRUE,
na.strings = "?")
na.omit(auto)## mpg cylinders displacement horsepower weight acceleration year origin
## 1 18.0 8 307.0 130 3504 12.0 70 1
## 2 15.0 8 350.0 165 3693 11.5 70 1
## 3 18.0 8 318.0 150 3436 11.0 70 1
## 4 16.0 8 304.0 150 3433 12.0 70 1
## 5 17.0 8 302.0 140 3449 10.5 70 1
## 6 15.0 8 429.0 198 4341 10.0 70 1
## 7 14.0 8 454.0 220 4354 9.0 70 1
## 8 14.0 8 440.0 215 4312 8.5 70 1
## 9 14.0 8 455.0 225 4425 10.0 70 1
## 10 15.0 8 390.0 190 3850 8.5 70 1
## 11 15.0 8 383.0 170 3563 10.0 70 1
## 12 14.0 8 340.0 160 3609 8.0 70 1
## 13 15.0 8 400.0 150 3761 9.5 70 1
## 14 14.0 8 455.0 225 3086 10.0 70 1
## 15 24.0 4 113.0 95 2372 15.0 70 3
## 16 22.0 6 198.0 95 2833 15.5 70 1
## 17 18.0 6 199.0 97 2774 15.5 70 1
## 18 21.0 6 200.0 85 2587 16.0 70 1
## 19 27.0 4 97.0 88 2130 14.5 70 3
## 20 26.0 4 97.0 46 1835 20.5 70 2
## 21 25.0 4 110.0 87 2672 17.5 70 2
## 22 24.0 4 107.0 90 2430 14.5 70 2
## 23 25.0 4 104.0 95 2375 17.5 70 2
## 24 26.0 4 121.0 113 2234 12.5 70 2
## 25 21.0 6 199.0 90 2648 15.0 70 1
## 26 10.0 8 360.0 215 4615 14.0 70 1
## 27 10.0 8 307.0 200 4376 15.0 70 1
## 28 11.0 8 318.0 210 4382 13.5 70 1
## 29 9.0 8 304.0 193 4732 18.5 70 1
## 30 27.0 4 97.0 88 2130 14.5 71 3
## 31 28.0 4 140.0 90 2264 15.5 71 1
## 32 25.0 4 113.0 95 2228 14.0 71 3
## 34 19.0 6 232.0 100 2634 13.0 71 1
## 35 16.0 6 225.0 105 3439 15.5 71 1
## 36 17.0 6 250.0 100 3329 15.5 71 1
## 37 19.0 6 250.0 88 3302 15.5 71 1
## 38 18.0 6 232.0 100 3288 15.5 71 1
## 39 14.0 8 350.0 165 4209 12.0 71 1
## 40 14.0 8 400.0 175 4464 11.5 71 1
## 41 14.0 8 351.0 153 4154 13.5 71 1
## 42 14.0 8 318.0 150 4096 13.0 71 1
## 43 12.0 8 383.0 180 4955 11.5 71 1
## 44 13.0 8 400.0 170 4746 12.0 71 1
## 45 13.0 8 400.0 175 5140 12.0 71 1
## 46 18.0 6 258.0 110 2962 13.5 71 1
## 47 22.0 4 140.0 72 2408 19.0 71 1
## 48 19.0 6 250.0 100 3282 15.0 71 1
## 49 18.0 6 250.0 88 3139 14.5 71 1
## 50 23.0 4 122.0 86 2220 14.0 71 1
## 51 28.0 4 116.0 90 2123 14.0 71 2
## 52 30.0 4 79.0 70 2074 19.5 71 2
## 53 30.0 4 88.0 76 2065 14.5 71 2
## 54 31.0 4 71.0 65 1773 19.0 71 3
## 55 35.0 4 72.0 69 1613 18.0 71 3
## 56 27.0 4 97.0 60 1834 19.0 71 2
## 57 26.0 4 91.0 70 1955 20.5 71 1
## 58 24.0 4 113.0 95 2278 15.5 72 3
## 59 25.0 4 97.5 80 2126 17.0 72 1
## 60 23.0 4 97.0 54 2254 23.5 72 2
## 61 20.0 4 140.0 90 2408 19.5 72 1
## 62 21.0 4 122.0 86 2226 16.5 72 1
## 63 13.0 8 350.0 165 4274 12.0 72 1
## 64 14.0 8 400.0 175 4385 12.0 72 1
## 65 15.0 8 318.0 150 4135 13.5 72 1
## 66 14.0 8 351.0 153 4129 13.0 72 1
## 67 17.0 8 304.0 150 3672 11.5 72 1
## 68 11.0 8 429.0 208 4633 11.0 72 1
## 69 13.0 8 350.0 155 4502 13.5 72 1
## 70 12.0 8 350.0 160 4456 13.5 72 1
## 71 13.0 8 400.0 190 4422 12.5 72 1
## 72 19.0 3 70.0 97 2330 13.5 72 3
## 73 15.0 8 304.0 150 3892 12.5 72 1
## 74 13.0 8 307.0 130 4098 14.0 72 1
## 75 13.0 8 302.0 140 4294 16.0 72 1
## 76 14.0 8 318.0 150 4077 14.0 72 1
## 77 18.0 4 121.0 112 2933 14.5 72 2
## 78 22.0 4 121.0 76 2511 18.0 72 2
## 79 21.0 4 120.0 87 2979 19.5 72 2
## 80 26.0 4 96.0 69 2189 18.0 72 2
## 81 22.0 4 122.0 86 2395 16.0 72 1
## 82 28.0 4 97.0 92 2288 17.0 72 3
## 83 23.0 4 120.0 97 2506 14.5 72 3
## 84 28.0 4 98.0 80 2164 15.0 72 1
## 85 27.0 4 97.0 88 2100 16.5 72 3
## 86 13.0 8 350.0 175 4100 13.0 73 1
## 87 14.0 8 304.0 150 3672 11.5 73 1
## 88 13.0 8 350.0 145 3988 13.0 73 1
## 89 14.0 8 302.0 137 4042 14.5 73 1
## 90 15.0 8 318.0 150 3777 12.5 73 1
## 91 12.0 8 429.0 198 4952 11.5 73 1
## 92 13.0 8 400.0 150 4464 12.0 73 1
## 93 13.0 8 351.0 158 4363 13.0 73 1
## 94 14.0 8 318.0 150 4237 14.5 73 1
## 95 13.0 8 440.0 215 4735 11.0 73 1
## 96 12.0 8 455.0 225 4951 11.0 73 1
## 97 13.0 8 360.0 175 3821 11.0 73 1
## 98 18.0 6 225.0 105 3121 16.5 73 1
## 99 16.0 6 250.0 100 3278 18.0 73 1
## 100 18.0 6 232.0 100 2945 16.0 73 1
## 101 18.0 6 250.0 88 3021 16.5 73 1
## 102 23.0 6 198.0 95 2904 16.0 73 1
## 103 26.0 4 97.0 46 1950 21.0 73 2
## 104 11.0 8 400.0 150 4997 14.0 73 1
## 105 12.0 8 400.0 167 4906 12.5 73 1
## 106 13.0 8 360.0 170 4654 13.0 73 1
## 107 12.0 8 350.0 180 4499 12.5 73 1
## 108 18.0 6 232.0 100 2789 15.0 73 1
## 109 20.0 4 97.0 88 2279 19.0 73 3
## 110 21.0 4 140.0 72 2401 19.5 73 1
## 111 22.0 4 108.0 94 2379 16.5 73 3
## 112 18.0 3 70.0 90 2124 13.5 73 3
## 113 19.0 4 122.0 85 2310 18.5 73 1
## 114 21.0 6 155.0 107 2472 14.0 73 1
## 115 26.0 4 98.0 90 2265 15.5 73 2
## 116 15.0 8 350.0 145 4082 13.0 73 1
## 117 16.0 8 400.0 230 4278 9.5 73 1
## 118 29.0 4 68.0 49 1867 19.5 73 2
## 119 24.0 4 116.0 75 2158 15.5 73 2
## 120 20.0 4 114.0 91 2582 14.0 73 2
## 121 19.0 4 121.0 112 2868 15.5 73 2
## 122 15.0 8 318.0 150 3399 11.0 73 1
## 123 24.0 4 121.0 110 2660 14.0 73 2
## 124 20.0 6 156.0 122 2807 13.5 73 3
## 125 11.0 8 350.0 180 3664 11.0 73 1
## 126 20.0 6 198.0 95 3102 16.5 74 1
## 128 19.0 6 232.0 100 2901 16.0 74 1
## 129 15.0 6 250.0 100 3336 17.0 74 1
## 130 31.0 4 79.0 67 1950 19.0 74 3
## 131 26.0 4 122.0 80 2451 16.5 74 1
## 132 32.0 4 71.0 65 1836 21.0 74 3
## 133 25.0 4 140.0 75 2542 17.0 74 1
## 134 16.0 6 250.0 100 3781 17.0 74 1
## 135 16.0 6 258.0 110 3632 18.0 74 1
## 136 18.0 6 225.0 105 3613 16.5 74 1
## 137 16.0 8 302.0 140 4141 14.0 74 1
## 138 13.0 8 350.0 150 4699 14.5 74 1
## 139 14.0 8 318.0 150 4457 13.5 74 1
## 140 14.0 8 302.0 140 4638 16.0 74 1
## 141 14.0 8 304.0 150 4257 15.5 74 1
## 142 29.0 4 98.0 83 2219 16.5 74 2
## 143 26.0 4 79.0 67 1963 15.5 74 2
## 144 26.0 4 97.0 78 2300 14.5 74 2
## 145 31.0 4 76.0 52 1649 16.5 74 3
## 146 32.0 4 83.0 61 2003 19.0 74 3
## 147 28.0 4 90.0 75 2125 14.5 74 1
## 148 24.0 4 90.0 75 2108 15.5 74 2
## 149 26.0 4 116.0 75 2246 14.0 74 2
## 150 24.0 4 120.0 97 2489 15.0 74 3
## 151 26.0 4 108.0 93 2391 15.5 74 3
## 152 31.0 4 79.0 67 2000 16.0 74 2
## 153 19.0 6 225.0 95 3264 16.0 75 1
## 154 18.0 6 250.0 105 3459 16.0 75 1
## 155 15.0 6 250.0 72 3432 21.0 75 1
## 156 15.0 6 250.0 72 3158 19.5 75 1
## 157 16.0 8 400.0 170 4668 11.5 75 1
## 158 15.0 8 350.0 145 4440 14.0 75 1
## 159 16.0 8 318.0 150 4498 14.5 75 1
## 160 14.0 8 351.0 148 4657 13.5 75 1
## 161 17.0 6 231.0 110 3907 21.0 75 1
## 162 16.0 6 250.0 105 3897 18.5 75 1
## 163 15.0 6 258.0 110 3730 19.0 75 1
## 164 18.0 6 225.0 95 3785 19.0 75 1
## 165 21.0 6 231.0 110 3039 15.0 75 1
## 166 20.0 8 262.0 110 3221 13.5 75 1
## 167 13.0 8 302.0 129 3169 12.0 75 1
## 168 29.0 4 97.0 75 2171 16.0 75 3
## 169 23.0 4 140.0 83 2639 17.0 75 1
## 170 20.0 6 232.0 100 2914 16.0 75 1
## 171 23.0 4 140.0 78 2592 18.5 75 1
## 172 24.0 4 134.0 96 2702 13.5 75 3
## 173 25.0 4 90.0 71 2223 16.5 75 2
## 174 24.0 4 119.0 97 2545 17.0 75 3
## 175 18.0 6 171.0 97 2984 14.5 75 1
## 176 29.0 4 90.0 70 1937 14.0 75 2
## 177 19.0 6 232.0 90 3211 17.0 75 1
## 178 23.0 4 115.0 95 2694 15.0 75 2
## 179 23.0 4 120.0 88 2957 17.0 75 2
## 180 22.0 4 121.0 98 2945 14.5 75 2
## 181 25.0 4 121.0 115 2671 13.5 75 2
## 182 33.0 4 91.0 53 1795 17.5 75 3
## 183 28.0 4 107.0 86 2464 15.5 76 2
## 184 25.0 4 116.0 81 2220 16.9 76 2
## 185 25.0 4 140.0 92 2572 14.9 76 1
## 186 26.0 4 98.0 79 2255 17.7 76 1
## 187 27.0 4 101.0 83 2202 15.3 76 2
## 188 17.5 8 305.0 140 4215 13.0 76 1
## 189 16.0 8 318.0 150 4190 13.0 76 1
## 190 15.5 8 304.0 120 3962 13.9 76 1
## 191 14.5 8 351.0 152 4215 12.8 76 1
## 192 22.0 6 225.0 100 3233 15.4 76 1
## 193 22.0 6 250.0 105 3353 14.5 76 1
## 194 24.0 6 200.0 81 3012 17.6 76 1
## 195 22.5 6 232.0 90 3085 17.6 76 1
## 196 29.0 4 85.0 52 2035 22.2 76 1
## 197 24.5 4 98.0 60 2164 22.1 76 1
## 198 29.0 4 90.0 70 1937 14.2 76 2
## 199 33.0 4 91.0 53 1795 17.4 76 3
## 200 20.0 6 225.0 100 3651 17.7 76 1
## 201 18.0 6 250.0 78 3574 21.0 76 1
## 202 18.5 6 250.0 110 3645 16.2 76 1
## 203 17.5 6 258.0 95 3193 17.8 76 1
## 204 29.5 4 97.0 71 1825 12.2 76 2
## 205 32.0 4 85.0 70 1990 17.0 76 3
## 206 28.0 4 97.0 75 2155 16.4 76 3
## 207 26.5 4 140.0 72 2565 13.6 76 1
## 208 20.0 4 130.0 102 3150 15.7 76 2
## 209 13.0 8 318.0 150 3940 13.2 76 1
## 210 19.0 4 120.0 88 3270 21.9 76 2
## 211 19.0 6 156.0 108 2930 15.5 76 3
## 212 16.5 6 168.0 120 3820 16.7 76 2
## 213 16.5 8 350.0 180 4380 12.1 76 1
## 214 13.0 8 350.0 145 4055 12.0 76 1
## 215 13.0 8 302.0 130 3870 15.0 76 1
## 216 13.0 8 318.0 150 3755 14.0 76 1
## 217 31.5 4 98.0 68 2045 18.5 77 3
## 218 30.0 4 111.0 80 2155 14.8 77 1
## 219 36.0 4 79.0 58 1825 18.6 77 2
## 220 25.5 4 122.0 96 2300 15.5 77 1
## 221 33.5 4 85.0 70 1945 16.8 77 3
## 222 17.5 8 305.0 145 3880 12.5 77 1
## 223 17.0 8 260.0 110 4060 19.0 77 1
## 224 15.5 8 318.0 145 4140 13.7 77 1
## 225 15.0 8 302.0 130 4295 14.9 77 1
## 226 17.5 6 250.0 110 3520 16.4 77 1
## 227 20.5 6 231.0 105 3425 16.9 77 1
## 228 19.0 6 225.0 100 3630 17.7 77 1
## 229 18.5 6 250.0 98 3525 19.0 77 1
## 230 16.0 8 400.0 180 4220 11.1 77 1
## 231 15.5 8 350.0 170 4165 11.4 77 1
## 232 15.5 8 400.0 190 4325 12.2 77 1
## 233 16.0 8 351.0 149 4335 14.5 77 1
## 234 29.0 4 97.0 78 1940 14.5 77 2
## 235 24.5 4 151.0 88 2740 16.0 77 1
## 236 26.0 4 97.0 75 2265 18.2 77 3
## 237 25.5 4 140.0 89 2755 15.8 77 1
## 238 30.5 4 98.0 63 2051 17.0 77 1
## 239 33.5 4 98.0 83 2075 15.9 77 1
## 240 30.0 4 97.0 67 1985 16.4 77 3
## 241 30.5 4 97.0 78 2190 14.1 77 2
## 242 22.0 6 146.0 97 2815 14.5 77 3
## 243 21.5 4 121.0 110 2600 12.8 77 2
## 244 21.5 3 80.0 110 2720 13.5 77 3
## 245 43.1 4 90.0 48 1985 21.5 78 2
## 246 36.1 4 98.0 66 1800 14.4 78 1
## 247 32.8 4 78.0 52 1985 19.4 78 3
## 248 39.4 4 85.0 70 2070 18.6 78 3
## 249 36.1 4 91.0 60 1800 16.4 78 3
## 250 19.9 8 260.0 110 3365 15.5 78 1
## 251 19.4 8 318.0 140 3735 13.2 78 1
## 252 20.2 8 302.0 139 3570 12.8 78 1
## 253 19.2 6 231.0 105 3535 19.2 78 1
## 254 20.5 6 200.0 95 3155 18.2 78 1
## 255 20.2 6 200.0 85 2965 15.8 78 1
## 256 25.1 4 140.0 88 2720 15.4 78 1
## 257 20.5 6 225.0 100 3430 17.2 78 1
## 258 19.4 6 232.0 90 3210 17.2 78 1
## 259 20.6 6 231.0 105 3380 15.8 78 1
## 260 20.8 6 200.0 85 3070 16.7 78 1
## 261 18.6 6 225.0 110 3620 18.7 78 1
## 262 18.1 6 258.0 120 3410 15.1 78 1
## 263 19.2 8 305.0 145 3425 13.2 78 1
## 264 17.7 6 231.0 165 3445 13.4 78 1
## 265 18.1 8 302.0 139 3205 11.2 78 1
## 266 17.5 8 318.0 140 4080 13.7 78 1
## 267 30.0 4 98.0 68 2155 16.5 78 1
## 268 27.5 4 134.0 95 2560 14.2 78 3
## 269 27.2 4 119.0 97 2300 14.7 78 3
## 270 30.9 4 105.0 75 2230 14.5 78 1
## 271 21.1 4 134.0 95 2515 14.8 78 3
## 272 23.2 4 156.0 105 2745 16.7 78 1
## 273 23.8 4 151.0 85 2855 17.6 78 1
## 274 23.9 4 119.0 97 2405 14.9 78 3
## 275 20.3 5 131.0 103 2830 15.9 78 2
## 276 17.0 6 163.0 125 3140 13.6 78 2
## 277 21.6 4 121.0 115 2795 15.7 78 2
## 278 16.2 6 163.0 133 3410 15.8 78 2
## 279 31.5 4 89.0 71 1990 14.9 78 2
## 280 29.5 4 98.0 68 2135 16.6 78 3
## 281 21.5 6 231.0 115 3245 15.4 79 1
## 282 19.8 6 200.0 85 2990 18.2 79 1
## 283 22.3 4 140.0 88 2890 17.3 79 1
## 284 20.2 6 232.0 90 3265 18.2 79 1
## 285 20.6 6 225.0 110 3360 16.6 79 1
## 286 17.0 8 305.0 130 3840 15.4 79 1
## 287 17.6 8 302.0 129 3725 13.4 79 1
## 288 16.5 8 351.0 138 3955 13.2 79 1
## 289 18.2 8 318.0 135 3830 15.2 79 1
## 290 16.9 8 350.0 155 4360 14.9 79 1
