Midterm Exam
113035131 Baljingarav
2025-04-18
Chapter 2 Exercise 10
a)
# Load the library and dataset
library(ISLR2)
data(Boston)
# View the dataset
Boston## crim zn indus chas nox rm age dis rad tax ptratio lstat
## 1 0.00632 18.0 2.31 0 0.5380 6.575 65.2 4.0900 1 296 15.3 4.98
## 2 0.02731 0.0 7.07 0 0.4690 6.421 78.9 4.9671 2 242 17.8 9.14
## 3 0.02729 0.0 7.07 0 0.4690 7.185 61.1 4.9671 2 242 17.8 4.03
## 4 0.03237 0.0 2.18 0 0.4580 6.998 45.8 6.0622 3 222 18.7 2.94
## 5 0.06905 0.0 2.18 0 0.4580 7.147 54.2 6.0622 3 222 18.7 5.33
## 6 0.02985 0.0 2.18 0 0.4580 6.430 58.7 6.0622 3 222 18.7 5.21
## 7 0.08829 12.5 7.87 0 0.5240 6.012 66.6 5.5605 5 311 15.2 12.43
## 8 0.14455 12.5 7.87 0 0.5240 6.172 96.1 5.9505 5 311 15.2 19.15
## 9 0.21124 12.5 7.87 0 0.5240 5.631 100.0 6.0821 5 311 15.2 29.93
## 10 0.17004 12.5 7.87 0 0.5240 6.004 85.9 6.5921 5 311 15.2 17.10
## 11 0.22489 12.5 7.87 0 0.5240 6.377 94.3 6.3467 5 311 15.2 20.45
## 12 0.11747 12.5 7.87 0 0.5240 6.009 82.9 6.2267 5 311 15.2 13.27
## 13 0.09378 12.5 7.87 0 0.5240 5.889 39.0 5.4509 5 311 15.2 15.71
## 14 0.62976 0.0 8.14 0 0.5380 5.949 61.8 4.7075 4 307 21.0 8.26
## 15 0.63796 0.0 8.14 0 0.5380 6.096 84.5 4.4619 4 307 21.0 10.26
## 16 0.62739 0.0 8.14 0 0.5380 5.834 56.5 4.4986 4 307 21.0 8.47
## 17 1.05393 0.0 8.14 0 0.5380 5.935 29.3 4.4986 4 307 21.0 6.58
## 18 0.78420 0.0 8.14 0 0.5380 5.990 81.7 4.2579 4 307 21.0 14.67
## 19 0.80271 0.0 8.14 0 0.5380 5.456 36.6 3.7965 4 307 21.0 11.69
## 20 0.72580 0.0 8.14 0 0.5380 5.727 69.5 3.7965 4 307 21.0 11.28
## 21 1.25179 0.0 8.14 0 0.5380 5.570 98.1 3.7979 4 307 21.0 21.02
## 22 0.85204 0.0 8.14 0 0.5380 5.965 89.2 4.0123 4 307 21.0 13.83
## 23 1.23247 0.0 8.14 0 0.5380 6.142 91.7 3.9769 4 307 21.0 18.72
## 24 0.98843 0.0 8.14 0 0.5380 5.813 100.0 4.0952 4 307 21.0 19.88
## 25 0.75026 0.0 8.14 0 0.5380 5.924 94.1 4.3996 4 307 21.0 16.30
## 26 0.84054 0.0 8.14 0 0.5380 5.599 85.7 4.4546 4 307 21.0 16.51
## 27 0.67191 0.0 8.14 0 0.5380 5.813 90.3 4.6820 4 307 21.0 14.81
## 28 0.95577 0.0 8.14 0 0.5380 6.047 88.8 4.4534 4 307 21.0 17.28
## 29 0.77299 0.0 8.14 0 0.5380 6.495 94.4 4.4547 4 307 21.0 12.80
## 30 1.00245 0.0 8.14 0 0.5380 6.674 87.3 4.2390 4 307 21.0 11.98
## 31 1.13081 0.0 8.14 0 0.5380 5.713 94.1 4.2330 4 307 21.0 22.60
## 32 1.35472 0.0 8.14 0 0.5380 6.072 100.0 4.1750 4 307 21.0 13.04
## 33 1.38799 0.0 8.14 0 0.5380 5.950 82.0 3.9900 4 307 21.0 27.71
## 34 1.15172 0.0 8.14 0 0.5380 5.701 95.0 3.7872 4 307 21.0 18.35
## 35 1.61282 0.0 8.14 0 0.5380 6.096 96.9 3.7598 4 307 21.0 20.34
## 36 0.06417 0.0 5.96 0 0.4990 5.933 68.2 3.3603 5 279 19.2 9.68
## 37 0.09744 0.0 5.96 0 0.4990 5.841 61.4 3.3779 5 279 19.2 11.41
## 38 0.08014 0.0 5.96 0 0.4990 5.850 41.5 3.9342 5 279 19.2 8.77
## 39 0.17505 0.0 5.96 0 0.4990 5.966 30.2 3.8473 5 279 19.2 10.13
## 40 0.02763 75.0 2.95 0 0.4280 6.595 21.8 5.4011 3 252 18.3 4.32
## 41 0.03359 75.0 2.95 0 0.4280 7.024 15.8 5.4011 3 252 18.3 1.98
## 42 0.12744 0.0 6.91 0 0.4480 6.770 2.9 5.7209 3 233 17.9 4.84
## 43 0.14150 0.0 6.91 0 0.4480 6.169 6.6 5.7209 3 233 17.9 5.81
## 44 0.15936 0.0 6.91 0 0.4480 6.211 6.5 5.7209 3 233 17.9 7.44
## 45 0.12269 0.0 6.91 0 0.4480 6.069 40.0 5.7209 3 233 17.9 9.55
## 46 0.17142 0.0 6.91 0 0.4480 5.682 33.8 5.1004 3 233 17.9 10.21
## 47 0.18836 0.0 6.91 0 0.4480 5.786 33.3 5.1004 3 233 17.9 14.15
## 48 0.22927 0.0 6.91 0 0.4480 6.030 85.5 5.6894 3 233 17.9 18.80
## 49 0.25387 0.0 6.91 0 0.4480 5.399 95.3 5.8700 3 233 17.9 30.81
## 50 0.21977 0.0 6.91 0 0.4480 5.602 62.0 6.0877 3 233 17.9 16.20
## 51 0.08873 21.0 5.64 0 0.4390 5.963 45.7 6.8147 4 243 16.8 13.45
## 52 0.04337 21.0 5.64 0 0.4390 6.115 63.0 6.8147 4 243 16.8 9.43
## 53 0.05360 21.0 5.64 0 0.4390 6.511 21.1 6.8147 4 243 16.8 5.28
## 54 0.04981 21.0 5.64 0 0.4390 5.998 21.4 6.8147 4 243 16.8 8.43
## 55 0.01360 75.0 4.00 0 0.4100 5.888 47.6 7.3197 3 469 21.1 14.80
## 56 0.01311 90.0 1.22 0 0.4030 7.249 21.9 8.6966 5 226 17.9 4.81
## 57 0.02055 85.0 0.74 0 0.4100 6.383 35.7 9.1876 2 313 17.3 5.77
## 58 0.01432 100.0 1.32 0 0.4110 6.816 40.5 8.3248 5 256 15.1 3.95
## 59 0.15445 25.0 5.13 0 0.4530 6.145 29.2 7.8148 8 284 19.7 6.86
## 60 0.10328 25.0 5.13 0 0.4530 5.927 47.2 6.9320 8 284 19.7 9.22
## 61 0.14932 25.0 5.13 0 0.4530 5.741 66.2 7.2254 8 284 19.7 13.15
## 62 0.17171 25.0 5.13 0 0.4530 5.966 93.4 6.8185 8 284 19.7 14.44
## 63 0.11027 25.0 5.13 0 0.4530 6.456 67.8 7.2255 8 284 19.7 6.73
## 64 0.12650 25.0 5.13 0 0.4530 6.762 43.4 7.9809 8 284 19.7 9.50
## 65 0.01951 17.5 1.38 0 0.4161 7.104 59.5 9.2229 3 216 18.6 8.05
## 66 0.03584 80.0 3.37 0 0.3980 6.290 17.8 6.6115 4 337 16.1 4.67
## 67 0.04379 80.0 3.37 0 0.3980 5.787 31.1 6.6115 4 337 16.1 10.24
## 68 0.05789 12.5 6.07 0 0.4090 5.878 21.4 6.4980 4 345 18.9 8.10
## 69 0.13554 12.5 6.07 0 0.4090 5.594 36.8 6.4980 4 345 18.9 13.09
## 70 0.12816 12.5 6.07 0 0.4090 5.885 33.0 6.4980 4 345 18.9 8.79
## 71 0.08826 0.0 10.81 0 0.4130 6.417 6.6 5.2873 4 305 19.2 6.72
## 72 0.15876 0.0 10.81 0 0.4130 5.961 17.5 5.2873 4 305 19.2 9.88
## 73 0.09164 0.0 10.81 0 0.4130 6.065 7.8 5.2873 4 305 19.2 5.52
## 74 0.19539 0.0 10.81 0 0.4130 6.245 6.2 5.2873 4 305 19.2 7.54
## 75 0.07896 0.0 12.83 0 0.4370 6.273 6.0 4.2515 5 398 18.7 6.78
## 76 0.09512 0.0 12.83 0 0.4370 6.286 45.0 4.5026 5 398 18.7 8.94
## 77 0.10153 0.0 12.83 0 0.4370 6.279 74.5 4.0522 5 398 18.7 11.97
## 78 0.08707 0.0 12.83 0 0.4370 6.140 45.8 4.0905 5 398 18.7 10.27
## 79 0.05646 0.0 12.83 0 0.4370 6.232 53.7 5.0141 5 398 18.7 12.34
## 80 0.08387 0.0 12.83 0 0.4370 5.874 36.6 4.5026 5 398 18.7 9.10
## 81 0.04113 25.0 4.86 0 0.4260 6.727 33.5 5.4007 4 281 19.0 5.29
## 82 0.04462 25.0 4.86 0 0.4260 6.619 70.4 5.4007 4 281 19.0 7.22
## 83 0.03659 25.0 4.86 0 0.4260 6.302 32.2 5.4007 4 281 19.0 6.72
## 84 0.03551 25.0 4.86 0 0.4260 6.167 46.7 5.4007 4 281 19.0 7.51
## 85 0.05059 0.0 4.49 0 0.4490 6.389 48.0 4.7794 3 247 18.5 9.62
## 86 0.05735 0.0 4.49 0 0.4490 6.630 56.1 4.4377 3 247 18.5 6.53
## 87 0.05188 0.0 4.49 0 0.4490 6.015 45.1 4.4272 3 247 18.5 12.86
## 88 0.07151 0.0 4.49 0 0.4490 6.121 56.8 3.7476 3 247 18.5 8.44
## 89 0.05660 0.0 3.41 0 0.4890 7.007 86.3 3.4217 2 270 17.8 5.50
## 90 0.05302 0.0 3.41 0 0.4890 7.079 63.1 3.4145 2 270 17.8 5.70
## 91 0.04684 0.0 3.41 0 0.4890 6.417 66.1 3.0923 2 270 17.8 8.81
## 92 0.03932 0.0 3.41 0 0.4890 6.405 73.9 3.0921 2 270 17.8 8.20
## 93 0.04203 28.0 15.04 0 0.4640 6.442 53.6 3.6659 4 270 18.2 8.16
## 94 0.02875 28.0 15.04 0 0.4640 6.211 28.9 3.6659 4 270 18.2 6.21
## 95 0.04294 28.0 15.04 0 0.4640 6.249 77.3 3.6150 4 270 18.2 10.59
## 96 0.12204 0.0 2.89 0 0.4450 6.625 57.8 3.4952 2 276 18.0 6.65
## 97 0.11504 0.0 2.89 0 0.4450 6.163 69.6 3.4952 2 276 18.0 11.34
## 98 0.12083 0.0 2.89 0 0.4450 8.069 76.0 3.4952 2 276 18.0 4.21
## 99 0.08187 0.0 2.89 0 0.4450 7.820 36.9 3.4952 2 276 18.0 3.57
## 100 0.06860 0.0 2.89 0 0.4450 7.416 62.5 3.4952 2 276 18.0 6.19
## 101 0.14866 0.0 8.56 0 0.5200 6.727 79.9 2.7778 5 384 20.9 9.42
## 102 0.11432 0.0 8.56 0 0.5200 6.781 71.3 2.8561 5 384 20.9 7.67
## 103 0.22876 0.0 8.56 0 0.5200 6.405 85.4 2.7147 5 384 20.9 10.63
## 104 0.21161 0.0 8.56 0 0.5200 6.137 87.4 2.7147 5 384 20.9 13.44
## 105 0.13960 0.0 8.56 0 0.5200 6.167 90.0 2.4210 5 384 20.9 12.33
## 106 0.13262 0.0 8.56 0 0.5200 5.851 96.7 2.1069 5 384 20.9 16.47
## 107 0.17120 0.0 8.56 0 0.5200 5.836 91.9 2.2110 5 384 20.9 18.66
## 108 0.13117 0.0 8.56 0 0.5200 6.127 85.2 2.1224 5 384 20.9 14.09
## 109 0.12802 0.0 8.56 0 0.5200 6.474 97.1 2.4329 5 384 20.9 12.27
## 110 0.26363 0.0 8.56 0 0.5200 6.229 91.2 2.5451 5 384 20.9 15.55
## 111 0.10793 0.0 8.56 0 0.5200 6.195 54.4 2.7778 5 384 20.9 13.00
## 112 0.10084 0.0 10.01 0 0.5470 6.715 81.6 2.6775 6 432 17.8 10.16
## 113 0.12329 0.0 10.01 0 0.5470 5.913 92.9 2.3534 6 432 17.8 16.21
## 114 0.22212 0.0 10.01 0 0.5470 6.092 95.4 2.5480 6 432 17.8 17.09
## 115 0.14231 0.0 10.01 0 0.5470 6.254 84.2 2.2565 6 432 17.8 10.45
## 116 0.17134 0.0 10.01 0 0.5470 5.928 88.2 2.4631 6 432 17.8 15.76
## 117 0.13158 0.0 10.01 0 0.5470 6.176 72.5 2.7301 6 432 17.8 12.04
## 118 0.15098 0.0 10.01 0 0.5470 6.021 82.6 2.7474 6 432 17.8 10.30
## 119 0.13058 0.0 10.01 0 0.5470 5.872 73.1 2.4775 6 432 17.8 15.37
## 120 0.14476 0.0 10.01 0 0.5470 5.731 65.2 2.7592 6 432 17.8 13.61
## 121 0.06899 0.0 25.65 0 0.5810 5.870 69.7 2.2577 2 188 19.1 14.37
## 122 0.07165 0.0 25.65 0 0.5810 6.004 84.1 2.1974 2 188 19.1 14.27
## 123 0.09299 0.0 25.65 0 0.5810 5.961 92.9 2.0869 2 188 19.1 17.93
## 124 0.15038 0.0 25.65 0 0.5810 5.856 97.0 1.9444 2 188 19.1 25.41
## 125 0.09849 0.0 25.65 0 0.5810 5.879 95.8 2.0063 2 188 19.1 17.58
## 126 0.16902 0.0 25.65 0 0.5810 5.986 88.4 1.9929 2 188 19.1 14.81
## 127 0.38735 0.0 25.65 0 0.5810 5.613 95.6 1.7572 2 188 19.1 27.26
## 128 0.25915 0.0 21.89 0 0.6240 5.693 96.0 1.7883 