## 291 15.5 8 351.0 142 4054 14.3 79 1
## 292 19.2 8 267.0 125 3605 15.0 79 1
## 293 18.5 8 360.0 150 3940 13.0 79 1
## 294 31.9 4 89.0 71 1925 14.0 79 2
## 295 34.1 4 86.0 65 1975 15.2 79 3
## 296 35.7 4 98.0 80 1915 14.4 79 1
## 297 27.4 4 121.0 80 2670 15.0 79 1
## 298 25.4 5 183.0 77 3530 20.1 79 2
## 299 23.0 8 350.0 125 3900 17.4 79 1
## 300 27.2 4 141.0 71 3190 24.8 79 2
## 301 23.9 8 260.0 90 3420 22.2 79 1
## 302 34.2 4 105.0 70 2200 13.2 79 1
## 303 34.5 4 105.0 70 2150 14.9 79 1
## 304 31.8 4 85.0 65 2020 19.2 79 3
## 305 37.3 4 91.0 69 2130 14.7 79 2
## 306 28.4 4 151.0 90 2670 16.0 79 1
## 307 28.8 6 173.0 115 2595 11.3 79 1
## 308 26.8 6 173.0 115 2700 12.9 79 1
## 309 33.5 4 151.0 90 2556 13.2 79 1
## 310 41.5 4 98.0 76 2144 14.7 80 2
## 311 38.1 4 89.0 60 1968 18.8 80 3
## 312 32.1 4 98.0 70 2120 15.5 80 1
## 313 37.2 4 86.0 65 2019 16.4 80 3
## 314 28.0 4 151.0 90 2678 16.5 80 1
## 315 26.4 4 140.0 88 2870 18.1 80 1
## 316 24.3 4 151.0 90 3003 20.1 80 1
## 317 19.1 6 225.0 90 3381 18.7 80 1
## 318 34.3 4 97.0 78 2188 15.8 80 2
## 319 29.8 4 134.0 90 2711 15.5 80 3
## 320 31.3 4 120.0 75 2542 17.5 80 3
## 321 37.0 4 119.0 92 2434 15.0 80 3
## 322 32.2 4 108.0 75 2265 15.2 80 3
## 323 46.6 4 86.0 65 2110 17.9 80 3
## 324 27.9 4 156.0 105 2800 14.4 80 1
## 325 40.8 4 85.0 65 2110 19.2 80 3
## 326 44.3 4 90.0 48 2085 21.7 80 2
## 327 43.4 4 90.0 48 2335 23.7 80 2
## 328 36.4 5 121.0 67 2950 19.9 80 2
## 329 30.0 4 146.0 67 3250 21.8 80 2
## 330 44.6 4 91.0 67 1850 13.8 80 3
## 332 33.8 4 97.0 67 2145 18.0 80 3
## 333 29.8 4 89.0 62 1845 15.3 80 2
## 334 32.7 6 168.0 132 2910 11.4 80 3
## 335 23.7 3 70.0 100 2420 12.5 80 3
## 336 35.0 4 122.0 88 2500 15.1 80 2
## 338 32.4 4 107.0 72 2290 17.0 80 3
## 339 27.2 4 135.0 84 2490 15.7 81 1
## 340 26.6 4 151.0 84 2635 16.4 81 1
## 341 25.8 4 156.0 92 2620 14.4 81 1
## 342 23.5 6 173.0 110 2725 12.6 81 1
## 343 30.0 4 135.0 84 2385 12.9 81 1
## 344 39.1 4 79.0 58 1755 16.9 81 3
## 345 39.0 4 86.0 64 1875 16.4 81 1
## 346 35.1 4 81.0 60 1760 16.1 81 3
## 347 32.3 4 97.0 67 2065 17.8 81 3
## 348 37.0 4 85.0 65 1975 19.4 81 3
## 349 37.7 4 89.0 62 2050 17.3 81 3
## 350 34.1 4 91.0 68 1985 16.0 81 3
## 351 34.7 4 105.0 63 2215 14.9 81 1
## 352 34.4 4 98.0 65 2045 16.2 81 1
## 353 29.9 4 98.0 65 2380 20.7 81 1
## 354 33.0 4 105.0 74 2190 14.2 81 2
## 356 33.7 4 107.0 75 2210 14.4 81 3
## 357 32.4 4 108.0 75 2350 16.8 81 3
## 358 32.9 4 119.0 100 2615 14.8 81 3
## 359 31.6 4 120.0 74 2635 18.3 81 3
## 360 28.1 4 141.0 80 3230 20.4 81 2
## 361 30.7 6 145.0 76 3160 19.6 81 2
## 362 25.4 6 168.0 116 2900 12.6 81 3
## 363 24.2 6 146.0 120 2930 13.8 81 3
## 364 22.4 6 231.0 110 3415 15.8 81 1
## 365 26.6 8 350.0 105 3725 19.0 81 1
## 366 20.2 6 200.0 88 3060 17.1 81 1
## 367 17.6 6 225.0 85 3465 16.6 81 1
## 368 28.0 4 112.0 88 2605 19.6 82 1
## 369 27.0 4 112.0 88 2640 18.6 82 1
## 370 34.0 4 112.0 88 2395 18.0 82 1
## 371 31.0 4 112.0 85 2575 16.2 82 1
## 372 29.0 4 135.0 84 2525 16.0 82 1
## 373 27.0 4 151.0 90 2735 18.0 82 1
## 374 24.0 4 140.0 92 2865 16.4 82 1
## 375 36.0 4 105.0 74 1980 15.3 82 2
## 376 37.0 4 91.0 68 2025 18.2 82 3
## 377 31.0 4 91.0 68 1970 17.6 82 3
## 378 38.0 4 105.0 63 2125 14.7 82 1
## 379 36.0 4 98.0 70 2125 17.3 82 1
## 380 36.0 4 120.0 88 2160 14.5 82 3
## 381 36.0 4 107.0 75 2205 14.5 82 3
## 382 34.0 4 108.0 70 2245 16.9 82 3
## 383 38.0 4 91.0 67 1965 15.0 82 3
## 384 32.0 4 91.0 67 1965 15.7 82 3
## 385 38.0 4 91.0 67 1995 16.2 82 3
## 386 25.0 6 181.0 110 2945 16.4 82 1
## 387 38.0 6 262.0 85 3015 17.0 82 1
## 388 26.0 4 156.0 92 2585 14.5 82 1
## 389 22.0 6 232.0 112 2835 14.7 82 1
## 390 32.0 4 144.0 96 2665 13.9 82 3
## 391 36.0 4 135.0 84 2370 13.0 82 1
## 392 27.0 4 151.0 90 2950 17.3 82 1
## 393 27.0 4 140.0 86 2790 15.6 82 1
## 394 44.0 4 97.0 52 2130 24.6 82 2
## 395 32.0 4 135.0 84 2295 11.6 82 1
## 396 28.0 4 120.0 79 2625 18.6 82 1
## 397 31.0 4 119.0 82 2720 19.4 82 1
## name
## 1 chevrolet chevelle malibu
## 2 buick skylark 320
## 3 plymouth satellite
## 4 amc rebel sst
## 5 ford torino
## 6 ford galaxie 500
## 7 chevrolet impala
## 8 plymouth fury iii
## 9 pontiac catalina
## 10 amc ambassador dpl
## 11 dodge challenger se
## 12 plymouth 'cuda 340
## 13 chevrolet monte carlo
## 14 buick estate wagon (sw)
## 15 toyota corona mark ii
## 16 plymouth duster
## 17 amc hornet
## 18 ford maverick
## 19 datsun pl510
## 20 volkswagen 1131 deluxe sedan
## 21 peugeot 504
## 22 audi 100 ls
## 23 saab 99e
## 24 bmw 2002
## 25 amc gremlin
## 26 ford f250
## 27 chevy c20
## 28 dodge d200
## 29 hi 1200d
## 30 datsun pl510
## 31 chevrolet vega 2300
## 32 toyota corona
## 34 amc gremlin
## 35 plymouth satellite custom
## 36 chevrolet chevelle malibu
## 37 ford torino 500
## 38 amc matador
## 39 chevrolet impala
## 40 pontiac catalina brougham
## 41 ford galaxie 500
## 42 plymouth fury iii
## 43 dodge monaco (sw)
## 44 ford country squire (sw)
## 45 pontiac safari (sw)
## 46 amc hornet sportabout (sw)
## 47 chevrolet vega (sw)
## 48 pontiac firebird
## 49 ford mustang
## 50 mercury capri 2000
## 51 opel 1900
## 52 peugeot 304
## 53 fiat 124b
## 54 toyota corolla 1200
## 55 datsun 1200
## 56 volkswagen model 111
## 57 plymouth cricket
## 58 toyota corona hardtop
## 59 dodge colt hardtop
## 60 volkswagen type 3
## 61 chevrolet vega
## 62 ford pinto runabout
## 63 chevrolet impala
## 64 pontiac catalina
## 65 plymouth fury iii
## 66 ford galaxie 500
## 67 amc ambassador sst
## 68 mercury marquis
## 69 buick lesabre custom
## 70 oldsmobile delta 88 royale
## 71 chrysler newport royal
## 72 mazda rx2 coupe
## 73 amc matador (sw)
## 74 chevrolet chevelle concours (sw)
## 75 ford gran torino (sw)
## 76 plymouth satellite custom (sw)
## 77 volvo 145e (sw)
## 78 volkswagen 411 (sw)
## 79 peugeot 504 (sw)
## 80 renault 12 (sw)
## 81 ford pinto (sw)
## 82 datsun 510 (sw)
## 83 toyouta corona mark ii (sw)
## 84 dodge colt (sw)
## 85 toyota corolla 1600 (sw)
## 86 buick century 350
## 87 amc matador
## 88 chevrolet malibu
## 89 ford gran torino
## 90 dodge coronet custom
## 91 mercury marquis brougham
## 92 chevrolet caprice classic
## 93 ford ltd
## 94 plymouth fury gran sedan
## 95 chrysler new yorker brougham
## 96 buick electra 225 custom
## 97 amc ambassador brougham
## 98 plymouth valiant
## 99 chevrolet nova custom
## 100 amc hornet
## 101 ford maverick
## 102 plymouth duster
## 103 volkswagen super beetle
## 104 chevrolet impala
## 105 ford country
## 106 plymouth custom suburb
## 107 oldsmobile vista cruiser
## 108 amc gremlin
## 109 toyota carina
## 110 chevrolet vega
## 111 datsun 610
## 112 maxda rx3
## 113 ford pinto
## 114 mercury capri v6
## 115 fiat 124 sport coupe
## 116 chevrolet monte carlo s
## 117 pontiac grand prix
## 118 fiat 128
## 119 opel manta
## 120 audi 100ls
## 121 volvo 144ea
## 122 dodge dart custom
## 123 saab 99le
## 124 toyota mark ii
## 125 oldsmobile omega
## 126 plymouth duster
## 128 amc hornet
## 129 chevrolet nova
## 130 datsun b210
## 131 ford pinto
## 132 toyota corolla 1200
## 133 chevrolet vega
## 134 chevrolet chevelle malibu classic
## 135 amc matador
## 136 plymouth satellite sebring
## 137 ford gran torino
## 138 buick century luxus (sw)
## 139 dodge coronet custom (sw)
## 140 ford gran torino (sw)
## 141 amc matador (sw)
## 142 audi fox
## 143 volkswagen dasher
## 144 opel manta
## 145 toyota corona
## 146 datsun 710
## 147 dodge colt
## 148 fiat 128
## 149 fiat 124 tc
## 150 honda civic
## 151 subaru
## 152 fiat x1.9
## 153 plymouth valiant custom
## 154 chevrolet nova
## 155 mercury monarch
## 156 ford maverick
## 157 pontiac catalina
## 158 chevrolet bel air
## 159 plymouth grand fury
## 160 ford ltd
## 161 buick century
## 162 chevroelt chevelle malibu
## 163 amc matador
## 164 plymouth fury
## 165 buick skyhawk
## 166 chevrolet monza 2+2
## 167 ford mustang ii
## 168 toyota corolla
## 169 ford pinto
## 170 amc gremlin
## 171 pontiac astro
## 172 toyota corona
## 173 volkswagen dasher
## 174 datsun 710
## 175 ford pinto
## 176 volkswagen rabbit
## 177 amc pacer
## 178 audi 100ls
## 179 peugeot 504
## 180 volvo 244dl
## 181 saab 99le
## 182 honda civic cvcc
## 183 fiat 131
## 184 opel 1900
## 185 capri ii
## 186 dodge colt
## 187 renault 12tl
## 188 chevrolet chevelle malibu classic
## 189 dodge coronet brougham
## 190 amc matador
## 191 ford gran torino
## 192 plymouth valiant
## 193 chevrolet nova
## 194 ford maverick
## 195 amc hornet
## 196 chevrolet chevette
## 197 chevrolet woody
## 198 vw rabbit
## 199 honda civic
## 200 dodge aspen se
## 201 ford granada ghia
## 202 pontiac ventura sj
## 203 amc pacer d/l
## 204 volkswagen rabbit
## 205 datsun b-210
## 206 toyota corolla
## 207 ford pinto
## 208 volvo 245
## 209 plymouth volare premier v8
## 210 peugeot 504
## 211 toyota mark ii
## 212 mercedes-benz 280s
## 213 cadillac seville
## 214 chevy c10
## 215 ford f108
## 216 dodge d100
## 217 honda accord cvcc
## 218 buick opel isuzu deluxe
## 219 renault 5 gtl
## 220 plymouth arrow gs
## 221 datsun f-10 hatchback
## 222 chevrolet caprice classic
## 223 oldsmobile cutlass supreme
## 224 dodge monaco brougham
## 225 mercury cougar brougham
## 226 chevrolet concours
## 227 buick skylark
## 228 plymouth volare custom
## 229 ford granada
## 230 pontiac grand prix lj
## 231 chevrolet monte carlo landau
## 232 chrysler cordoba
## 233 ford thunderbird
## 234 volkswagen rabbit custom
## 235 pontiac sunbird coupe
## 236 toyota corolla liftback
## 237 ford mustang ii 2+2
## 238 chevrolet chevette
## 239 dodge colt m/m
## 240 subaru dl
## 241 volkswagen dasher
## 242 datsun 810
## 243 bmw 320i
## 244 mazda rx-4
## 245 volkswagen rabbit custom diesel
## 246 ford fiesta
## 247 mazda glc deluxe
## 248 datsun b210 gx
## 249 honda civic cvcc
## 250 oldsmobile cutlass salon brougham
## 251 dodge diplomat
## 252 mercury monarch ghia
## 253 pontiac phoenix lj
## 254 chevrolet malibu
## 255 ford fairmont (auto)
## 256 ford fairmont (man)
## 257 plymouth volare
## 258 amc concord
## 259 buick century special
## 260 mercury zephyr
## 261 dodge aspen
## 262 amc concord d/l
## 263 chevrolet monte carlo landau
## 264 buick regal sport coupe (turbo)
## 265 ford futura
## 266 dodge magnum xe
## 267 chevrolet chevette
## 268 toyota corona
## 269 datsun 510
## 270 dodge omni
## 271 toyota celica gt liftback
## 272 plymouth sapporo
## 273 oldsmobile starfire sx
## 274 datsun 200-sx
## 275 audi 5000
## 276 volvo 264gl
## 277 saab 99gle
## 278 peugeot 604sl
## 279 volkswagen scirocco
## 280 honda accord lx
## 281 pontiac lemans v6
## 282 mercury zephyr 6
## 283 ford fairmont 4
## 284 amc concord dl 6
## 285 dodge aspen 6
## 286 chevrolet caprice classic
## 287 ford ltd landau
## 288 mercury grand marquis
## 289 dodge st. regis
## 290 buick estate wagon (sw)
## 291 ford country squire (sw)
## 292 chevrolet malibu classic (sw)
## 293 chrysler lebaron town @ country (sw)
## 294 vw rabbit custom
## 295 maxda glc deluxe
## 296 dodge colt hatchback custom
## 297 amc spirit dl
## 298 mercedes benz 300d
## 299 cadillac eldorado
## 300 peugeot 