4 437 21.2 17.19
## 129 0.32543 0.0 21.89 0 0.6240 6.431 98.8 1.8125 4 437 21.2 15.39
## 130 0.88125 0.0 21.89 0 0.6240 5.637 94.7 1.9799 4 437 21.2 18.34
## 131 0.34006 0.0 21.89 0 0.6240 6.458 98.9 2.1185 4 437 21.2 12.60
## 132 1.19294 0.0 21.89 0 0.6240 6.326 97.7 2.2710 4 437 21.2 12.26
## 133 0.59005 0.0 21.89 0 0.6240 6.372 97.9 2.3274 4 437 21.2 11.12
## 134 0.32982 0.0 21.89 0 0.6240 5.822 95.4 2.4699 4 437 21.2 15.03
## 135 0.97617 0.0 21.89 0 0.6240 5.757 98.4 2.3460 4 437 21.2 17.31
## 136 0.55778 0.0 21.89 0 0.6240 6.335 98.2 2.1107 4 437 21.2 16.96
## 137 0.32264 0.0 21.89 0 0.6240 5.942 93.5 1.9669 4 437 21.2 16.90
## 138 0.35233 0.0 21.89 0 0.6240 6.454 98.4 1.8498 4 437 21.2 14.59
## 139 0.24980 0.0 21.89 0 0.6240 5.857 98.2 1.6686 4 437 21.2 21.32
## 140 0.54452 0.0 21.89 0 0.6240 6.151 97.9 1.6687 4 437 21.2 18.46
## 141 0.29090 0.0 21.89 0 0.6240 6.174 93.6 1.6119 4 437 21.2 24.16
## 142 1.62864 0.0 21.89 0 0.6240 5.019 100.0 1.4394 4 437 21.2 34.41
## 143 3.32105 0.0 19.58 1 0.8710 5.403 100.0 1.3216 5 403 14.7 26.82
## 144 4.09740 0.0 19.58 0 0.8710 5.468 100.0 1.4118 5 403 14.7 26.42
## 145 2.77974 0.0 19.58 0 0.8710 4.903 97.8 1.3459 5 403 14.7 29.29
## 146 2.37934 0.0 19.58 0 0.8710 6.130 100.0 1.4191 5 403 14.7 27.80
## 147 2.15505 0.0 19.58 0 0.8710 5.628 100.0 1.5166 5 403 14.7 16.65
## 148 2.36862 0.0 19.58 0 0.8710 4.926 95.7 1.4608 5 403 14.7 29.53
## 149 2.33099 0.0 19.58 0 0.8710 5.186 93.8 1.5296 5 403 14.7 28.32
## 150 2.73397 0.0 19.58 0 0.8710 5.597 94.9 1.5257 5 403 14.7 21.45
## 151 1.65660 0.0 19.58 0 0.8710 6.122 97.3 1.6180 5 403 14.7 14.10
## 152 1.49632 0.0 19.58 0 0.8710 5.404 100.0 1.5916 5 403 14.7 13.28
## 153 1.12658 0.0 19.58 1 0.8710 5.012 88.0 1.6102 5 403 14.7 12.12
## 154 2.14918 0.0 19.58 0 0.8710 5.709 98.5 1.6232 5 403 14.7 15.79
## 155 1.41385 0.0 19.58 1 0.8710 6.129 96.0 1.7494 5 403 14.7 15.12
## 156 3.53501 0.0 19.58 1 0.8710 6.152 82.6 1.7455 5 403 14.7 15.02
## 157 2.44668 0.0 19.58 0 0.8710 5.272 94.0 1.7364 5 403 14.7 16.14
## 158 1.22358 0.0 19.58 0 0.6050 6.943 97.4 1.8773 5 403 14.7 4.59
## 159 1.34284 0.0 19.58 0 0.6050 6.066 100.0 1.7573 5 403 14.7 6.43
## 160 1.42502 0.0 19.58 0 0.8710 6.510 100.0 1.7659 5 403 14.7 7.39
## 161 1.27346 0.0 19.58 1 0.6050 6.250 92.6 1.7984 5 403 14.7 5.50
## 162 1.46336 0.0 19.58 0 0.6050 7.489 90.8 1.9709 5 403 14.7 1.73
## 163 1.83377 0.0 19.58 1 0.6050 7.802 98.2 2.0407 5 403 14.7 1.92
## 164 1.51902 0.0 19.58 1 0.6050 8.375 93.9 2.1620 5 403 14.7 3.32
## 165 2.24236 0.0 19.58 0 0.6050 5.854 91.8 2.4220 5 403 14.7 11.64
## 166 2.92400 0.0 19.58 0 0.6050 6.101 93.0 2.2834 5 403 14.7 9.81
## 167 2.01019 0.0 19.58 0 0.6050 7.929 96.2 2.0459 5 403 14.7 3.70
## 168 1.80028 0.0 19.58 0 0.6050 5.877 79.2 2.4259 5 403 14.7 12.14
## 169 2.30040 0.0 19.58 0 0.6050 6.319 96.1 2.1000 5 403 14.7 11.10
## 170 2.44953 0.0 19.58 0 0.6050 6.402 95.2 2.2625 5 403 14.7 11.32
## 171 1.20742 0.0 19.58 0 0.6050 5.875 94.6 2.4259 5 403 14.7 14.43
## 172 2.31390 0.0 19.58 0 0.6050 5.880 97.3 2.3887 5 403 14.7 12.03
## 173 0.13914 0.0 4.05 0 0.5100 5.572 88.5 2.5961 5 296 16.6 14.69
## 174 0.09178 0.0 4.05 0 0.5100 6.416 84.1 2.6463 5 296 16.6 9.04
## 175 0.08447 0.0 4.05 0 0.5100 5.859 68.7 2.7019 5 296 16.6 9.64
## 176 0.06664 0.0 4.05 0 0.5100 6.546 33.1 3.1323 5 296 16.6 5.33
## 177 0.07022 0.0 4.05 0 0.5100 6.020 47.2 3.5549 5 296 16.6 10.11
## 178 0.05425 0.0 4.05 0 0.5100 6.315 73.4 3.3175 5 296 16.6 6.29
## 179 0.06642 0.0 4.05 0 0.5100 6.860 74.4 2.9153 5 296 16.6 6.92
## 180 0.05780 0.0 2.46 0 0.4880 6.980 58.4 2.8290 3 193 17.8 5.04
## 181 0.06588 0.0 2.46 0 0.4880 7.765 83.3 2.7410 3 193 17.8 7.56
## 182 0.06888 0.0 2.46 0 0.4880 6.144 62.2 2.5979 3 193 17.8 9.45
## 183 0.09103 0.0 2.46 0 0.4880 7.155 92.2 2.7006 3 193 17.8 4.82
## 184 0.10008 0.0 2.46 0 0.4880 6.563 95.6 2.8470 3 193 17.8 5.68
## 185 0.08308 0.0 2.46 0 0.4880 5.604 89.8 2.9879 3 193 17.8 13.98
## 186 0.06047 0.0 2.46 0 0.4880 6.153 68.8 3.2797 3 193 17.8 13.15
## 187 0.05602 0.0 2.46 0 0.4880 7.831 53.6 3.1992 3 193 17.8 4.45
## 188 0.07875 45.0 3.44 0 0.4370 6.782 41.1 3.7886 5 398 15.2 6.68
## 189 0.12579 45.0 3.44 0 0.4370 6.556 29.1 4.5667 5 398 15.2 4.56
## 190 0.08370 45.0 3.44 0 0.4370 7.185 38.9 4.5667 5 398 15.2 5.39
## 191 0.09068 45.0 3.44 0 0.4370 6.951 21.5 6.4798 5 398 15.2 5.10
## 192 0.06911 45.0 3.44 0 0.4370 6.739 30.8 6.4798 5 398 15.2 4.69
## 193 0.08664 45.0 3.44 0 0.4370 7.178 26.3 6.4798 5 398 15.2 2.87
## 194 0.02187 60.0 2.93 0 0.4010 6.800 9.9 6.2196 1 265 15.6 5.03
## 195 0.01439 60.0 2.93 0 0.4010 6.604 18.8 6.2196 1 265 15.6 4.38
## 196 0.01381 80.0 0.46 0 0.4220 7.875 32.0 5.6484 4 255 14.4 2.97
## 197 0.04011 80.0 1.52 0 0.4040 7.287 34.1 7.3090 2 329 12.6 4.08
## 198 0.04666 80.0 1.52 0 0.4040 7.107 36.6 7.3090 2 329 12.6 8.61
## 199 0.03768 80.0 1.52 0 0.4040 7.274 38.3 7.3090 2 329 12.6 6.62
## 200 0.03150 95.0 1.47 0 0.4030 6.975 15.3 7.6534 3 402 17.0 4.56
## 201 0.01778 95.0 1.47 0 0.4030 7.135 13.9 7.6534 3 402 17.0 4.45
## 202 0.03445 82.5 2.03 0 0.4150 6.162 38.4 6.2700 2 348 14.7 7.43
## 203 0.02177 82.5 2.03 0 0.4150 7.610 15.7 6.2700 2 348 14.7 3.11
## 204 0.03510 95.0 2.68 0 0.4161 7.853 33.2 5.1180 4 224 14.7 3.81
## 205 0.02009 95.0 2.68 0 0.4161 8.034 31.9 5.1180 4 224 14.7 2.88
## 206 0.13642 0.0 10.59 0 0.4890 5.891 22.3 3.9454 4 277 18.6 10.87
## 207 0.22969 0.0 10.59 0 0.4890 6.326 52.5 4.3549 4 277 18.6 10.97
## 208 0.25199 0.0 10.59 0 0.4890 5.783 72.7 4.3549 4 277 18.6 18.06
## 209 0.13587 0.0 10.59 1 0.4890 6.064 59.1 4.2392 4 277 18.6 14.66
## 210 0.43571 0.0 10.59 1 0.4890 5.344 100.0 3.8750 4 277 18.6 23.09
## 211 0.17446 0.0 10.59 1 0.4890 5.960 92.1 3.8771 4 277 18.6 17.27
## 212 0.37578 0.0 10.59 1 0.4890 5.404 88.6 3.6650 4 277 18.6 23.98
## 213 0.21719 0.0 10.59 1 0.4890 5.807 53.8 3.6526 4 277 18.6 16.03
## 214 0.14052 0.0 10.59 0 0.4890 6.375 32.3 3.9454 4 277 18.6 9.38
## 215 0.28955 0.0 10.59 0 0.4890 5.412 9.8 3.5875 4 277 18.6 29.55
## 216 0.19802 0.0 10.59 0 0.4890 6.182 42.4 3.9454 4 277 18.6 9.47
## 217 0.04560 0.0 13.89 1 0.5500 5.888 56.0 3.1121 5 276 16.4 13.51
## 218 0.07013 0.0 13.89 0 0.5500 6.642 85.1 3.4211 5 276 16.4 9.69
## 219 0.11069 0.0 13.89 1 0.5500 5.951 93.8 2.8893 5 276 16.4 17.92
## 220 0.11425 0.0 13.89 1 0.5500 6.373 92.4 3.3633 5 276 16.4 10.50
## 221 0.35809 0.0 6.20 1 0.5070 6.951 88.5 2.8617 8 307 17.4 9.71
## 222 0.40771 0.0 6.20 1 0.5070 6.164 91.3 3.0480 8 307 17.4 21.46
## 223 0.62356 0.0 6.20 1 0.5070 6.879 77.7 3.2721 8 307 17.4 9.93
## 224 0.61470 0.0 6.20 0 0.5070 6.618 80.8 3.2721 8 307 17.4 7.60
## 225 0.31533 0.0 6.20 0 0.5040 8.266 78.3 2.8944 8 307 17.4 4.14
## 226 0.52693 0.0 6.20 0 0.5040 8.725 83.0 2.8944 8 307 17.4 4.63
## 227 0.38214 0.0 6.20 0 0.5040 8.040 86.5 3.2157 8 307 17.4 3.13
## 228 0.41238 0.0 6.20 0 0.5040 7.163 79.9 3.2157 8 307 17.4 6.36
## 229 0.29819 0.0 6.20 0 0.5040 7.686 17.0 3.3751 8 307 17.4 3.92
## 230 0.44178 0.0 6.20 0 0.5040 6.552 21.4 3.3751 8 307 17.4 3.76
## 231 0.53700 0.0 6.20 0 0.5040 5.981 68.1 3.6715 8 307 17.4 11.65
## 232 0.46296 0.0 6.20 0 0.5040 7.412 76.9 3.6715 8 307 17.4 5.25
## 233 0.57529 0.0 6.20 0 0.5070 8.337 73.3 3.8384 8 307 17.4 2.47
## 234 0.33147 0.0 6.20 0 0.5070 8.247 70.4 3.6519 8 307 17.4 3.95
## 235 0.44791 0.0 6.20 1 0.5070 6.726 66.5 3.6519 8 307 17.4 8.05
## 236 0.33045 0.0 6.20 0 0.5070 6.086 61.5 3.6519 8 307 17.4 10.88
## 237 0.52058 0.0 6.20 1 0.5070 6.631 76.5 4.1480 8 307 17.4 9.54
## 238 0.51183 0.0 6.20 0 0.5070 7.358 71.6 4.1480 8 307 17.4 4.73
## 239 0.08244 30.0 4.93 0 0.4280 6.481 18.5 6.1899 6 300 16.6 6.36
## 240 0.09252 30.0 4.93 0 0.4280 6.606 42.2 6.1899 6 300 16.6 7.37
## 241 0.11329 30.0 4.93 0 0.4280 6.897 54.3 6.3361 6 300 16.6 11.38
## 242 0.10612 30.0 4.93 0 0.4280 6.095 65.1 6.3361 6 300 16.6 12.40
## 243 0.10290 30.0 4.93 0 0.4280 6.358 52.9 7.0355 6 300 16.6 11.22
## 244 0.12757 30.0 4.93 0 0.4280 6.393 7.8 7.0355 6 300 16.6 5.19
## 245 0.20608 22.0 5.86 0 0.4310 5.593 76.5 7.9549 7 330 19.1 12.50
## 246 0.19133 22.0 5.86 0 0.4310 5.605 70.2 7.9549 7 330 19.1 18.46
## 247 0.33983 22.0 5.86 0 0.4310 6.108 34.9 8.0555 7 330 19.1 9.16
## 248 0.19657 22.0 5.86 0 0.4310 6.226 79.2 8.0555 7 330 19.1 10.15
## 249 0.16439 22.0 5.86 0 0.4310 6.433 49.1 7.8265 7 330 19.1 9.52
## 250 0.19073 22.0 5.86 0 0.4310 6.718 17.5 7.8265 7 330 19.1 6.56
## 251 0.14030 22.0 5.86 0 0.4310 6.487 13.0 7.3967 7 330 19.1 5.90
## 252 0.21409 22.0 5.86 0 0.4310 6.438 8.9 7.3967 7 330 19.1 3.59
## 253 0.08221 22.0 5.86 0 0.4310 6.957 6.8 8.9067 7 330 19.1 3.53
## 254 0.36894 22.0 5.86 0 0.4310 8.259 8.4 8.9067 7 330 19.1 3.54
## 255 0.04819 80.0 3.64 0 0.3920 6.108 32.0 9.2203 1 315 16.4 6.57
## 256 0.03548 80.0 3.64 0 0.3920 5.876 19.1 9.2203 1 315 16.4 9.25
## 257 0.01538 90.0 3.75 0 0.3940 7.454 34.2 