504
## 301 oldsmobile cutlass salon brougham
## 302 plymouth horizon
## 303 plymouth horizon tc3
## 304 datsun 210
## 305 fiat strada custom
## 306 buick skylark limited
## 307 chevrolet citation
## 308 oldsmobile omega brougham
## 309 pontiac phoenix
## 310 vw rabbit
## 311 toyota corolla tercel
## 312 chevrolet chevette
## 313 datsun 310
## 314 chevrolet citation
## 315 ford fairmont
## 316 amc concord
## 317 dodge aspen
## 318 audi 4000
## 319 toyota corona liftback
## 320 mazda 626
## 321 datsun 510 hatchback
## 322 toyota corolla
## 323 mazda glc
## 324 dodge colt
## 325 datsun 210
## 326 vw rabbit c (diesel)
## 327 vw dasher (diesel)
## 328 audi 5000s (diesel)
## 329 mercedes-benz 240d
## 330 honda civic 1500 gl
## 332 subaru dl
## 333 vokswagen rabbit
## 334 datsun 280-zx
## 335 mazda rx-7 gs
## 336 triumph tr7 coupe
## 338 honda accord
## 339 plymouth reliant
## 340 buick skylark
## 341 dodge aries wagon (sw)
## 342 chevrolet citation
## 343 plymouth reliant
## 344 toyota starlet
## 345 plymouth champ
## 346 honda civic 1300
## 347 subaru
## 348 datsun 210 mpg
## 349 toyota tercel
## 350 mazda glc 4
## 351 plymouth horizon 4
## 352 ford escort 4w
## 353 ford escort 2h
## 354 volkswagen jetta
## 356 honda prelude
## 357 toyota corolla
## 358 datsun 200sx
## 359 mazda 626
## 360 peugeot 505s turbo diesel
## 361 volvo diesel
## 362 toyota cressida
## 363 datsun 810 maxima
## 364 buick century
## 365 oldsmobile cutlass ls
## 366 ford granada gl
## 367 chrysler lebaron salon
## 368 chevrolet cavalier
## 369 chevrolet cavalier wagon
## 370 chevrolet cavalier 2-door
## 371 pontiac j2000 se hatchback
## 372 dodge aries se
## 373 pontiac phoenix
## 374 ford fairmont futura
## 375 volkswagen rabbit l
## 376 mazda glc custom l
## 377 mazda glc custom
## 378 plymouth horizon miser
## 379 mercury lynx l
## 380 nissan stanza xe
## 381 honda accord
## 382 toyota corolla
## 383 honda civic
## 384 honda civic (auto)
## 385 datsun 310 gx
## 386 buick century limited
## 387 oldsmobile cutlass ciera (diesel)
## 388 chrysler lebaron medallion
## 389 ford granada l
## 390 toyota celica gt
## 391 dodge charger 2.2
## 392 chevrolet camaro
## 393 ford mustang gl
## 394 vw pickup
## 395 dodge rampage
## 396 ford ranger
## 397 chevy s-10mod<-lm(auto$mpg~auto$horsepower)
summary(mod)##
## Call:
## lm(formula = auto$mpg ~ auto$horsepower)
##
## Residuals:
## Min 1Q Median 3Q Max
## -13.5710 -3.2592 -0.3435 2.7630 16.9240
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 39.935861 0.717499 55.66 <2e-16 ***
## auto$horsepower -0.157845 0.006446 -24.49 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.906 on 390 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.6059, Adjusted R-squared: 0.6049
## F-statistic: 599.7 on 1 and 390 DF, p-value: < 2.2e-16## – (i) Is there a relationship between the predictor and the response?
# Yes.
## – (ii) How strong is the relationship between the predictor and the response?
# Strong
## – (iii) Is the relationship between the predictor and the response positive or negative?
# negative
## – (iv) What is the predicted mpg associated with a horsepower of 98? What are the associated 95% confidence and prediction intervals?
attach(auto)
newdata<-data.frame(horsepower=c(98))
predict(mod, newdata,
interval="predict")## Warning: 'newdata' had 1 row but variables found have 397 rows## fit lwr upr
## 1 19.416046 9.7532948 29.07880
## 2 13.891480 4.2037318 23.57923
## 3 16.259151 6.5845976 25.93370
## 4 16.259151 6.5845976 25.93370
## 5 17.837598 8.1697749 27.50542
## 6 8.682604 -1.0471901 18.41240
## 7 5.210020 -4.5576558 14.97770
## 8 5.999243 -3.7591362 15.75762
## 9 4.420796 -5.3565771 14.19817
## 10 9.945362 0.2273963 19.66333
## 11 13.102256 3.4092855 22.79523
## 12 14.680704 4.9977664 24.36364
## 13 16.259151 6.5845976 25.93370
## 14 4.420796 -5.3565771 14.19817
## 15 24.940611 15.2825327 34.59869
## 16 24.940611 15.2825327 34.59869
## 17 24.624922 14.9671249 34.28272
## 18 26.519059 16.8585745 36.17954
## 19 26.045524 16.3859365 35.70511
## 20 32.675003 22.9892883 42.36072
## 21 26.203369 16.5434991 35.86324
## 22 25.729835 16.0707614 35.38891
## 23 24.940611 15.2825327 34.59869
## 24 22.099406 12.4414680 31.75734
## 25 25.729835 16.0707614 35.38891
## 26 5.999243 -3.7591362 15.75762
## 27 8.366914 -1.3659995 18.09983
## 28 6.788467 -2.9610196 16.53795
## 29 9.471827 -0.2504514 19.19411
## 30 26.045524 16.3859365 35.70511
## 31 25.729835 16.0707614 35.38891
## 32 24.940611 15.2825327 34.59869
## 33 NA NA NA
## 34 24.151388 14.4938885 33.80889
## 35 23.362164 13.7048286 33.01950
## 36 24.151388 14.4938885 33.80889
## 37 26.045524 16.3859365 35.70511
## 38 24.151388 14.4938885 33.80889
## 39 13.891480 4.2037318 23.57923
## 40 12.313033 2.6144281 22.01164
## 41 15.785617 6.1087217 25.46251
## 42 16.259151 6.5845976 25.93370
## 43 11.523809 1.8191602 21.22846
## 44 13.102256 3.4092855 22.79523
## 45 12.313033 2.6144281 22.01164
## 46 22.572940 12.9153529 32.23053
## 47 28.571040 18.9049457 38.23713
## 48 24.151388 14.4938885 33.80889
## 49 26.045524 16.3859365 35.70511
## 50 26.361214 16.7010451 36.02138
## 51 25.729835 16.0707614 35.38891
## 52 28.886730 19.2195232 38.55394
## 53 27.939661 18.2755918 37.60373
## 54 29.675953 20.0056767 39.34623
## 55 29.044574 19.3767870 38.71236
## 56 30.465177 20.7914163 40.13894
## 57 28.886730 19.2195232 38.55394
## 58 24.940611 15.2825327 34.59869
## 59 27.308282 17.6459724 36.97059
## 60 31.412245 21.7337578 41.09073
## 61 25.729835 16.0707614 35.38891
## 62 26.361214 16.7010451 36.02138
## 63 13.891480 4.2037318 23.57923
## 64 12.313033 2.6144281 22.01164
## 65 16.259151 6.5845976 25.93370
## 66 15.785617 6.1087217 25.46251
## 67 16.259151 6.5845976 25.93370
## 68 7.104156 -2.6418860 16.85020
## 69 15.469927 5.7913884 25.14847
## 70 14.680704 4.9977664 24.36364
## 71 9.945362 0.2273963 19.66333
## 72 24.624922 14.9671249 34.28272
## 73 16.259151 6.5845976 25.93370
## 74 19.416046 9.7532948 29.07880
## 75 17.837598 8.1697749 27.50542
## 76 16.259151 6.5845976 25.93370
## 77 22.257251 12.5994463 31.91506
## 78 27.939661 18.2755918 37.60373
## 79 26.203369 16.5434991 35.86324
## 80 29.044574 19.3767870 38.71236
## 81 26.361214 16.7010451 36.02138
## 82 25.414146 15.7555198 35.07277
## 83 24.624922 14.9671249 34.28272
## 84 27.308282 17.6459724 36.97059
## 85 26.045524 16.3859365 35.70511
## 86 12.313033 2.6144281 22.01164
## 87 16.259151 6.5845976 25.93370
## 88 17.048375 7.3773932 26.71936
## 89 18.311133 8.6450050 27.97726
## 90 16.259151 6.5845976 25.93370
## 91 8.682604 -1.0471901 18.41240
## 92 16.259151 6.5845976 25.93370
## 93 14.996393 5.3152647 24.67752
## 94 16.259151 6.5845976 25.93370
## 95 5.999243 -3.7591362 15.75762
## 96 4.420796 -5.3565771 14.19817
## 97 12.313033 2.6144281 22.01164
## 98 23.362164 13.7048286 33.01950
## 99 24.151388 14.4938885 33.80889
## 100 24.151388 14.4938885 33.80889
## 101 26.045524 16.3859365 35.70511
## 102 24.940611 15.2825327 34.59869
## 103 32.675003 22.9892883 42.36072
## 104 16.259151 6.5845976 25.93370
## 105 13.575791 3.8860027 23.26558
## 106 13.102256 3.4092855 22.79523
## 107 11.523809 1.8191602 21.22846
## 108 24.151388 14.4938885 33.80889
## 109 26.045524 16.3859365 35.70511
## 110 28.571040 18.9049457 38.23713
## 111 25.098456 15.4402117 34.75670
## 112 25.729835 16.0707614 35.38891
## 113 26.519059 16.8585745 36.17954
## 114 23.046475 13.3890882 32.70386
## 115 25.729835 16.0707614 35.38891
## 116 17.048375 7.3773932 26.71936
## 117 3.631572 -6.1558990 13.41904
## 118 32.201469 22.5185881 41.88435
## 119 28.097506 18.4329552 37.76206
## 120 25.571990 15.9131489 35.23083
## 121 22.257251 12.5994463 31.91506
## 122 16.259151 6.5845976 25.93370
## 123 22.572940 12.9153529 32.23053
## 124 20.678804 11.0189156 30.33869
## 125 11.523809 1.8191602 21.22846
## 126 24.940611 15.2825327 34.59869
## 127 NA NA NA
## 128 24.151388 14.4938885 33.80889
## 129 24.151388 14.4938885 33.80889
## 130 29.360264 19.6912650 39.02926
## 131 27.308282 17.6459724 36.97059
## 132 29.675953 20.0056767 39.34623
## 133 28.097506 18.4329552 37.76206
## 134 24.151388 14.4938885 33.80889
## 135 22.572940 12.9153529 32.23053
## 136 23.362164 13.7048286 33.01950
## 137 17.837598 8.1697749 27.50542
## 138 16.259151 6.5845976 25.93370
## 139 16.259151 6.5845976 25.93370
## 140 17.837598 8.1697749 27.50542
## 141 16.259151 6.5845976 25.93370
## 142 26.834748 17.1735835 36.49591
## 143 29.360264 19.6912650 39.02926
## 144 27.623972 17.9608153 37.28713
## 145 31.727935 22.0477395 41.40813
## 146 30.307332 20.6343015 39.98036
## 147 28.097506 18.4329552 37.76206
## 148 28.097506 18.4329552 37.76206
## 149 28.097506 18.4329552 37.76206
## 150 24.624922 14.9671249 34.28272
## 151 25.256301 15.5978741 34.91473
## 152 29.360264 19.6912650 39.02926
## 153 24.940611 15.2825327 34.59869
## 154 23.362164 13.7048286 33.01950
## 155 28.571040 18.9049457 38.23713
## 156 28.571040 18.9049457 38.23713
## 157 13.102256 3.4092855 22.79523
## 158 17.048375 7.3773932 26.71936
## 159 16.259151 6.5845976 25.93370
## 160 16.574840 6.9017655 26.24792
## 161 22.572940 12.9153529 32.23053
## 162 23.362164 13.7048286 33.01950
## 163 22.572940 12.9153529 32.23053
## 164 24.940611 15.2825327 34.59869
## 165 22.572940 12.9153529 32.23053
## 166 22.572940 12.9153529 32.23053
## 167 19.573890 9.9115555 29.23623
## 168 28.097506 18.4329552 37.76206
## 169 26.834748 17.1735835 36.49591
## 170 24.151388 14.4938885 33.80889
## 171 27.623972 17.9608153 37.28713
## 172 24.782767 15.1248371 34.44070
## 173 28.728885 19.0622427 38.39553
## 174 24.624922 14.9671249 34.28272
## 175 24.624922 14.9671249 34.28272
## 176 28.886730 19.2195232 38.55394
## 177 25.729835 16.0707614 35.38891
## 178 24.940611 15.2825327 34.59869
## 179 26.045524 16.3859365 35.70511
## 180 24.467077 14.8093961 34.12476
## 181 21.783717 12.1254616 31.44197
## 182 31.570090 21.8907569 41.24942
## 183 26.361214 16.7010451 36.02138
## 184 27.150438 17.4885260 36.81235
## 185 25.414146 15.7555198 35.07277
## 186 27.466127 17.8034021 37.12885
## 187 26.834748 17.1735835 36.49591
## 188 17.837598 8.1697749 27.50542
## 189 16.259151 6.5845976 25.93370
## 190 20.994493 11.3351547 30.65383
## 191 15.943462 6.2673635 25.61956
## 192 24.151388 14.4938885 33.80889
## 193 23.362164 13.7048286 33.01950
## 194 27.150438 17.4885260 36.81235
## 195 25.729835 16.0707614 35.38891
## 196 31.727935 22.0477395 41.40813
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## 202 22.572940 12.9153529 32.23053
## 203 24.940611 15.2825327 34.59869
## 204 28.728885 19.0622427 38.39553
## 205 28.886730 19.2195232 38.55394
## 206 28.097506 18.4329552 37.76206
## 207 28.571040 18.9049457 38.23713
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## 209 16.259151 6.5845976 25.93370
## 210 26.045524 16.3859365 35.70511
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## 212 20.994493 11.3351547 30.65383
## 213 11.523809 1.8191602 21.22846
## 214 17.048375 7.3773932 26.71936
## 215 19.416046 9.7532948 29.07880
## 216 16.259151 6.5845976 25.93370
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## 219 30.780866 21.1055963 40.45614