6.3361 3 244 15.9 3.11
## 258 0.61154 20.0 3.97 0 0.6470 8.704 86.9 1.8010 5 264 13.0 5.12
## 259 0.66351 20.0 3.97 0 0.6470 7.333 100.0 1.8946 5 264 13.0 7.79
## 260 0.65665 20.0 3.97 0 0.6470 6.842 100.0 2.0107 5 264 13.0 6.90
## 261 0.54011 20.0 3.97 0 0.6470 7.203 81.8 2.1121 5 264 13.0 9.59
## 262 0.53412 20.0 3.97 0 0.6470 7.520 89.4 2.1398 5 264 13.0 7.26
## 263 0.52014 20.0 3.97 0 0.6470 8.398 91.5 2.2885 5 264 13.0 5.91
## 264 0.82526 20.0 3.97 0 0.6470 7.327 94.5 2.0788 5 264 13.0 11.25
## 265 0.55007 20.0 3.97 0 0.6470 7.206 91.6 1.9301 5 264 13.0 8.10
## 266 0.76162 20.0 3.97 0 0.6470 5.560 62.8 1.9865 5 264 13.0 10.45
## 267 0.78570 20.0 3.97 0 0.6470 7.014 84.6 2.1329 5 264 13.0 14.79
## 268 0.57834 20.0 3.97 0 0.5750 8.297 67.0 2.4216 5 264 13.0 7.44
## 269 0.54050 20.0 3.97 0 0.5750 7.470 52.6 2.8720 5 264 13.0 3.16
## 270 0.09065 20.0 6.96 1 0.4640 5.920 61.5 3.9175 3 223 18.6 13.65
## 271 0.29916 20.0 6.96 0 0.4640 5.856 42.1 4.4290 3 223 18.6 13.00
## 272 0.16211 20.0 6.96 0 0.4640 6.240 16.3 4.4290 3 223 18.6 6.59
## 273 0.11460 20.0 6.96 0 0.4640 6.538 58.7 3.9175 3 223 18.6 7.73
## 274 0.22188 20.0 6.96 1 0.4640 7.691 51.8 4.3665 3 223 18.6 6.58
## 275 0.05644 40.0 6.41 1 0.4470 6.758 32.9 4.0776 4 254 17.6 3.53
## 276 0.09604 40.0 6.41 0 0.4470 6.854 42.8 4.2673 4 254 17.6 2.98
## 277 0.10469 40.0 6.41 1 0.4470 7.267 49.0 4.7872 4 254 17.6 6.05
## 278 0.06127 40.0 6.41 1 0.4470 6.826 27.6 4.8628 4 254 17.6 4.16
## 279 0.07978 40.0 6.41 0 0.4470 6.482 32.1 4.1403 4 254 17.6 7.19
## 280 0.21038 20.0 3.33 0 0.4429 6.812 32.2 4.1007 5 216 14.9 4.85
## 281 0.03578 20.0 3.33 0 0.4429 7.820 64.5 4.6947 5 216 14.9 3.76
## 282 0.03705 20.0 3.33 0 0.4429 6.968 37.2 5.2447 5 216 14.9 4.59
## 283 0.06129 20.0 3.33 1 0.4429 7.645 49.7 5.2119 5 216 14.9 3.01
## 284 0.01501 90.0 1.21 1 0.4010 7.923 24.8 5.8850 1 198 13.6 3.16
## 285 0.00906 90.0 2.97 0 0.4000 7.088 20.8 7.3073 1 285 15.3 7.85
## 286 0.01096 55.0 2.25 0 0.3890 6.453 31.9 7.3073 1 300 15.3 8.23
## 287 0.01965 80.0 1.76 0 0.3850 6.230 31.5 9.0892 1 241 18.2 12.93
## 288 0.03871 52.5 5.32 0 0.4050 6.209 31.3 7.3172 6 293 16.6 7.14
## 289 0.04590 52.5 5.32 0 0.4050 6.315 45.6 7.3172 6 293 16.6 7.60
## 290 0.04297 52.5 5.32 0 0.4050 6.565 22.9 7.3172 6 293 16.6 9.51
## 291 0.03502 80.0 4.95 0 0.4110 6.861 27.9 5.1167 4 245 19.2 3.33
## 292 0.07886 80.0 4.95 0 0.4110 7.148 27.7 5.1167 4 245 19.2 3.56
## 293 0.03615 80.0 4.95 0 0.4110 6.630 23.4 5.1167 4 245 19.2 4.70
## 294 0.08265 0.0 13.92 0 0.4370 6.127 18.4 5.5027 4 289 16.0 8.58
## 295 0.08199 0.0 13.92 0 0.4370 6.009 42.3 5.5027 4 289 16.0 10.40
## 296 0.12932 0.0 13.92 0 0.4370 6.678 31.1 5.9604 4 289 16.0 6.27
## 297 0.05372 0.0 13.92 0 0.4370 6.549 51.0 5.9604 4 289 16.0 7.39
## 298 0.14103 0.0 13.92 0 0.4370 5.790 58.0 6.3200 4 289 16.0 15.84
## 299 0.06466 70.0 2.24 0 0.4000 6.345 20.1 7.8278 5 358 14.8 4.97
## 300 0.05561 70.0 2.24 0 0.4000 7.041 10.0 7.8278 5 358 14.8 4.74
## 301 0.04417 70.0 2.24 0 0.4000 6.871 47.4 7.8278 5 358 14.8 6.07
## 302 0.03537 34.0 6.09 0 0.4330 6.590 40.4 5.4917 7 329 16.1 9.50
## 303 0.09266 34.0 6.09 0 0.4330 6.495 18.4 5.4917 7 329 16.1 8.67
## 304 0.10000 34.0 6.09 0 0.4330 6.982 17.7 5.4917 7 329 16.1 4.86
## 305 0.05515 33.0 2.18 0 0.4720 7.236 41.1 4.0220 7 222 18.4 6.93
## 306 0.05479 33.0 2.18 0 0.4720 6.616 58.1 3.3700 7 222 18.4 8.93
## 307 0.07503 33.0 2.18 0 0.4720 7.420 71.9 3.0992 7 222 18.4 6.47
## 308 0.04932 33.0 2.18 0 0.4720 6.849 70.3 3.1827 7 222 18.4 7.53
## 309 0.49298 0.0 9.90 0 0.5440 6.635 82.5 3.3175 4 304 18.4 4.54
## 310 0.34940 0.0 9.90 0 0.5440 5.972 76.7 3.1025 4 304 18.4 9.97
## 311 2.63548 0.0 9.90 0 0.5440 4.973 37.8 2.5194 4 304 18.4 12.64
## 312 0.79041 0.0 9.90 0 0.5440 6.122 52.8 2.6403 4 304 18.4 5.98
## 313 0.26169 0.0 9.90 0 0.5440 6.023 90.4 2.8340 4 304 18.4 11.72
## 314 0.26938 0.0 9.90 0 0.5440 6.266 82.8 3.2628 4 304 18.4 7.90
## 315 0.36920 0.0 9.90 0 0.5440 6.567 87.3 3.6023 4 304 18.4 9.28
## 316 0.25356 0.0 9.90 0 0.5440 5.705 77.7 3.9450 4 304 18.4 11.50
## 317 0.31827 0.0 9.90 0 0.5440 5.914 83.2 3.9986 4 304 18.4 18.33
## 318 0.24522 0.0 9.90 0 0.5440 5.782 71.7 4.0317 4 304 18.4 15.94
## 319 0.40202 0.0 9.90 0 0.5440 6.382 67.2 3.5325 4 304 18.4 10.36
## 320 0.47547 0.0 9.90 0 0.5440 6.113 58.8 4.0019 4 304 18.4 12.73
## 321 0.16760 0.0 7.38 0 0.4930 6.426 52.3 4.5404 5 287 19.6 7.20
## 322 0.18159 0.0 7.38 