## 220 24.782767 15.1248371 34.44070
## 221 28.886730 19.2195232 38.55394
## 222 17.048375 7.3773932 26.71936
## 223 22.572940 12.9153529 32.23053
## 224 17.048375 7.3773932 26.71936
## 225 19.416046 9.7532948 29.07880
## 226 22.572940 12.9153529 32.23053
## 227 23.362164 13.7048286 33.01950
## 228 24.151388 14.4938885 33.80889
## 229 24.467077 14.8093961 34.12476
## 230 11.523809 1.8191602 21.22846
## 231 13.102256 3.4092855 22.79523
## 232 9.945362 0.2273963 19.66333
## 233 16.416996 6.7431898 26.09080
## 234 27.623972 17.9608153 37.28713
## 235 26.045524 16.3859365 35.70511
## 236 28.097506 18.4329552 37.76206
## 237 25.887680 16.2283572 35.54700
## 238 29.991643 20.3200222 39.66326
## 239 26.834748 17.1735835 36.49591
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## 241 27.623972 17.9608153 37.28713
## 242 24.624922 14.9671249 34.28272
## 243 22.572940 12.9153529 32.23053
## 244 22.572940 12.9153529 32.23053
## 245 32.359314 22.6755047 42.04312
## 246 29.518109 19.8484792 39.18774
## 247 31.727935 22.0477395 41.40813
## 248 28.886730 19.2195232 38.55394
## 249 30.465177 20.7914163 40.13894
## 250 22.572940 12.9153529 32.23053
## 251 17.837598 8.1697749 27.50542
## 252 17.995443 8.3282015 27.66268
## 253 23.362164 13.7048286 33.01950
## 254 24.940611 15.2825327 34.59869
## 255 26.519059 16.8585745 36.17954
## 256 26.045524 16.3859365 35.70511
## 257 24.151388 14.4938885 33.80889
## 258 25.729835 16.0707614 35.38891
## 259 23.362164 13.7048286 33.01950
## 260 26.519059 16.8585745 36.17954
## 261 22.572940 12.9153529 32.23053
## 262 20.994493 11.3351547 30.65383
## 263 17.048375 7.3773932 26.71936
## 264 13.891480 4.2037318 23.57923
## 265 17.995443 8.3282015 27.66268
## 266 17.837598 8.1697749 27.50542
## 267 29.202419 19.5340343 38.87080
## 268 24.940611 15.2825327 34.59869
## 269 24.624922 14.9671249 34.28272
## 270 28.097506 18.4329552 37.76206
## 271 24.940611 15.2825327 34.59869
## 272 23.362164 13.7048286 33.01950
## 273 26.519059 16.8585745 36.17954
## 274 24.624922 14.9671249 34.28272
## 275 23.677853 14.0205024 33.33520
## 276 20.205269 10.5444324 29.86611
## 277 21.783717 12.1254616 31.44197
## 278 18.942511 9.2784130 28.60661
## 279 28.728885 19.0622427 38.39553
## 280 29.202419 19.5340343 38.87080
## 281 21.783717 12.1254616 31.44197
## 282 26.519059 16.8585745 36.17954
## 283 26.045524 16.3859365 35.70511
## 284 25.729835 16.0707614 35.38891
## 285 22.572940 12.9153529 32.23053
## 286 19.416046 9.7532948 29.07880
## 287 19.573890 9.9115555 29.23623
## 288 18.153288 8.4866115 27.81996
## 289 18.626822 8.9617422 28.29190
## 290 15.469927 5.7913884 25.14847
## 291 17.521909 7.8528719 27.19095
## 292 20.205269 10.5444324 29.86611
## 293 16.259151 6.5845976 25.93370
## 294 28.728885 19.0622427 38.39553
## 295 29.675953 20.0056767 39.34623
## 296 27.308282 17.6459724 36.97059
## 297 27.308282 17.6459724 36.97059
## 298 27.781817 18.1182118 37.44542
## 299 20.205269 10.5444324 29.86611
## 300 28.728885 19.0622427 38.39553
## 301 25.729835 16.0707614 35.38891
## 302 28.886730 19.2195232 38.55394
## 303 28.886730 19.2195232 38.55394
## 304 29.675953 20.0056767 39.34623
## 305 29.044574 19.3767870 38.71236
## 306 25.729835 16.0707614 35.38891
## 307 21.783717 12.1254616 31.44197
## 308 21.783717 12.1254616 31.44197
## 309 25.729835 16.0707614 35.38891
## 310 27.939661 18.2755918 37.60373
## 311 30.465177 20.7914163 40.13894
## 312 28.886730 19.2195232 38.55394
## 313 29.675953 20.0056767 39.34623
## 314 25.729835 16.0707614 35.38891
## 315 26.045524 16.3859365 35.70511
## 316 25.729835 16.0707614 35.38891
## 317 25.729835 16.0707614 35.38891
## 318 27.623972 17.9608153 37.28713
## 319 25.729835 16.0707614 35.38891
## 320 28.097506 18.4329552 37.76206
## 321 25.414146 15.7555198 35.07277
## 322 28.097506 18.4329552 37.76206
## 323 29.675953 20.0056767 39.34623
## 324 23.362164 13.7048286 33.01950
## 325 29.675953 20.0056767 39.34623
## 326 32.359314 22.6755047 42.04312
## 327 32.359314 22.6755047 42.04312
## 328 29.360264 19.6912650 39.02926
## 329 29.360264 19.6912650 39.02926
## 330 29.360264 19.6912650 39.02926
## 331 NA NA NA
## 332 29.360264 19.6912650 39.02926
## 333 30.149488 20.4771701 39.82180
## 334 19.100356 9.4367235 28.76399
## 335 24.151388 14.4938885 33.80889
## 336 26.045524 16.3859365 35.70511
## 337 NA NA NA
## 338 28.571040 18.9049457 38.23713
## 339 26.676903 17.0160873 36.33772
## 340 26.676903 17.0160873 36.33772
## 341 25.414146 15.7555198 35.07277
## 342 22.572940 12.9153529 32.23053
## 343 26.676903 17.0160873 36.33772
## 344 30.780866 21.1055963 40.45614
## 345 29.833798 20.1628578 39.50474
## 346 30.465177 20.7914163 40.13894
## 347 29.360264 19.6912650 39.02926
## 348 29.675953 20.0056767 39.34623
## 349 30.149488 20.4771701 39.82180
## 350 29.202419 19.5340343 38.87080
## 351 29.991643 20.3200222 39.66326
## 352 29.675953 20.0056767 39.34623
## 353 29.675953 20.0056767 39.34623
## 354 28.255351 18.5903019 37.92040
## 355 NA NA NA
## 356 28.097506 18.4329552 37.76206
## 357 28.097506 18.4329552 37.76206
## 358 24.151388 14.4938885 33.80889
## 359 28.255351 18.5903019 37.92040
## 360 27.308282 17.6459724 36.97059
## 361 27.939661 18.2755918 37.60373
## 362 21.625872 11.9674335 31.28431
## 363 20.994493 11.3351547 30.65383
## 364 22.572940 12.9153529 32.23053
## 365 23.362164 13.7048286 33.01950
## 366 26.045524 16.3859365 35.70511
## 367 26.519059 16.8585745 36.17954
## 368 26.045524 16.3859365 35.70511
## 369 26.045524 16.3859365 35.70511
## 370 26.045524 16.3859365 35.70511
## 371 26.519059 16.8585745 36.17954
## 372 26.676903 17.0160873 36.33772
## 373 25.729835 16.0707614 35.38891
## 374 25.414146 15.7555198 35.07277
## 375 28.255351 18.5903019 37.92040
## 376 29.202419 19.5340343 38.87080
## 377 29.202419 19.5340343 38.87080
## 378 29.991643 20.3200222 39.66326
## 379 28.886730 19.2195232 38.55394
## 380 26.045524 16.3859365 35.70511
## 381 28.097506 18.4329552 37.76206
## 382 28.886730 19.2195232 38.55394
## 383 29.360264 19.6912650 39.02926
## 384 29.360264 19.6912650 39.02926
## 385 29.360264 19.6912650 39.02926
## 386 22.572940 12.9153529 32.23053
## 387 26.519059 16.8585745 36.17954
## 388 25.414146 15.7555198 35.07277
## 389 22.257251 12.5994463 31.91506
## 390 24.782767 15.1248371 34.44070
## 391 26.676903 17.0160873 36.33772
## 392 25.729835 16.0707614 35.38891
## 393 26.361214 16.7010451 36.02138
## 394 31.727935 22.0477395 41.40813
## 395 26.676903 17.0160873 36.33772
## 396 27.466127 17.8034021 37.12885
## 397 26.992593 17.3310631 36.65412# Horsepower of 98 prediction --> fit lwr upr
# 125.37566 77.7200072 173.03131
predict(mod, newdata,
interval="confidence")## Warning: 'newdata' had 1 row but variables found have 397 rows## fit lwr upr
## 1 19.416046 18.831250 20.000841
## 2 13.891480 12.982802 14.800158
## 3 16.259151 15.504025 17.014277
## 4 16.259151 15.504025 17.014277
## 5 17.837598 17.174242 18.500955
## 6 8.682604 7.401151 9.964056
## 7 5.210020 3.667064 6.752976
## 8 5.999243 4.516273 7.482214
## 9 4.420796 2.817595 6.023998
## 10 9.945362 8.757051 11.133672
## 11 13.102256 12.139493 14.065020
## 12 14.680704 13.824838 15.536569
## 13 16.259151 15.504025 17.014277
## 14 4.420796 2.817595 6.023998
## 15 24.940611 24.438901 25.442321
## 16 24.940611 24.438901 25.442321
## 17 24.624922 24.128663 25.121180
## 18 26.519059 25.972996 27.065122
## 19 26.045524 25.515552 26.575497
## 20 32.675003 31.788265 33.561742
## 21 26.203369 25.668280 26.738459
## 22 25.729835 25.209322 26.250348
## 23 24.940611 24.438901 25.442321
## 24 22.099406 21.600408 22.598404
## 25 25.729835 25.209322 26.250348
## 26 5.999243 4.516273 7.482214
## 27 8.366914 7.061985 9.671844
## 28 6.788467 5.365189 8.211745
## 29 9.471827 8.248742 10.694913
## 30 26.045524 25.515552 26.575497
## 31 25.729835 25.209322 26.250348
## 32 24.940611 24.438901 25.442321
## 33 NA NA NA
## 34 24.151388 23.660958 24.641817
## 35 23.362164 22.874970 23.849358
## 36 24.151388 23.660958 24.641817
## 37 26.045524 25.515552 26.575497
## 38 24.151388 23.660958 24.641817
## 39 13.891480 12.982802 14.800158
## 40 12.313033 11.295112 13.330953
## 41 15.785617 15.001060 16.570174
## 42 16.259151 15.504025 17.014277
## 43 11.523809 10.449826 12.597792
## 44 13.102256 12.139493 14.065020
## 45 12.313033 11.295112 13.330953
## 46 22.572940 22.080777 23.065104
## 47 28.571040 27.933378 29.208703
## 48 24.151388 23.660958 24.641817
## 49 26.045524 25.515552 26.575497
## 50 26.361214 25.820758 26.901669
## 51 25.729835 25.209322 26.250348
## 52 28.886730 28.232427 29.541033
## 53 27.939661 27.333469 28.545854
## 54 29.675953 28.977756 30.374151
## 55 29.044574 28.381744 29.707404
## 56 30.465177 29.720276 31.210078
## 57 28.886730 28.232427 29.541033
## 58 24.940611 24.438901 25.442321
## 59 27.308282 26.730818 27.885746
## 60 31.412245 30.608285 32.216206
## 61 25.729835 25.209322 26.250348
## 62 26.361214 25.820758 26.901669
## 63 13.891480 12.982802 14.800158
## 64 12.313033 11.295112 13.330953
## 65 16.259151 15.504025 17.014277
## 66 15.785617 15.001060 16.570174
## 67 16.259151 15.504025 17.014277
## 68 7.104156 5.704666 8.503647
## 69 15.469927 14.665349 16.274505
## 70 14.680704 13.824838 15.536569
## 71 9.945362 8.757051 11.133672
## 72 24.624922 24.128663 25.121180
## 73 16.259151 15.504025 17.014277
## 74 19.416046 18.831250 20.000841
## 75 17.837598 17.174242 18.500955
## 76 16.259151 15.504025 17.014277
## 77 22.257251 21.760844 22.753658
## 78 27.939661 27.333469 28.545854
## 79 26.203369 25.668280 26.738459
## 80 29.044574 28.381744 29.707404
## 81 26.361214 25.820758 26.901669
## 82 25.414146 24.902011 25.926280
## 83 24.624922 24.128663 25.121180
## 84 27.308282 26.730818 27.885746
## 85 26.045524 25.515552 26.575497
## 86 12.313033 11.295112 13.330953
## 87 16.259151 15.504025 17.014277
## 88 17.048375 16.340481 17.756268
## 89 18.311133 17.672969 18.949296
## 90 16.259151 15.504025 17.014277
## 91 8.682604 7.401151 9.964056
## 92 16.259151 15.504025 17.014277
## 93 14.996393 14.161242 15.831545
## 94 16.259151 15.504025 17.014277
## 95 5.999243 4.516273 7.482214
## 96 4.420796 2.817595 6.023998
## 97 12.313033 11.295112 13.330953
## 98 23.362164 22.874970 23.849358
## 99 24.151388 23.660958 24.641817
## 100 24.151388 23.660958 24.641817
## 101 26.045524 25.515552 26.575497
## 102 24.940611 24.438901 25.442321
## 103 32.675003 31.788265 33.561742
## 104 16.259151 15.504025 17.014277
## 105 13.575791 12.645619 14.505962
## 106 13.102256 12.139493 14.065020
## 107 11.523809 10.449826 12.597792
## 108 24.151388 23.660958 24.641817
## 109 26.045524 25.515552 26.575497
## 110 28.571040 27.933378 29.208703
## 111 25.098456 24.593565 25.603347
## 112 25.729835 25.209322 26.250348
## 113 26.519059 25.972996 27.065122
## 114 23.046475 22.558272 23.534677
## 115 25.729835 25.209322 26.250348
## 116 17.048375 16.340481 17.756268
## 117 3.631572 1.967894 5.295251
## 118 32.201469 31.346241 33.056697
## 119 28.097506 27.483687 28.711325
## 120 25.571990 25.055805 26.088176
## 121 22.257251 21.760844 22.753658
## 122 16.259151 15.504025 17.014277
## 123 22.572940 22.080777 23.065104
## 124 20.678804 20.143392 21.214215
## 125 11.523809 10.449826 12.597792
## 126 24.940611 24.438901 25.442321
## 127 NA NA NA
## 128 24.151388 23.660958 24.641817
## 129 24.151388 23.660958 24.641817
## 130 29.360264 28.679992 30.040536
## 131 27.308282 26.730818 27.885746
## 132 29.675953 28.977756 30.374151
## 133 28.097506 27.483687 28.711325
## 134 24.151388 23.660958 24.641817
## 135 22.572940 22.080777 23.065104
## 136 23.362164 22.874970 23.849358
## 137 17.837598 17.174242 18.500955
## 138 16.259151 15.504025 17.014277
## 139 16.259151 15.504025 17.014277
## 140 17.837598 17.174242 18.500955
## 141 16.259151 15.504025 17.014277
## 142 26.834748 26.276776 27.392720
## 143 29.360264 28.679992 30.040536
## 144 27.623972 27.032512 28.215432
## 145 31.727935 30.903669 32.552201
## 146 30.307332 29.571971 31.042693
## 147 28.097506 27.483687 28.711325
## 148 28.097506 27.483687 28.711325
## 149 28.097506 27.483687 28.711325
## 150 24.624922 24.128663 25.121180
## 151 25.256301 24.747933 25.764669
## 152 29.360264 28.679992 30.040536
## 153 24.940611 24.438901 25.442321
## 154 23.362164 22.874970 23.849358
## 155 28.571040 27.933378 29.208703
## 156 28.571040 27.933378 29.208703
## 157 13.102256 12.139493 14.065020
## 158 17.048375 16.340481 17.756268
## 159 16.259151 15.504025 17.014277
## 160 16.574840 15.838898 17.310783
## 161 22.572940 22.080777 23.065104
## 162 23.362164 22.874970 23.849358
## 163 22.572940 22.080777 23.065104
## 164 24.940611 24.438901 25.442321
## 165 22.572940 22.080777 23.065104
## 166 22.572940 22.080777 23.065104
## 167 19.573890 18.996010 20.151771
## 168 28.097506 27.483687 28.711325
## 169 26.834748 26.276776 27.392720
## 170 24.151388 23.660958 24.641817
## 171 27.623972 27.032512 28.215432
## 172 24.782767 24.283936 25.281597
## 173 28.728885 28.082973 29.374797
## 174 24.624922 24.128663 25.121180
## 175 24.624922 24.128663 25.121180
## 176 28.886730 28.232427 29.541033
## 177 25.729835 25.209322 26.250348
## 178 24.940611 24.438901 25.442321
## 179 26.045524 25.515552 26.575497
## 180 24.467077 23.973079 24.961075
## 181 21.783717 21.278621 22.288812
## 182 31.570090 30.756012 32.384168
## 183 26.361214 25.820758 26.901669
## 184 27.150438 26.579678 27.721197
## 185 25.414146 24.902011 25.926280
## 186 27.466127 26.881761 28.050493
## 187 26.834748 26.276776 27.392720
## 188 17.837598 17.174242 18.500955
## 189 16.259151 15.504025 17.014277
## 190 20.994493 20.469092 21.519894
## 191 15.943462 15.168798 16.718125
## 192 24.151388 23.660958 24.641817
## 193 23.362164 22.874970 23.849358
## 194 27.150438 26.579678 27.721197
## 195 25.729835 25.209322 26.250348
## 196 31.727935 30.903669 32.552201
## 197 30.465177 29.720276 31.210078
## 198 28.886730 28.232427 29.541033
## 199 31.570090 30.756012 32.384168
## 200 24.151388 23.660958 24.641817
## 201 27.623972 27.032512 28.215432
## 202 22.572940 22.080777 23.065104
## 203 24.940611 24.438901 25.442321
## 204 28.728885 28.082973 29.374797
## 205 28.886730 28.232427 29.541033
## 206 28.097506 27.483687 28.711325
## 207 28.571040 27.933378 29.208703
## 208 23.835698 23.347546 24.323850
## 209 16.259151 15.504025 17.014277
## 210 26.045524 25.515552 26.575497
## 211 22.888630 22.399432 23.377828
## 212 20.994493 20.469092 21.519894
## 213 11.523809 10.449826 12.597792
## 214 17.048375 16.340481 17.756268
## 215 19.416046 18.831250 20.000841
## 216 16.259151 15.504025 17.014277
## 217 29.202419 28.530931 29.873907
## 218 27.308282 26.730818 27.885746
## 219 30.780866 30.016612 31.545121
## 220 24.782767 24.283936 25.281597
## 221 28.886730 28.232427 29.541033
## 222 17.048375 16.340481 17.756268
## 223 22.572940 22.080777 23.065104
## 224 17.048375 16.340481 17.756268
## 225 19.416046 18.831250 20.000841
## 226 22.572940 22.080777 23.065104
## 227 23.362164 22.874970 23.849358
## 228 24.151388 23.660958 24.641817
## 229 24.467077 23.973079 24.961075
## 230 11.523809 10.449826 12.597792
## 231 13.102256 12.139493 14.065020
## 232 9.945362 8.757051 11.133672
## 233 16.416996 15.671508 17.162484
## 234 27.623972 27.032512 28.215432
## 235 26.045524 25.515552 26.575497
## 236 28.097506 27.483687 28.711325
## 237 25.887680 25.362568 26.412791
## 238 29.991643 29.275071 30.708215
## 239 26.834748 26.276776 27.392720
## 240 29.360264 28.679992 30.040536
## 241 27.623972 27.032512 28.215432
## 242 24.624922 24.128663 25.121180
## 243 22.572940 22.080777 23.065104
## 244 22.572940 22.080777 23.065104
## 245 32.359314 31.493641 33.224987
## 246 29.518109 28.828932 30.207285
## 247 31.727935 30.903669 32.552201
## 248 28.886730 28.232427 29.541033
## 249 30.465177 29.720276 31.210078
## 250 22.572940 22.080777 23.065104
## 251 17.837598 17.174242 18.500955
## 252 17.995443 17.340622 18.650264
## 253 23.362164 22.874970 23.849358
## 254 24.940611 24.438901 25.442321
## 255 26.519059 25.972996 27.065122
## 256 26.045524 25.515552 26.575497
## 257 24.151388 23.660958 24.641817
## 258 25.729835 25.209322 26.250348
## 259 23.362164 22.874970 23.849358
## 260 26.519059 25.972996 27.065122
## 261 22.572940 22.080777 23.065104
## 262 20.994493 20.469092 21.519894
## 263 17.048375 16.340481 17.756268
## 264 13.891480 12.982802 14.800158
## 265 17.995443 17.340622 18.650264
## 266 17.837598 17.174242 18.500955
## 267 29.202419 28.530931 29.873907
## 268 24.940611 24.438901 25.442321
## 269 24.624922 24.128663 25.121180
## 270 28.097506 27.483687 28.711325
## 271 24.940611 24.438901 25.442321
## 272 23.362164 22.874970 23.849358
## 273 26.519059 25.972996 27.065122
## 274 24.624922 24.128663 25.121180
## 275 23.677853 23.190350 24.165357
## 276 20.205269 19.653000 20.757539
## 277 21.783717 21.278621 22.288812
## 278 18.942511 18.335857 19.549166
## 279 28.728885 28.082973 29.374797
## 280 29.202419 28.530931 29.873907
## 281 21.783717 21.278621 22.288812
## 282 26.519059 25.972996 27.065122
## 283 26.045524 25.515552 26.575497
## 284 25.729835 25.209322 26.250348
## 285 22.572940 22.080777 23.065104
## 286 19.416046 18.831250 20.000841
## 287 19.573890 18.996010 20.151771
## 288 18.153288 17.506866 18.799709
## 289 18.626822 18.004730 19.248914
## 290 15.469927 14.665349 16.274505
## 291 17.521909 16.841095 18.202722
## 292 20.205269 19.653000 20.757539
## 293 16.259151 15.504025 17.014277
## 294 28.728885 28.082973 29.374797
## 295 29.675953 28.977756 30.374151
## 296 27.308282 26.730818 27.885746
## 297 27.308282 26.730818 27.885746
## 298 27.781817 27.183079 28.380554
## 299 20.205269 19.653000 20.757539
## 300 28.728885 28.082973 29.374797
## 301 25.729835 25.209322 26.250348
## 302 28.886730 28.232427 29.541033
## 303 28.886730 28.232427 29.541033
## 304 29.675953 28.977756 30.374151
## 305 29.044574 28.381744 29.707404
## 306 25.729835 25.209322 26.250348
## 307 21.783717 21.278621 22.288812
## 308 21.783717 21.278621 22.288812
## 309 25.729835 25.209322 26.250348
## 310 27.939661 27.333469 28.545854
## 311 30.465177 29.720276 31.210078
## 312 28.886730 28.232427 29.541033
## 313 29.675953 28.977756 30.374151
## 314 25.729835 25.209322 26.250348
## 315 26.045524 25.515552 26.575497
## 316 25.729835 25.209322 26.250348
## 317 25.729835 25.209322 26.250348
## 318 27.623972 27.032512 28.215432
## 319 25.729835 25.209322 26.250348
## 320 28.097506 27.483687 28.711325
## 321 25.414146 24.902011 25.926280
## 322 28.097506 27.483687 28.711325
## 323 29.675953 28.977756 30.374151
## 324 23.362164 22.874970 23.849358
## 325 29.675953 28.977756 30.374151
## 326 32.359314 31.493641 33.224987
## 327 32.359314 31.493641 33.224987
## 328 29.360264 28.679992 30.040536
## 329 29.360264 28.679992 30.040536
## 330 29.360264 28.679992 30.040536
## 331 NA NA NA
## 332 29.360264 28.679992 30.040536
## 333 30.149488 29.423571 30.875404
## 334 19.100356 18.501167 19.699545
## 335 24.151388 23.660958 24.641817
## 336 26.045524 25.515552 26.575497
## 337 NA NA NA
## 338 28.571040 27.933378 29.208703
## 339 26.676903 26.124999 27.228808
## 340 26.676903 26.124999 27.228808
## 341 25.414146 24.902011 25.926280
## 342 22.572940 22.080777 23.065104
## 343 26.676903 26.124999 27.228808
## 344 30.780866 30.016612 31.545121
## 345 29.833798 29.126467 30.541129
## 346 30.465177 29.720276 31.210078
## 347 29.360264 28.679992 30.040536
## 348 29.675953 28.977756 30.374151
## 349 30.149488 29.423571 30.875404
## 350 29.202419 28.530931 29.873907
## 351 29.991643 29.275071 30.708215
## 352 29.675953 28.977756 30.374151
## 353 29.675953 28.977756 30.374151
## 354 28.255351 27.633741 28.876961
## 355 NA NA NA
## 356 28.097506 27.483687 28.711325
## 357 28.097506 27.483687 28.711325
## 358 24.151388 23.660958 24.641817
## 359 28.255351 27.633741 28.876961
## 360 27.308282 26.730818 27.885746
## 361 27.939661 27.333469 28.545854
## 362 21.625872 21.117282 22.134462
## 363 20.994493 20.469092 21.519894
## 364 22.572940 22.080777 23.065104
## 365 23.362164 22.874970 23.849358
## 366 26.045524 25.515552 26.575497
## 367 26.519059 25.972996 27.065122
## 368 26.045524 25.515552 26.575497
## 369 26.045524 25.515552 26.575497
## 370 26.045524 25.515552 26.575497
## 371 26.519059 25.972996 27.065122
## 372 26.676903 26.124999 27.228808
## 373 25.729835 25.209322 26.250348
## 374 25.414146 24.902011 25.926280
## 375 28.255351 27.633741 28.876961
## 376 29.202419 28.530931 29.873907
## 377 29.202419 28.530931 29.873907
## 378 29.991643 29.275071 30.708215
## 379 28.886730 28.232427 29.541033
## 380 26.045524 25.515552 26.575497
## 381 28.097506 27.483687 28.711325
## 382 28.886730 28.232427 29.541033
## 383 29.360264 28.679992 30.040536
## 384 29.360264 28.679992 30.040536
## 385 29.360264 28.679992 30.040536
## 386 22.572940 22.080777 23.065104
## 387 26.519059 25.972996 27.065122
## 388 25.414146 24.902011 25.926280
## 389 22.257251 21.760844 22.753658
## 390 24.782767 24.283936 25.281597
## 391 26.676903 26.124999 27.228808
## 392 25.729835 25.209322 26.250348
## 393 26.361214 25.820758 26.901669
## 394 31.727935 30.903669 32.552201
## 395 26.676903 26.124999 27.228808
## 396 27.466127 26.881761 28.050493
## 397 26.992593 26.428333 27.556853#Horsepower of 98 confidence --> fit lwr upr
# 125.37566 122.44501 128.306312(b) Plot the response and the predictor. Use the abline() function to display the least squares regression line.
library(tidyverse)## -- Attaching packages ---------------------------------------------------------- tidyverse 1.2.1 --## v ggplot2 3.2.1 v purrr 0.3.2
## v tibble 2.1.3 v dplyr 0.8.3
## v tidyr 1.0.0 v stringr 1.4.0
## v readr 1.3.1 v forcats 0.4.0## -- Conflicts ------------------------------------------------------------- tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag() masks stats::lag()ggplot(auto, aes(x=mpg, y=horsepower))+
geom_point()+
geom_abline(slope=mod$coefficients[2], intercept=mod$coefficients[1],
color="blue", lty=2, lwd=1)## Warning: Removed 5 rows containing missing values (geom_point).