0 0.4930 6.376 54.3 4.5404 5 287 19.6 6.87
## 323 0.35114 0.0 7.38 0 0.4930 6.041 49.9 4.7211 5 287 19.6 7.70
## 324 0.28392 0.0 7.38 0 0.4930 5.708 74.3 4.7211 5 287 19.6 11.74
## 325 0.34109 0.0 7.38 0 0.4930 6.415 40.1 4.7211 5 287 19.6 6.12
## 326 0.19186 0.0 7.38 0 0.4930 6.431 14.7 5.4159 5 287 19.6 5.08
## 327 0.30347 0.0 7.38 0 0.4930 6.312 28.9 5.4159 5 287 19.6 6.15
## 328 0.24103 0.0 7.38 0 0.4930 6.083 43.7 5.4159 5 287 19.6 12.79
## 329 0.06617 0.0 3.24 0 0.4600 5.868 25.8 5.2146 4 430 16.9 9.97
## 330 0.06724 0.0 3.24 0 0.4600 6.333 17.2 5.2146 4 430 16.9 7.34
## 331 0.04544 0.0 3.24 0 0.4600 6.144 32.2 5.8736 4 430 16.9 9.09
## 332 0.05023 35.0 6.06 0 0.4379 5.706 28.4 6.6407 1 304 16.9 12.43
## 333 0.03466 35.0 6.06 0 0.4379 6.031 23.3 6.6407 1 304 16.9 7.83
## 334 0.05083 0.0 5.19 0 0.5150 6.316 38.1 6.4584 5 224 20.2 5.68
## 335 0.03738 0.0 5.19 0 0.5150 6.310 38.5 6.4584 5 224 20.2 6.75
## 336 0.03961 0.0 5.19 0 0.5150 6.037 34.5 5.9853 5 224 20.2 8.01
## 337 0.03427 0.0 5.19 0 0.5150 5.869 46.3 5.2311 5 224 20.2 9.80
## 338 0.03041 0.0 5.19 0 0.5150 5.895 59.6 5.6150 5 224 20.2 10.56
## 339 0.03306 0.0 5.19 0 0.5150 6.059 37.3 4.8122 5 224 20.2 8.51
## 340 0.05497 0.0 5.19 0 0.5150 5.985 45.4 4.8122 5 224 20.2 9.74
## 341 0.06151 0.0 5.19 0 0.5150 5.968 58.5 4.8122 5 224 20.2 9.29
## 342 0.01301 35.0 1.52 0 0.4420 7.241 49.3 7.0379 1 284 15.5 5.49
## 343 0.02498 0.0 1.89 0 0.5180 6.540 59.7 6.2669 1 422 15.9 8.65
## 344 0.02543 55.0 3.78 0 0.4840 6.696 56.4 5.7321 5 370 17.6 7.18
## 345 0.03049 55.0 3.78 0 0.4840 6.874 28.1 6.4654 5 370 17.6 4.61
## 346 0.03113 0.0 4.39 0 0.4420 6.014 48.5 8.0136 3 352 18.8 10.53
## 347 0.06162 0.0 4.39 0 0.4420 5.898 52.3 8.0136 3 352 18.8 12.67
## 348 0.01870 85.0 4.15 0 0.4290 6.516 27.7 8.5353 4 351 17.9 6.36
## 349 0.01501 80.0 2.01 0 0.4350 6.635 29.7 8.3440 4 280 17.0 5.99
## 350 0.02899 40.0 1.25 0 0.4290 6.939 34.5 8.7921 1 335 19.7 5.89
## 351 0.06211 40.0 1.25 0 0.4290 6.490 44.4 8.7921 1 335 19.7 5.98
## 352 0.07950 60.0 1.69 0 0.4110 6.579 35.9 10.7103 4 411 18.3 5.49
## 353 0.07244 60.0 1.69 0 0.4110 5.884 18.5 10.7103 4 411 18.3 7.79
## 354 0.01709 90.0 2.02 0 0.4100 6.728 36.1 12.1265 5 187 17.0 4.50
## 355 0.04301 80.0 1.91 0 0.4130 5.663 21.9 10.5857 4 334 22.0 8.05
## 356 0.10659 80.0 1.91 0 0.4130 5.936 19.5 10.5857 4 334 22.0 5.57
## 357 8.98296 0.0 18.10 1 0.7700 6.212 97.4 2.1222 24 666 20.2 17.60
## 358 3.84970 0.0 18.10 1 0.7700 6.395 91.0 2.5052 24 666 20.2 13.27
## 359 5.20177 0.0 18.10 1 0.7700 6.127 83.4 2.7227 24 666 20.2 11.48
## 360 4.26131 0.0 18.10 0 0.7700 6.112 81.3 2.5091 24 666 20.2 12.67
## 361 4.54192 0.0 18.10 0 0.7700 6.398 88.0 2.5182 24 666 20.2 7.79
## 362 3.83684 0.0 18.10 0 0.7700 6.251 91.1 2.2955 24 666 20.2 14.19
## 363 3.67822 0.0 18.10 0 0.7700 5.362 96.2 2.1036 24 666 20.2 10.19
## 364 4.22239 0.0 18.10 1 0.7700 5.803 89.0 1.9047 24 666 20.2 14.64
## 365 3.47428 0.0 18.10 1 0.7180 8.780 82.9 1.9047 24 666 20.2 5.29
## 366 4.55587 0.0 18.10 0 0.7180 3.561 87.9 1.6132 24 666 20.2 7.12
## 367 3.69695 0.0 18.10 0 0.7180 4.963 91.4 1.7523 24 666 20.2 14.00
## 368 13.52220 0.0 18.10 0 0.6310 3.863 100.0 1.5106 24 666 20.2 13.33
## 369 4.89822 0.0 18.10 0 0.6310 4.970 100.0 1.3325 24 666 20.2 3.26
## 370 5.66998 0.0 18.10 1 0.6310 6.683 96.8 1.3567 24 666 20.2 3.73
## 371 6.53876 0.0 18.10 1 0.6310 7.016 97.5 1.2024 24 666 20.2 2.96
## 372 9.23230 0.0 18.10 0 0.6310 6.216 100.0 1.1691 24 666 20.2 9.53
## 373 8.26725 0.0 18.10 1 0.6680 5.875 89.6 1.1296 24 666 20.2 8.88
## 374 11.10810 0.0 18.10 0 0.6680 4.906 100.0 1.1742 24 666 20.2 34.77
## 375 18.49820 0.0 18.10 0 0.6680 4.138 100.0 1.1370 24 666 20.2 37.97
## 376 19.60910 0.0 18.10 0 0.6710 7.313 97.9 1.3163 24 666 20.2 13.44
## 377 15.28800 0.0 18.10 0 0.6710 6.649 93.3 1.3449 24 666 20.2 23.24
## 378 9.82349 0.0 18.10 0 0.6710 6.794 98.8 1.3580 24 666 20.2 21.24
## 379 23.64820 0.0 18.10 0 0.6710 6.380 96.2 1.3861 24 666 20.2 23.69
## 380 17.86670 0.0 18.10 0 0.6710 6.223 100.0 1.3861 24 666 20.2 21.78
## 381 88.97620 0.0 18.10 0 0.6710 6.968 91.9 1.4165 24 666 20.2 17.21
## 382 15.87440 0.0 18.10 0 0.6710 6.545 99.1 1.5192 24 666 20.2 21.08
## 383 9.18702 0.0 18.10 0 0.7000 5.536 100.0 1.5804 24 666 20.2 23.60
## 384 7.99248 0.0 18.10 0 0.7000 5.520 100.0 1.5331 24 666 20.2 24.56
## 385 20.08490 0.0 18.10 0 0.7000 4.368 91.2 1.4395 24 666 20.2 30.63
## 386 16.81180 0.0 18.10 0 0.7000 5.277 98.1 1.4261 24 666 20.2 30.81
## 387 24.39380 0.0 18.10 0 0.7000 4.652 100.0 1.4672 24 666 20.2 28.28
## 388 22.59710 0.0 18.10 0 0.7000 5.000 89.5 1.5184 24 666 20.2 31.99
## 389 14.33370 0.0 18.10 0 0.7000 4.880 100.0 1.5895 24 666 20.2 30.62
## 390 8.15174 0.0 18.10 0 0.7000 5.390 98.9 1.7281 24 666 20.2 20.85
## 391 6.96215 0.0 18.10 0 0.7000 5.713 97.0 1.9265 24 666 20.2 17.11
## 392 5.29305 0.0 18.10 0 0.7000 6.051 82.5 2.1678 24 666 20.2 18.76
## 393 11.57790 0.0 18.10 0 0.7000 5.036 97.0 1.7700 24 666 20.2 25.68
## 394 8.64476 0.0 18.10 0 0.6930 6.193 92.6 1.7912 24 666 20.2 15.17
## 395 13.35980 0.0 18.10 0 0.6930 5.887 94.7 1.7821 24 666 20.2 16.35
## 396 8.71675 0.0 18.10 0 0.6930 6.471 98.8 1.7257 24 666 20.2 17.12
## 397 5.87205 0.0 18.10 0 0.6930 6.405 96.0 1.6768 24 666 20.2 19.37
## 398 7.67202 0.0 18.10 0 0.6930 5.747 98.9 1.6334 24 666 20.2 19.92
## 399 38.35180 0.0 18.10 0 0.6930 5.453 100.0 1.4896 24 666 20.2 30.59
## 400 9.91655 0.0 18.10 0 0.6930 5.852 77.8 1.5004 24 666 20.2 29.97
## 401 25.04610 0.0 18.10 0 0.6930 5.987 100.0 1.5888 24 666 20.2 26.77
## 402 14.23620 0.0 18.10 0 0.6930 6.343 100.0 1.5741 24 666 20.2 20.32
## 403 9.59571 0.0 18.10 0 0.6930 6.404 100.0 1.6390 24 666 20.2 20.31
## 404 24.80170 0.0 18.10 0 0.6930 5.349 96.0 1.7028 24 666 20.2 19.77
## 405 41.52920 0.0 18.10 0 0.6930 5.531 85.4 1.6074 24 666 20.2 27.38
## 406 67.92080 0.0 18.10 0 0.6930 5.683 100.0 1.4254 24 666 20.2 22.98
## 407 20.71620 0.0 18.10 0 0.6590 4.138 100.0 1.1781 24 666 20.2 23.34
## 408 11.95110 0.0 18.10 0 0.6590 5.608 100.0 1.2852 24 666 20.2 12.13
## 409 7.40389 0.0 18.10 0 0.5970 5.617 97.9 1.4547 24 666 20.2 26.40
## 410 14.43830 0.0 18.10 0 0.5970 6.852 100.0 1.4655 24 666 20.2 19.78
## 411 51.13580 0.0 18.10 0 0.5970 5.757 100.0 1.4130 24 666 20.2 10.11
## 412 14.05070 0.0 18.10 0 0.5970 6.657 100.0 1.5275 24 666 20.2 21.22
## 413 18.81100 0.0 18.10 0 0.5970 4.628 100.0 1.5539 24 666 20.2 34.37
## 414 28.65580 0.0 18.10 0 0.5970 5.155 100.0 1.5894 24 666 20.2 20.08
## 415 45.74610 0.0 18.10 0 0.6930 4.519 100.0 1.6582 24 666 20.2 36.98
## 416 18.08460 0.0 18.10 0 0.6790 6.434 100.0 1.8347 24 666 20.2 29.05
## 417 10.83420 0.0 18.10 0 0.6790 6.782 90.8 1.8195 24 666 20.2 25.79
## 418 25.94060 0.0 18.10 0 0.6790 5.304 89.1 1.6475 24 666 20.2 26.64
## 419 73.53410 0.0 18.10 0 0.6790 5.957 100.0 1.8026 24 666 20.2 20.62
## 420 11.81230 0.0 18.10 0 0.7180 6.824 76.5 1.7940 24 666 20.2 22.74
## 421 11.08740 0.0 18.10 0 0.7180 6.411 100.0 1.8589 24 666 20.2 15.02
## 422 7.02259 0.0 18.10 0 0.7180 6.006 95.3 1.8746 24 666 20.2 15.70
## 423 12.04820 0.0 18.10 0 0.6140 5.648 87.6 1.9512 24 666 20.2 14.10
## 424 7.05042 0.0 18.10 0 0.6140 6.103 85.1 2.0218 24 666 20.2 23.29
## 425 8.79212 0.0 18.10 0 0.5840 5.565 70.6 2.0635 24 666 20.2 17.16
## 426 15.86030 0.0 18.10 0 0.6790 5.896 95.4 1.9096 24 666 20.2 24.39
## 427 12.24720 0.0 18.10 0 0.5840 5.837 59.7 1.9976 24 666 20.2 15.69
## 428 37.66190 0.0 18.10 0 0.6790 6.202 78.7 1.8629 24 666 20.2 14.52
## 429 7.36711 0.0 18.10 0 0.6790 6.193 78.1 1.9356 24 666 20.2 21.52
## 430 9.33889 0.0 18.10 0 0.6790 6.380 95.6 1.9682 24 666 20.2 24.08
## 431 8.49213 0.0 18.10 0 0.5840 6.348 86.1 2.0527 24 666 20.2 17.64
## 432 10.06230 0.0 18.10 0 0.5840 6.833 94.3 2.0882 24 666 20.2 19.69
## 433 6.44405 0.0 18.10 0 0.5840 6.425 74.8 2.2004 24 666 20.2 12.03
## 434 5.58107 0.0 18.10 0 0.7130 6.436 87.9 2.3158 24 666 20.2 16.22
## 435 13.91340 0.0 18.10 0 0.7130 6.208 95.0 2.2222 24 666 20.2 15.17
## 436 11.16040 0.0 18.10 0 0.7400 6.629 94.6 2.1247 24 666 20.2 23.27
## 437 14.42080 0.0 18.10 0 0.7400 6.461 93.3 2.0026 24 666 20.2 18.05
## 438 15.17720 0.0 18.10 0 0.7400 6.152 100.0 1.9142 24 666 20.2 26.45
## 439 13.67810 0.0 18.10 0 0.7400 5.935 87.9 1.8206 24 666 20.2 34.02
## 440 9.39063 0.0 18.10 0 0.7400 5.627 93.9 1.8172 24 666 20.2 22.88
## 441 22.05110 0.0 18.10 0 0.7400 5.818 92.4 1.8662 24 666 20.2 22.11
## 442 9.72418 0.0 18.10 0 0.7400 6.406 97.2 2.0651 24 666 20.2 19.52
## 443 5.66637 0.0 18.10 0 0.7400 6.219 100.0 2.0048 24 666 20.2 16.59
## 444 9.96654 0.0 18.10 0 0.7400 6.485 100.0 1.9784 24 666 20.2 18.85
## 445 12.80230 0.0 18.10 0 0.7400 5.854 96.6 1.8956 24 666 20.2 23.79
## 446 10.67180 0.0 18.10 0 0.7400 6.459 94.8 1.9879 24 666 20.2 23.98
## 447 6.28807 0.0 18.10 0 0.7400 6.341 96.4 2.0720 24 666 20.2 17.79
## 448 9.92485 