2(c) Use the plot() function to produce diagnostic plots of the least squares regression fit. Comment on any problems you see with the fit.
plot(mod)
#It has a large curve upwards near the end of the depicted data, which would indicate a skew.Problem 3 (textbook 3.9 ) This question involves the use of multiple linear regression on the Auto dataset from the textbook website. Load in the data and take out the quantitative variables for origin and name:
3(A) Produce a scatter plot matrix which includes all of the variables in the dataset
auto<-auto[,-c(8:9)]
pairs(auto)
3(B) Compute the matrix of correlations between the variables using the function
cor(auto)## mpg cylinders displacement horsepower weight
## mpg 1.0000000 -0.7762599 -0.8044430 NA -0.8317389
## cylinders -0.7762599 1.0000000 0.9509199 NA 0.8970169
## displacement -0.8044430 0.9509199 1.0000000 NA 0.9331044
## horsepower NA NA NA 1 NA
## weight -0.8317389 0.8970169 0.9331044 NA 1.0000000
## acceleration 0.4222974 -0.5040606 -0.5441618 NA -0.4195023
## year 0.5814695 -0.3467172 -0.3698041 NA -0.3079004
## acceleration year
## mpg 0.4222974 0.5814695
## cylinders -0.5040606 -0.3467172
## displacement -0.5441618 -0.3698041
## horsepower NA NA
## weight -0.4195023 -0.3079004
## acceleration 1.0000000 0.2829009
## year 0.2829009 1.00000003(C) Use the lm() function to perform a multiple linear regression with mpg as the response and all other variables (except origin and name) as predictors. Use the summary() function to print the results. Comment on the output.
names(auto)## [1] "mpg" "cylinders" "displacement" "horsepower"
## [5] "weight" "acceleration" "year"mlr_mod<-lm(mpg ~ cylinders+displacement+horsepower+weight+acceleration+year, data=auto)
summary(mlr_mod)##
## Call:
## lm(formula = mpg ~ cylinders + displacement + horsepower + weight +
## acceleration + year, data = auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.6927 -2.3864 -0.0801 2.0291 14.3607
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.454e+01 4.764e+00 -3.051 0.00244 **
## cylinders -3.299e-01 3.321e-01 -0.993 0.32122
## displacement 7.678e-03 7.358e-03 1.044 0.29733
## horsepower -3.914e-04 1.384e-02 -0.028 0.97745
## weight -6.795e-03 6.700e-04 -10.141 < 2e-16 ***
## acceleration 8.527e-02 1.020e-01 0.836 0.40383
## year 7.534e-01 5.262e-02 14.318 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.435 on 385 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.8093, Adjusted R-squared: 0.8063
## F-statistic: 272.2 on 6 and 385 DF, p-value: < 2.2e-16# 1. Is there a relationship between the predictors and the response?
# Yes.
# 2. Which predictors appear to have a statistically significant relationship to the response?
# Weight and year.
# 3. What does the coefficient for the year variable suggest?
# I am unsure and would like to review this.3(D): Write code using matrix algebra to produce the summary output (ie. ˆ vector and SE(ˆ), you’ll need to find the response vector and design matrix to do this).
Y<-as.matrix(mpg)
n<-dim(Y)[1]
X<-matrix(c(rep(1, n),
cylinders,
displacement,
horsepower,
weight,
acceleration,
year),
ncol=6,
byrow=FALSE)
dim(X)## [1] 436 63(E) Use the plot function in base R or use ggplot to produce diagnostic plots of linear regression fit. Comment on any problems you see with the fit. Do the residual plots suggest any unusually large outliers (in the y direction)?
plot(mlr_mod)
#It appears to have a skew in the upper right part of the data.HW #5
Problem 1
1A) Use the * and : symbols to fit linear regression models with interaction effects. Do any interactions appear to be statistically significant? (at the 0.05 level)
names(auto)## [1] "mpg" "cylinders" "displacement" "horsepower"
## [5] "weight" "acceleration" "year"autoMod1<-lm(mpg~cylinders*year, data=auto)
summary(autoMod1)#The interaction between cylinders and year does appear to be statistically significant##
## Call:
## lm(formula = mpg ~ cylinders * year, data = auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.2564 -2.5806 -0.1016 2.2535 15.2226
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -61.84732 15.06957 -4.104 4.94e-05 ***
## cylinders 5.54877 2.74057 2.025 0.04358 *
## year 1.34426 0.19865 6.767 4.79e-11 ***
## cylinders:year -0.11410 0.03652 -3.124 0.00191 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.154 on 393 degrees of freedom
## Multiple R-squared: 0.7204, Adjusted R-squared: 0.7183
## F-statistic: 337.5 on 3 and 393 DF, p-value: < 2.2e-16autoMod2<-lm(mpg~cylinders:displacement, data=auto)
summary(autoMod2)#Cylinders and displacement's relationship does appear to be statistically significant##
## Call:
## lm(formula = mpg ~ cylinders:displacement, data = auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.7498 -3.4685 -0.4891 2.7089 17.6733
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 31.0397951 0.3891622 79.76 <2e-16 ***
## cylinders:displacement -0.0061427 0.0002467 -24.90 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.888 on 395 degrees of freedom
## Multiple R-squared: 0.6109, Adjusted R-squared: 0.6099
## F-statistic: 620.2 on 1 and 395 DF, p-value: < 2.2e-16autoMod3<-lm(weight~horsepower:displacement, data=auto)
summary(autoMod3) #The interaction between horsepower and displacement appears significant. ##
## Call:
## lm(formula = weight ~ horsepower:displacement, data = auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2596.78 -264.87 -25.09 270.75 1027.21
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.153e+03 3.040e+01 70.82 <2e-16 ***
## horsepower:displacement 3.448e-02 9.423e-04 36.59 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 404 on 390 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.7744, Adjusted R-squared: 0.7738
## F-statistic: 1339 on 1 and 390 DF, p-value: < 2.2e-16(B) Try a few different transformations of the variables, such as log(X), √X, X2. Comment on your findings.
log(weight,10)#Answers appear to be between 3-4, which is approximately 1/1000 of the original responses.## [1] 3.544564 3.567379 3.536053 3.535674 3.537693 3.637590 3.638888
## [8] 3.634679 3.645913 3.585461 3.551816 3.557387 3.575303 3.489396
## [15] 3.375115 3.452247 3.443106 3.412796 3.328380 3.263636 3.426836
## [22] 3.385606 3.375664 3.349083 3.422918 3.664172 3.641077 3.641672
## [29] 3.675045 3.328380 3.354876 3.347915 3.310906 3.420616 3.536432
## [36] 3.522314 3.518777 3.516932 3.624179 3.649724 3.618466 3.612360
## [43] 3.695044 3.676328 3.710963 3.471585 3.381656 3.516139 3.496791
## [50] 3.346353 3.326950 3.316809 3.314920 3.248709 3.207634 3.263399
## [57] 3.291147 3.357554 3.327563 3.352954 3.381656 3.347525 3.630835
## [64] 3.641970 3.616476 3.615845 3.564903 3.665862 3.653405 3.648945
## [71] 3.645619 3.367356 3.590173 3.612572 3.632862 3.610341 3.467312
## [78] 3.399847 3.474071 3.340246 3.379306 3.359456 3.398981 3.335257
## [85] 3.322219 3.612784 3.564903 3.600755 3.606596 3.577147 3.694781
## [92] 3.649724 3.639785 3.627058 3.675320 3.694693 3.582177 3.494294
## [99] 3.515609 3.469085 3.480151 3.462997 3.290035 3.698709 3.690728
## [106] 3.667826 3.653116 3.445449 3.357744 3.380392 3.376394 3.327155
## [113] 3.363612 3.393048 3.355068 3.610873 3.631241 3.271144 3.334051
## [120] 3.411956 3.457579 3.531351 3.424882 3.448242 3.563955 3.491642
## [127] 3.458638 3.462548 3.523226 3.290035 3.389343 3.263873 3.405176
## [134] 3.577607 3.560146 3.557868 3.617105 3.672005 3.649043 3.666331
## [141] 3.629104 3.346157 3.292920 3.361728 3.217221 3.301681 3.327359
## [148] 3.323871 3.351410 3.396025 3.378580 3.301030 3.513750 3.538951
## [155] 3.535547 3.499412 3.669131 3.647383 3.653019 3.668106 3.591843
## [162] 3.590730 3.571709 3.578066 3.482731 3.507991 3.500922 3.336660
## [169] 3.421439 3.464490 3.413635 3.431685 3.346939 3.405688 3.474799
## [176] 3.287130 3.506640 3.430398 3.470851 3.469085 3.426674 3.254064
## [183] 3.391641 3.346353 3.410271 3.353147 3.342817 3.624798 3.622214
## [190] 3.597914 3.624798 3.509606 3.525434 3.478855 3.489255 3.308564
## [197] 3.335257 3.287130 3.254064 3.562412 3.553155 3.561698 3.504199
## [204] 3.261263 3.298853 3.333447 3.409087 3.498311 3.595496 3.514548
## [211] 3.466868 3.582063 3.641474 3.607991 3.587711 3.574610 3.310693
## [218] 3.333447 3.261263 3.361728 3.288920 3.588832 3.608526 3.617000
## [225] 3.632963 3.546543 3.534661 3.559907 3.547159 3.625312 3.619615
## [232] 3.635986 3.636989 3.287802 3.437751 3.355068 3.440122 3.311966
## [239] 3.317018 3.297761 3.340444 3.449478 3.414973 3.434569 3.297761
## [246] 3.255273 3.297761 3.315970 3.255273 3.526985 3.572291 3.552668
## [253] 3.548389 3.498999 3.472025 3.434569 3.535294 3.506505 3.528917
## [260] 3.487138 3.558709 3.532754 3.534661 3.537189 3.505828 3.610660
## [267] 3.333447 3.408240 3.361728 3.348305 3.400538 3.438542 3.455606
## [274] 3.381115 3.451786 3.496930 3.446382 3.532754 3.298853 3.329398
## [281] 3.511215 3.475671 3.460898 3.513883 3.526339 3.584331 3.571126
## [288] 3.597146 3.583199 3.639486 3.607884 3.556905 3.595496 3.284431
## [295] 3.295567 3.282169 3.426511 3.547775 3.591065 3.503791 3.534026
## [302] 3.342423 3.332438 3.305351 3.328380 3.426511 3.414137 3.431364
## [309] 3.407561 3.331225 3.294025 3.326336 3.305136 3.427811 3.457882
## [316] 3.477555 3.529045 3.340047 3.433130 3.405176 3.386321 3.355068
## [323] 3.324282 3.447158 3.324282 3.319106 3.368287 3.469822 3.511883
## [330] 3.267172 3.263636 3.331427 3.265996 3.463893 3.383815 3.397940
## [337] 3.463146 3.359835 3.396199 3.420781 3.418301 3.435367 3.377488
## [344] 3.244277 3.273001 3.245513 3.314920 3.295567 3.311754 3.297761
## [351] 3.345374 3.310693 3.376577 3.340444 3.365488 3.344392 3.371068
## [358] 3.417472 3.420781 3.509203 3.499687 3.462398 3.466868 3.533391
## [365] 3.571126 3.485721 3.539703 3.415808 3.421604 3.379306 3.410777
## [372] 3.402261 3.436957 3.457125 3.296665 3.306425 3.294466 3.327359
## [379] 3.327359 3.334454 3.343409 3.351216 3.293363 3.293363 3.299943
## [386] 3.469085 3.479287 3.412461 3.452553 3.425697 3.374748 3.469822
## [393] 3.445604 3.328380 3.360783 3.419129 3.434569weight*2 #Each response has doubled. ## [1] 7008 7386 6872 6866 6898 8682 8708 8624 8850 7700 7126
## [12] 7218 7522 6172 4744 5666 5548 5174 4260 3670 5344 4860
## [23] 4750 4468 5296 9230 8752 8764 9464 4260 4528 4456 4092
## [34] 5268 6878 6658 6604 6576 8418 8928 8308 8192 9910 9492
## [45] 10280 5924 4816 6564 6278 4440 4246 4148 4130 3546 3226
## [56] 3668 3910 4556 4252 4508 4816 4452 8548 8770 8270 8258
## [67] 7344 9266 9004 8912 8844 4660 7784 8196 8588 8154 5866
## [78] 5022 5958 4378 4790 4576 5012 4328 4200 8200 7344 7976
## [89] 8084 7554 9904 8928 8726 8474 9470 9902 7642 6242 6556
## [100] 5890 6042 5808 3900 9994 9812 9308 8998 5578 4558 4802
## [111] 4758 4248 4620 4944 4530 8164 8556 3734 4316 5164 5736
## [122] 6798 5320 5614 7328 6204 5750 5802 6672 3900 4902 3672
## [133] 5084 7562 7264 7226 8282 9398 8914 9276 8514 4438 3926
## [144] 4600 3298 4006 4250 4216 4492 4978 4782 4000 6528 6918
## [155] 6864 6316 9336 8880 8996 9314 7814 7794 7460 7570 6078
## [166] 6442 6338 4342 5278 5828 5184 5404 4446 5090 5968 3874
## [177] 6422 5388 5914 5890 5342 3590 4928 4440 5144 4510 4404
## [188] 8430 8380 7924 8430 6466 6706 6024 6170 4070 4328 3874
## [199] 3590 7302 7148 7290 6386 3650 3980 4310 5130 6300 7880
## [210] 6540 5860 7640 8760 8110 7740 7510 4090 4310 3650 4600
## [221] 3890 7760 8120 8280 8590 7040 6850 7260 7050 8440 8330
## [232] 8650 8670 3880 5480 4530 5510 4102 4150 3970 4380 5630
## [243] 5200 5440 3970 3600 3970 4140 3600 6730 7470 7140 7070
## [254] 6310 5930 5440 6860 6420 6760 6140 7240 6820 6850 6890
## [265] 6410 8160 4310 5120 4600 4460 5030 5490 5710 4810 5660
## [276] 6280 5590 6820 3980 4270 6490 5980 5780 6530 6720 7680
## [287] 7450 7910 7660 8720 8108 7210 7880 3850 3950 3830 5340
## [298] 7060 7800 6380 6840 4400 4300 4040 4260 5340 5190 5400
## [309] 5112 4288 3936 4240 4038 5356 5740 6006 6762 4376 5422
## [320] 5084 4868 4530 4220 5600 4220 4170 4670 5900 6500 3700
## [331] 3670 4290 3690 5820 4840 5000 5810 4580 4980 5270 5240
## [342] 5450 4770 3510 3750 3520 4130 3950 4100 3970 4430 4090
## [353] 4760 4380 4640 4420 4700 5230 5270 6460 6320 5800 5860
## [364] 6830 7450 6120 6930 5210 5280 4790 5150 5050 5470 5730
## [375] 3960 4050 3940 4250 4250 4320 4410 4490 3930 3930 3990
## [386] 5890 6030 5170 5670 5330 4740 5900 5580 4260 4590 5250
## [397] 5440log(cylinders,10) #Answers appear to be between 1-3,which is approximately 1/100 of the original responses.## [1] 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900
## [8] 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900
## [15] 0.6020600 0.7781513 0.7781513 0.7781513 0.6020600 0.6020600 0.6020600
## [22] 0.6020600 0.6020600 0.6020600 0.7781513 0.9030900 0.9030900 0.9030900
## [29] 0.9030900 0.6020600 0.6020600 0.6020600 0.6020600 0.7781513 0.7781513
## [36] 0.7781513 0.7781513 0.7781513 0.9030900 0.9030900 0.9030900 0.9030900
## [43] 0.9030900 0.9030900 0.9030900 0.7781513 0.6020600 0.7781513 0.7781513
## [50] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [57] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.9030900
## [64] 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900
## [71] 0.9030900 0.4771213 0.9030900 0.9030900 0.9030900 0.9030900 0.6020600
## [78] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [85] 0.6020600 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900
## [92] 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.7781513
## [99] 0.7781513 0.7781513 0.7781513 0.7781513 0.6020600 0.9030900 0.9030900
## [106] 0.9030900 0.9030900 0.7781513 0.6020600 0.6020600 0.6020600 0.4771213
## [113] 0.6020600 0.7781513 0.6020600 0.9030900 0.9030900 0.6020600 0.6020600
## [120] 0.6020600 0.6020600 0.9030900 0.6020600 0.7781513 0.9030900 0.7781513
## [127] 0.7781513 0.7781513 0.7781513 0.6020600 0.6020600 0.6020600 0.6020600
## [134] 0.7781513 0.7781513 0.7781513 0.9030900 0.9030900 0.9030900 0.9030900
## [141] 0.9030900 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [148] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.7781513 0.7781513
## [155] 0.7781513 0.7781513 0.9030900 0.9030900 0.9030900 0.9030900 0.7781513
## [162] 0.7781513 0.7781513 0.7781513 0.7781513 0.9030900 0.9030900 0.6020600
## [169] 0.6020600 0.7781513 0.6020600 0.6020600 0.6020600 0.6020600 0.7781513
## [176] 0.6020600 0.7781513 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [183] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.9030900 0.9030900
## [190] 0.9030900 0.9030900 0.7781513 0.7781513 0.7781513 0.7781513 0.6020600
## [197] 0.6020600 0.6020600 0.6020600 0.7781513 0.7781513 0.7781513 0.7781513
## [204] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.9030900 0.6020600
## [211] 0.7781513 0.7781513 0.9030900 0.9030900 0.9030900 0.9030900 0.6020600
## [218] 0.6020600 0.6020600 0.6020600 0.6020600 0.9030900 0.9030900 0.9030900
## [225] 0.9030900 0.7781513 0.7781513 0.7781513 0.7781513 0.9030900 0.9030900
## [232] 0.9030900 0.9030900 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [239] 0.6020600 0.6020600 0.6020600 0.7781513 0.6020600 0.4771213 0.6020600
## [246] 0.6020600 0.6020600 0.6020600 0.6020600 0.9030900 0.9030900 0.9030900
## [253] 0.7781513 0.7781513 0.7781513 0.6020600 0.7781513 0.7781513 0.7781513
## [260] 0.7781513 0.7781513 0.7781513 0.9030900 0.7781513 0.9030900 0.9030900
## [267] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [274] 0.6020600 0.6989700 0.7781513 0.6020600 0.7781513 0.6020600 0.6020600
## [281] 0.7781513 0.7781513 0.6020600 0.7781513 0.7781513 0.9030900 0.9030900
## [288] 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.6020600
## [295] 0.6020600 0.6020600 0.6020600 0.6989700 0.9030900 0.6020600 0.9030900
## [302] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.7781513 0.7781513
## [309] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [316] 0.6020600 0.7781513 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [323] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6989700 0.6020600
## [330] 0.6020600 0.6020600 0.6020600 0.6020600 0.7781513 0.4771213 0.6020600
## [337] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.7781513 0.6020600
## [344] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [351] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [358] 0.6020600 0.6020600 0.6020600 0.7781513 0.7781513 0.7781513 0.7781513
## [365] 0.9030900 0.7781513 0.7781513 0.6020600 0.6020600 0.6020600 0.6020600
## [372] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [379] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [386] 0.7781513 0.7781513 0.6020600 0.7781513 0.6020600 0.6020600 0.6020600
## [393] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600sqrt(cylinders)#Answers appear to be betweeen 2-2.5, with a smaller range than the log responses.## [1] 2.828427 2.828427 2.828427 2.828427 2.828427 2.828427 2.828427
## [8] 2.828427 2.828427 2.828427 2.828427 2.828427 2.828427 2.828427
## [15] 2.000000 2.449490 2.449490 2.449490 2.000000 2.000000 2.000000
## [22] 2.000000 2.000000 2.000000 2.449490 2.828427 2.828427 2.828427
## [29] 2.828427 2.000000 2.000000 2.000000 2.000000 2.449490 2.449490
## [36] 2.449490 2.449490 2.449490 2.828427 2.828427 2.828427 2.828427
## [43] 2.828427 2.828427 2.828427 2.449490 2.000000 2.449490 2.449490
## [50] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [57] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.828427
## [64] 2.828427 2.828427 2.828427 2.828427 2.828427 2.828427 2.828427
## [71] 2.828427 1.732051 2.828427 2.828427 2.828427 2.828427 2.000000
## [78] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [85] 2.000000 2.828427 2.828427 2.828427 2.828427 2.828427 2.828427
## [92] 2.828427 2.828427 2.828427 2.828427 2.828427 2.828427 2.449490
## [99] 2.449490 2.449490 2.449490 2.449490 2.000000 2.828427 2.828427
## [106] 2.828427 2.828427 2.449490 2.000000 2.000000 2.000000 1.732051
## [113] 2.000000 2.449490 2.000000 2.828427 2.828427 2.000000 2.000000
## [120] 2.000000 2.000000 2.828427 2.000000 2.449490 2.828427 2.449490
## [127] 2.449490 2.449490 2.449490 2.000000 2.000000 2.000000 2.000000
## [134] 2.449490 2.449490 2.449490 2.828427 2.828427 2.828427 2.828427
## [141] 2.828427 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [148] 2.000000 2.000000 2.000000 2.000000 2.000000 2.449490 2.449490
## [155] 2.449490 2.449490 2.828427 2.828427 2.828427 2.828427 2.449490
## [162] 2.449490 2.449490 2.449490 2.449490 2.828427 2.828427 2.000000
## [169] 2.000000 2.449490 2.000000 2.000000 2.000000 2.000000 2.449490
## [176] 2.000000 2.449490 2.000000 2.000000 2.000000 2.000000 2.000000
## [183] 2.000000 2.000000 2.000000 2.000000 2.000000 2.828427 2.828427
## [190] 2.828427 2.828427 2.449490 2.449490 2.449490 2.449490 2.000000
## [197] 2.000000 2.000000 2.000000 2.449490 2.449490 2.449490 2.449490
## [204] 2.000000 2.000000 2.000000 2.000000 2.000000 2.828427 2.000000
## [211] 2.449490 2.449490 2.828427 2.828427 2.828427 2.828427 2.000000
## [218] 2.000000 2.000000 2.000000 2.000000 2.828427 2.828427 2.828427
## [225] 2.828427 2.449490 2.449490 2.449490 2.449490 2.828427 2.828427
## [232] 2.828427 2.828427 2.000000 2.000000 2.000000 2.000000 2.000000
## [239] 2.000000 2.000000 2.000000 2.449490 2.000000 1.732051 2.000000
## [246] 2.000000 2.000000 2.000000 2.000000 2.828427 2.828427 2.828427
## [253] 2.449490 2.449490 2.449490 2.000000 2.449490 2.449490 2.449490
## [260] 2.449490 2.449490 2.449490 2.828427 2.449490 2.828427 2.828427
## [267] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [274] 2.000000 2.236068 2.449490 2.000000 2.449490 2.000000 2.000000
## [281] 2.449490 2.449490 2.000000 2.449490 2.449490 2.828427 2.828427
## [288] 2.828427 2.828427 2.828427 2.828427 2.828427 2.828427 2.000000
## [295] 2.000000 2.000000 2.000000 2.236068 2.828427 2.000000 2.828427
## [302] 2.000000 2.000000 2.000000 2.000000 2.000000 2.449490 2.449490
## [309] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [316] 2.000000 2.449490 2.000000 2.000000 2.000000 2.000000 2.000000
## [323] 2.000000 2.000000 2.000000 2.000000 2.000000 2.236068 2.000000
## [330] 2.000000 2.000000 2.000000 2.000000 2.449490 1.732051 2.000000
## [337] 2.000000 2.000000 2.000000 2.000000 2.000000 2.449490 2.000000
## [344] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [351] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [358] 2.000000 2.000000 2.000000 2.449490 2.449490 2.449490 2.449490
## [365] 2.828427 2.449490 2.449490 2.000000 2.000000 2.000000 2.000000
## [372] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [379] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [386] 2.449490 2.449490 2.000000 2.449490 2.000000 2.000000 2.000000
## [393] 2.000000 2.000000 2.000000 2.000000 2.0000002B) Try a few different transformations of the variables, such as log(X), √X, X2. Comment on your findings.
log(weight,10)#Answers appear to be between 3-4, which is approximately 1/1000 of the original responses.## [1] 3.544564 3.567379 3.536053 3.535674 3.537693 3.637590 3.638888
## [8] 3.634679 3.645913 3.585461 3.551816 3.557387 3.575303 3.489396
## [15] 3.375115 3.452247 3.443106 3.412796 3.328380 3.263636 3.426836
## [22] 3.385606 3.375664 3.349083 3.422918 3.664172 3.641077 3.641672
## [29] 3.675045 3.328380 3.354876 3.347915 3.310906 3.420616 3.536432
## [36] 3.522314 3.518777 3.516932 3.624179 3.649724 3.618466 3.612360
## [43] 3.695044 3.676328 3.710963 3.471585 3.381656 3.516139 3.496791
## [50] 3.346353 3.326950 3.316809 3.314920 3.248709 3.207634 3.263399
## [57] 3.291147 3.357554 3.327563 3.352954 3.381656 3.347525 3.630835
## [64] 3.641970 3.616476 3.615845 3.564903 3.665862 3.653405 3.648945
## [71] 3.645619 3.367356 3.590173 3.612572 3.632862 3.610341 3.467312
## [78] 3.399847 3.474071 3.340246 3.379306 3.359456 3.398981 3.335257
## [85] 3.322219 3.612784 3.564903 3.600755 3.606596 3.577147 3.694781
## [92] 3.649724 3.639785 3.627058 3.675320 3.694693 3.582177 3.494294
## [99] 3.515609 3.469085 3.480151 3.462997 3.290035 3.698709 3.690728
## [106] 3.667826 3.653116 3.445449 3.357744 3.380392 3.376394 3.327155
## [113] 3.363612 3.393048 3.355068 3.610873 3.631241 3.271144 3.334051
## [120] 3.411956 3.457579 3.531351 3.424882 3.448242 3.563955 3.491642
## [127] 3.458638 3.462548 3.523226 3.290035 3.389343 3.263873 3.405176
## [134] 3.577607 3.560146 3.557868 3.617105 3.672005 3.649043 3.666331
## [141] 3.629104 3.346157 3.292920 3.361728 3.217221 3.301681 3.327359
## [148] 3.323871 3.351410 3.396025 3.378580 3.301030 3.513750 3.538951
## [155] 3.535547 3.499412 3.669131 3.647383 3.653019 3.668106 3.591843
## [162] 3.590730 3.571709 3.578066 3.482731 3.507991 3.500922 3.336660
## [169] 3.421439 3.464490 3.413635 3.431685 3.346939 3.405688 3.474799
## [176] 3.287130 3.506640 3.430398 3.470851 3.469085 3.426674 3.254064
## [183] 3.391641 3.346353 3.410271 3.353147 3.342817 3.624798 3.622214
## [190] 3.597914 3.624798 3.509606 3.525434 3.478855 3.489255 3.308564
## [197] 3.335257 3.287130 3.254064 3.562412 3.553155 3.561698 3.504199
## [204] 3.261263 3.298853 3.333447 3.409087 3.498311 3.595496 3.514548
## [211] 3.466868 3.582063 3.641474 3.607991 3.587711 3.574610 3.310693
## [218] 3.333447 3.261263 3.361728 3.288920 3.588832 3.608526 3.617000
## [225] 3.632963 3.546543 3.534661 3.559907 3.547159 3.625312 3.619615
## [232] 3.635986 3.636989 3.287802 3.437751 3.355068 3.440122 3.311966
## [239] 3.317018 3.297761 3.340444 3.449478 3.414973 3.434569 3.297761
## [246] 3.255273 3.297761 3.315970 3.255273 3.526985 3.572291 3.552668
## [253] 3.548389 3.498999 3.472025 3.434569 3.535294 3.506505 3.528917
## [260] 3.487138 3.558709 3.532754 3.534661 3.537189 3.505828 3.610660
## [267] 3.333447 3.408240 3.361728 3.348305 3.400538 3.438542 3.455606
## [274] 3.381115 3.451786 3.496930 3.446382 3.532754 3.298853 3.329398
## [281] 3.511215 3.475671 3.460898 3.513883 3.526339 3.584331 3.571126
## [288] 3.597146 3.583199 3.639486 3.607884 3.556905 3.595496 3.284431
## [295] 3.295567 3.282169 3.426511 3.547775 3.591065 3.503791 3.534026
## [302] 3.342423 3.332438 3.305351 3.328380 3.426511 3.414137 3.431364
## [309] 3.407561 3.331225 3.294025 3.326336 3.305136 3.427811 3.457882
## [316] 3.477555 3.529045 3.340047 3.433130 3.405176 3.386321 3.355068
## [323] 3.324282 3.447158 3.324282 3.319106 3.368287 3.469822 3.511883
## [330] 3.267172 3.263636 3.331427 3.265996 3.463893 3.383815 3.397940
## [337] 3.463146 3.359835 3.396199 3.420781 3.418301 3.435367 3.377488
## [344] 3.244277 3.273001 3.245513 3.314920 3.295567 3.311754 3.297761
## [351] 3.345374 3.310693 3.376577 3.340444 3.365488 3.344392 3.371068
## [358] 3.417472 3.420781 3.509203 3.499687 3.462398 3.466868 3.533391
## [365] 3.571126 3.485721 3.539703 3.415808 3.421604 3.379306 3.410777
## [372] 3.402261 3.436957 3.457125 3.296665 3.306425 3.294466 3.327359
## [379] 3.327359 3.334454 3.343409 3.351216 3.293363 3.293363 3.299943
## [386] 3.469085 3.479287 3.412461 3.452553 3.425697 3.374748 3.469822
## [393] 3.445604 3.328380 3.360783 3.419129 3.434569weight*2 #Each response has doubled. ## [1] 7008 7386 6872 6866 6898 8682 8708 8624 8850 7700 7126
## [12] 7218 7522 6172 4744 5666 5548 5174 4260 3670 5344 4860
## [23] 4750 4468 5296 9230 8752 8764 9464 4260 4528 4456 4092
## [34] 5268 6878 6658 6604 6576 8418 8928 8308 8192 9910 9492
## [45] 10280 5924 4816 6564 6278 4440 4246 4148 4130 3546 3226
## [56] 3668 3910 4556 4252 4508 4816 4452 8548 8770 8270 8258
## [67] 7344 9266 9004 8912 8844 4660 7784 8196 8588 8154 5866
## [78] 5022 5958 4378 4790 4576 5012 4328 4200 8200 7344 7976
## [89] 8084 7554 9904 8928 8726 8474 9470 9902 7642 6242 6556
## [100] 5890 6042 5808 3900 9994 9812 9308 8998 5578 4558 4802
## [111] 4758 4248 4620 4944 4530 8164 8556 3734 4316 5164 5736
## [122] 6798 5320 5614 7328 6204 5750 5802 6672 3900 4902 3672
## [133] 5084 7562 7264 7226 8282 9398 8914 9276 8514 4438 3926
## [144] 4600 3298 4006 4250 4216 4492 4978 4782 4000 6528 6918
## [155] 6864 6316 9336 8880 8996 9314 7814 7794 7460 7570 6078
## [166] 6442 6338 4342 5278 5828 5184 5404 4446 5090 5968 3874
## [177] 6422 5388 5914 5890 5342 3590 4928 4440 5144 4510 4404
## [188] 8430 8380 7924 8430 6466 6706 6024 6170 4070 4328 3874
## [199] 3590 7302 7148 7290 6386 3650 3980 4310 5130 6300 7880
## [210] 6540 5860 7640 8760 8110 7740 7510 4090 4310 3650 4600
## [221] 3890 7760 8120 8280 8590 7040 6850 7260 7050 8440 8330
## [232] 8650 8670 3880 5480 4530 5510 4102 4150 3970 4380 5630
## [243] 5200 5440 3970 3600 3970 4140 3600 6730 7470 7140 7070
## [254] 6310 5930 5440 6860 6420 6760 6140 7240 6820 6850 6890
## [265] 6410 8160 4310 5120 4600 4460 5030 5490 5710 4810 5660
## [276] 6280 5590 6820 3980 4270 6490 5980 5780 6530 6720 7680
## [287] 7450 7910 7660 8720 8108 7210 7880 3850 3950 3830 5340
## [298] 7060 7800 6380 6840 4400 4300 4040 4260 5340 5190 5400
## [309] 5112 4288 3936 4240 4038 5356 5740 6006 6762 4376 5422
## [320] 5084 4868 4530 4220 5600 4220 4170 4670 5900 6500 3700
## [331] 3670 4290 3690 5820 4840 5000 5810 4580 4980 5270 5240
## [342] 5450 4770 3510 3750 3520 4130 3950 4100 3970 4430 4090
## [353] 4760 4380 4640 4420 4700 5230 5270 6460 6320 5800 5860
## [364] 6830 7450 6120 6930 5210 5280 4790 5150 5050 5470 5730
## [375] 3960 4050 3940 4250 4250 4320 4410 4490 3930 3930 3990
## [386] 5890 6030 5170 5670 5330 4740 5900 5580 4260 4590 5250
## [397] 5440log(cylinders,10) #Answers appear to be between 1-3,which is approximately 1/100 of the original responses.## [1] 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900
## [8] 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900
## [15] 0.6020600 0.7781513 0.7781513 0.7781513 0.6020600 0.6020600 0.6020600
## [22] 0.6020600 0.6020600 0.6020600 0.7781513 0.9030900 0.9030900 0.9030900
## [29] 0.9030900 0.6020600 0.6020600 0.6020600 0.6020600 0.7781513 0.7781513
## [36] 0.7781513 0.7781513 0.7781513 0.9030900 0.9030900 0.9030900 0.9030900
## [43] 0.9030900 0.9030900 0.9030900 0.7781513 0.6020600 0.7781513 0.7781513
## [50] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [57] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.9030900
## [64] 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900
## [71] 0.9030900 0.4771213 0.9030900 0.9030900 0.9030900 0.9030900 0.6020600
## [78] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [85] 0.6020600 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900
## [92] 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.7781513
## [99] 0.7781513 0.7781513 0.7781513 0.7781513 0.6020600 0.9030900 0.9030900
## [106] 0.9030900 0.9030900 0.7781513 0.6020600 0.6020600 0.6020600 0.4771213
## [113] 0.6020600 0.7781513 0.6020600 0.9030900 0.9030900 0.6020600 0.6020600
## [120] 0.6020600 0.6020600 0.9030900 0.6020600 0.7781513 0.9030900 0.7781513
## [127] 0.7781513 0.7781513 0.7781513 0.6020600 0.6020600 0.6020600 0.6020600
## [134] 0.7781513 0.7781513 0.7781513 0.9030900 0.9030900 0.9030900 0.9030900
## [141] 0.9030900 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [148] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.7781513 0.7781513
## [155] 0.7781513 0.7781513 0.9030900 0.9030900 0.9030900 0.9030900 0.7781513
## [162] 0.7781513 0.7781513 0.7781513 0.7781513 0.9030900 0.9030900 0.6020600
## [169] 0.6020600 0.7781513 0.6020600 0.6020600 0.6020600 0.6020600 0.7781513
## [176] 0.6020600 0.7781513 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [183] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.9030900 0.9030900
## [190] 0.9030900 0.9030900 0.7781513 0.7781513 0.7781513 0.7781513 0.6020600
## [197] 0.6020600 0.6020600 0.6020600 0.7781513 0.7781513 0.7781513 0.7781513
## [204] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.9030900 0.6020600
## [211] 0.7781513 0.7781513 0.9030900 0.9030900 0.9030900 0.9030900 0.6020600
## [218] 0.6020600 0.6020600 0.6020600 0.6020600 0.9030900 0.9030900 0.9030900
## [225] 0.9030900 0.7781513 0.7781513 0.7781513 0.7781513 0.9030900 0.9030900
## [232] 0.9030900 0.9030900 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [239] 0.6020600 0.6020600 0.6020600 0.7781513 0.6020600 0.4771213 0.6020600
## [246] 0.6020600 0.6020600 0.6020600 0.6020600 0.9030900 0.9030900 0.9030900
## [253] 0.7781513 0.7781513 0.7781513 0.6020600 0.7781513 0.7781513 0.7781513
## [260] 0.7781513 0.7781513 0.7781513 0.9030900 0.7781513 0.9030900 0.9030900
## [267] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [274] 0.6020600 0.6989700 0.7781513 0.6020600 0.7781513 0.6020600 0.6020600
## [281] 0.7781513 0.7781513 0.6020600 0.7781513 0.7781513 0.9030900 0.9030900
## [288] 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.9030900 0.6020600
## [295] 0.6020600 0.6020600 0.6020600 0.6989700 0.9030900 0.6020600 0.9030900
## [302] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.7781513 0.7781513
## [309] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [316] 0.6020600 0.7781513 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [323] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6989700 0.6020600
## [330] 0.6020600 0.6020600 0.6020600 0.6020600 0.7781513 0.4771213 0.6020600
## [337] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.7781513 0.6020600
## [344] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [351] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [358] 0.6020600 0.6020600 0.6020600 0.7781513 0.7781513 0.7781513 0.7781513
## [365] 0.9030900 0.7781513 0.7781513 0.6020600 0.6020600 0.6020600 0.6020600
## [372] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [379] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600
## [386] 0.7781513 0.7781513 0.6020600 0.7781513 0.6020600 0.6020600 0.6020600
## [393] 0.6020600 0.6020600 0.6020600 0.6020600 0.6020600sqrt(cylinders)#Answers appear to be betweeen 2-2.5, with a smaller range than the log responses.## [1] 2.828427 2.828427 2.828427 2.828427 2.828427 2.828427 2.828427
## [8] 2.828427 2.828427 2.828427 2.828427 2.828427 2.828427 2.828427
## [15] 2.000000 2.449490 2.449490 2.449490 2.000000 2.000000 2.000000
## [22] 2.000000 2.000000 2.000000 2.449490 2.828427 2.828427 2.828427
## [29] 2.828427 2.000000 2.000000 2.000000 2.000000 2.449490 2.449490
## [36] 2.449490 2.449490 2.449490 2.828427 2.828427 2.828427 2.828427
## [43] 2.828427 2.828427 2.828427 2.449490 2.000000 2.449490 2.449490
## [50] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [57] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.828427
## [64] 2.828427 2.828427 2.828427 2.828427 2.828427 2.828427 2.828427
## [71] 2.828427 1.732051 2.828427 2.828427 2.828427 2.828427 2.000000
## [78] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [85] 2.000000 2.828427 2.828427 2.828427 2.828427 2.828427 2.828427
## [92] 2.828427 2.828427 2.828427 2.828427 2.828427 2.828427 2.449490
## [99] 2.449490 2.449490 2.449490 2.449490 2.000000 2.828427 2.828427
## [106] 2.828427 2.828427 2.449490 2.000000 2.000000 2.000000 1.732051
## [113] 2.000000 2.449490 2.000000 2.828427 2.828427 2.000000 2.000000
## [120] 2.000000 2.000000 2.828427 2.000000 2.449490 2.828427 2.449490
## [127] 2.449490 2.449490 2.449490 2.000000 2.000000 2.000000 2.000000
## [134] 2.449490 2.449490 2.449490 2.828427 2.828427 2.828427 2.828427
## [141] 2.828427 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [148] 2.000000 2.000000 2.000000 2.000000 2.000000 2.449490 2.449490
## [155] 2.449490 2.449490 2.828427 2.828427 2.828427 2.828427 2.449490
## [162] 2.449490 2.449490 2.449490 2.449490 2.828427 2.828427 2.000000
## [169] 2.000000 2.449490 2.000000 2.000000 2.000000 2.000000 2.449490
## [176] 2.000000 2.449490 2.000000 2.000000 2.000000 2.000000 2.000000
## [183] 2.000000 2.000000 2.000000 2.000000 2.000000 2.828427 2.828427
## [190] 2.828427 2.828427 2.449490 2.449490 2.449490 2.449490 2.000000
## [197] 2.000000 2.000000 2.000000 2.449490 2.449490 2.449490 2.449490
## [204] 2.000000 2.000000 2.000000 2.000000 2.000000 2.828427 2.000000
## [211] 2.449490 2.449490 2.828427 2.828427 2.828427 2.828427 2.000000
## [218] 2.000000 2.000000 2.000000 2.000000 2.828427 2.828427 2.828427
## [225] 2.828427 2.449490 2.449490 2.449490 2.449490 2.828427 2.828427
## [232] 2.828427 2.828427 2.000000 2.000000 2.000000 2.000000 2.000000
## [239] 2.000000 2.000000 2.000000 2.449490 2.000000 1.732051 2.000000
## [246] 2.000000 2.000000 2.000000 2.000000 2.828427 2.828427 2.828427
## [253] 2.449490 2.449490 2.449490 2.000000 2.449490 2.449490 2.449490
## [260] 2.449490 2.449490 2.449490 2.828427 2.449490 2.828427 2.828427
## [267] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [274] 2.000000 2.236068 2.449490 2.000000 2.449490 2.000000 2.000000
## [281] 2.449490 2.449490 2.000000 2.449490 2.449490 2.828427 2.828427
## [288] 2.828427 2.828427 2.828427 2.828427 2.828427 2.828427 2.000000
## [295] 2.000000 2.000000 2.000000 2.236068 2.828427 2.000000 2.828427
## [302] 2.000000 2.000000 2.000000 2.000000 2.000000 2.449490 2.449490
## [309] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [316] 2.000000 2.449490 2.000000 2.000000 2.000000 2.000000 2.000000
## [323] 2.000000 2.000000 2.000000 2.000000 2.000000 2.236068 2.000000
## [330] 2.000000 2.000000 2.000000 2.000000 2.449490 1.732051 2.000000
## [337] 2.000000 2.000000 2.000000 2.000000 2.000000 2.449490 2.000000
## [344] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [351] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [358] 2.000000 2.000000 2.000000 2.449490 2.449490 2.449490 2.449490
## [365] 2.828427 2.449490 2.449490 2.000000 2.000000 2.000000 2.000000
## [372] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [379] 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000
## [386] 2.449490 2.449490 2.000000 2.449490 2.000000 2.000000 2.000000
## [393] 2.000000 2.000000 2.000000 2.000000 2.000000Problem 2 (textbook 3.10). This question involves the use of multiple linear regression with categorical variables on the Carseat dataset from the textbook website.
2A) Fit a multiple regression model to predict Sales using Price, Urban, and US.
library(ISLR)
data(Carseats)
carMod<-lm(Sales~Price+Urban+US, data=Carseats)
summary(carMod)##
## Call:
## lm(formula = Sales ~ Price + Urban + US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9206 -1.6220 -0.0564 1.5786 7.0581
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.043469 0.651012 20.036 < 2e-16 ***
## Price -0.054459 0.005242 -10.389 < 2e-16 ***
## UrbanYes -0.021916 0.271650 -0.081 0.936
## USYes 1.200573 0.259042 4.635 4.86e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.472 on 396 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2335
## F-statistic: 41.52 on 3 and 396 DF, p-value: < 2.2e-162B) Provide an interpretation of each coefficient in the model. Be careful - some of the variables in the model are qualitative!
# I believe Urban and US are both qualitative variables. Sales slope coefficient is rather high, while the other 3 variables' slopes are relatively low (although for Urban and US that is likely in part due to them being qualitative). The standard errors are all small, implying each data set is made up of small values. The p-values are extremely small for all but Urban, implying that there is a significant difference or influence between at least two variables.2C) Write out the model in equation form, being careful to handle the qualitative variables properly. (Hint: You can write seperate equations)
#Sales = Bsales+Bprice1*Xprice1+Bprice2*Xprice2+Bprice3*Xprice3+....+Bpricen*Xpricen
#Sales = Bsales+BurbanY1*XurbanY1+BurbanY2*XurbanY2+BurbanY3*XurbanY3+...+BurbanYn*XurbanYn
#Sales = Bsales+BusY1*XusY1+BusY2*XusY2+BusY3*XusY3+...+BusYn*XusYn
#Sales = Bsales+BurbanN1*XurbanN1+BurbanN2*XurbanN2+BurbanN3*XurbanN3+...+BurbanNn*XurbanNn
#Sales = Bsales+BusN1*XusN1+BusN2*XusN2+BusN3*XusN3+...+BusNn*XusNn2D) For which of the predictors can you reject the null hypothesis H0 : βj = 0?
#Price, and USYes.
#Price, and USYes.2E) On the basis of your response to the previous question, fit a smaller model that only uses the predictors for which there is evidence of association with the outcome.
carMod2<-lm(Sales~Price+US, data=Carseats)
summary(carMod2)##
## Call:
## lm(formula = Sales ~ Price + US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9269 -1.6286 -0.0574 1.5766 7.0515
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.03079 0.63098 20.652 < 2e-16 ***
## Price -0.05448 0.00523 -10.416 < 2e-16 ***
## USYes 1.19964 0.25846 4.641 4.71e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.469 on 397 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2354
## F-statistic: 62.43 on 2 and 397 DF, p-value: < 2.2e-162F) How well do the models in (a) and (e) fit the data?
#carMod2 is a better fit than carMod to the data because it only includes predictors that affect the response.2G) Using the model from (e), obtain the 95% confidence intervals for the coefficient(s).
confint(carMod2)## 2.5 % 97.5 %
## (Intercept) 11.79032020 14.27126531
## Price -0.06475984 -0.04419543
## USYes 0.69151957 1.70776632