0.0 18.10 0 0.7400 6.251 96.6 2.1980 24 666 20.2 16.44
## 449 9.32909 0.0 18.10 0 0.7130 6.185 98.7 2.2616 24 666 20.2 18.13
## 450 7.52601 0.0 18.10 0 0.7130 6.417 98.3 2.1850 24 666 20.2 19.31
## 451 6.71772 0.0 18.10 0 0.7130 6.749 92.6 2.3236 24 666 20.2 17.44
## 452 5.44114 0.0 18.10 0 0.7130 6.655 98.2 2.3552 24 666 20.2 17.73
## 453 5.09017 0.0 18.10 0 0.7130 6.297 91.8 2.3682 24 666 20.2 17.27
## 454 8.24809 0.0 18.10 0 0.7130 7.393 99.3 2.4527 24 666 20.2 16.74
## 455 9.51363 0.0 18.10 0 0.7130 6.728 94.1 2.4961 24 666 20.2 18.71
## 456 4.75237 0.0 18.10 0 0.7130 6.525 86.5 2.4358 24 666 20.2 18.13
## 457 4.66883 0.0 18.10 0 0.7130 5.976 87.9 2.5806 24 666 20.2 19.01
## 458 8.20058 0.0 18.10 0 0.7130 5.936 80.3 2.7792 24 666 20.2 16.94
## 459 7.75223 0.0 18.10 0 0.7130 6.301 83.7 2.7831 24 666 20.2 16.23
## 460 6.80117 0.0 18.10 0 0.7130 6.081 84.4 2.7175 24 666 20.2 14.70
## 461 4.81213 0.0 18.10 0 0.7130 6.701 90.0 2.5975 24 666 20.2 16.42
## 462 3.69311 0.0 18.10 0 0.7130 6.376 88.4 2.5671 24 666 20.2 14.65
## 463 6.65492 0.0 18.10 0 0.7130 6.317 83.0 2.7344 24 666 20.2 13.99
## 464 5.82115 0.0 18.10 0 0.7130 6.513 89.9 2.8016 24 666 20.2 10.29
## 465 7.83932 0.0 18.10 0 0.6550 6.209 65.4 2.9634 24 666 20.2 13.22
## 466 3.16360 0.0 18.10 0 0.6550 5.759 48.2 3.0665 24 666 20.2 14.13
## 467 3.77498 0.0 18.10 0 0.6550 5.952 84.7 2.8715 24 666 20.2 17.15
## 468 4.42228 0.0 18.10 0 0.5840 6.003 94.5 2.5403 24 666 20.2 21.32
## 469 15.57570 0.0 18.10 0 0.5800 5.926 71.0 2.9084 24 666 20.2 18.13
## 470 13.07510 0.0 18.10 0 0.5800 5.713 56.7 2.8237 24 666 20.2 14.76
## 471 4.34879 0.0 18.10 0 0.5800 6.167 84.0 3.0334 24 666 20.2 16.29
## 472 4.03841 0.0 18.10 0 0.5320 6.229 90.7 3.0993 24 666 20.2 12.87
## 473 3.56868 0.0 18.10 0 0.5800 6.437 75.0 2.8965 24 666 20.2 14.36
## 474 4.64689 0.0 18.10 0 0.6140 6.980 67.6 2.5329 24 666 20.2 11.66
## 475 8.05579 0.0 18.10 0 0.5840 5.427 95.4 2.4298 24 666 20.2 18.14
## 476 6.39312 0.0 18.10 0 0.5840 6.162 97.4 2.2060 24 666 20.2 24.10
## 477 4.87141 0.0 18.10 0 0.6140 6.484 93.6 2.3053 24 666 20.2 18.68
## 478 15.02340 0.0 18.10 0 0.6140 5.304 97.3 2.1007 24 666 20.2 24.91
## 479 10.23300 0.0 18.10 0 0.6140 6.185 96.7 2.1705 24 666 20.2 18.03
## 480 14.33370 0.0 18.10 0 0.6140 6.229 88.0 1.9512 24 666 20.2 13.11
## 481 5.82401 0.0 18.10 0 0.5320 6.242 64.7 3.4242 24 666 20.2 10.74
## 482 5.70818 0.0 18.10 0 0.5320 6.750 74.9 3.3317 24 666 20.2 7.74
## 483 5.73116 0.0 18.10 0 0.5320 7.061 77.0 3.4106 24 666 20.2 7.01
## 484 2.81838 0.0 18.10 0 0.5320 5.762 40.3 4.0983 24 666 20.2 10.42
## 485 2.37857 0.0 18.10 0 0.5830 5.871 41.9 3.7240 24 666 20.2 13.34
## 486 3.67367 0.0 18.10 0 0.5830 6.312 51.9 3.9917 24 666 20.2 10.58
## 487 5.69175 0.0 18.10 0 0.5830 6.114 79.8 3.5459 24 666 20.2 14.98
## 488 4.83567 0.0 18.10 0 0.5830 5.905 53.2 3.1523 24 666 20.2 11.45
## 489 0.15086 0.0 27.74 0 0.6090 5.454 92.7 1.8209 4 711 20.1 18.06
## 490 0.18337 0.0 27.74 0 0.6090 5.414 98.3 1.7554 4 711 20.1 23.97
## 491 0.20746 0.0 27.74 0 0.6090 5.093 98.0 1.8226 4 711 20.1 29.68
## 492 0.10574 0.0 27.74 0 0.6090 5.983 98.8 1.8681 4 711 20.1 18.07
## 493 0.11132 0.0 27.74 0 0.6090 5.983 83.5 2.1099 4 711 20.1 13.35
## 494 0.17331 0.0 9.69 0 0.5850 5.707 54.0 2.3817 6 391 19.2 12.01
## 495 0.27957 0.0 9.69 0 0.5850 5.926 42.6 2.3817 6 391 19.2 13.59
## 496 0.17899 0.0 9.69 0 0.5850 5.670 28.8 2.7986 6 391 19.2 17.60
## 497 0.28960 0.0 9.69 0 0.5850 5.390 72.9 2.7986 6 391 19.2 21.14
## 498 0.26838 0.0 9.69 0 0.5850 5.794 70.6 2.8927 6 391 19.2 14.10
## 499 0.23912 0.0 9.69 0 0.5850 6.019 65.3 2.4091 6 391 19.2 12.92
## 500 0.17783 0.0 9.69 0 0.5850 5.569 73.5 2.3999 6 391 19.2 15.10
## 501 0.22438 0.0 9.69 0 0.5850 6.027 79.7 2.4982 6 391 19.2 14.33
## 502 0.06263 0.0 11.93 0 0.5730 6.593 69.1 2.4786 1 273 21.0 9.67
## 503 0.04527 0.0 11.93 0 0.5730 6.120 76.7 2.2875 1 273 21.0 9.08
## 504 0.06076 0.0 11.93 0 0.5730 6.976 91.0 2.1675 1 273 21.0 5.64
## 505 0.10959 0.0 11.93 0 0.5730 6.794 89.3 2.3889 1 273 21.0 6.48
## 506 0.04741 0.0 11.93 0 0.5730 6.030 80.8 2.5050 1 273 21.0 7.88
## medv
## 1 24.0
## 2 21.6
## 3 34.7
## 4 33.4
## 5 36.2
## 6 28.7
## 7 22.9
## 8 27.1
## 9 16.5
## 10 18.9
## 11 15.0
## 12 18.9
## 13 21.7
## 14 20.4
## 15 18.2
## 16 19.9
## 17 23.1
## 18 17.5
## 19 20.2
## 20 18.2
## 21 13.6
## 22 19.6
## 23 15.2
## 24 14.5
## 25 15.6
## 26 13.9
## 27 16.6
## 28 14.8
## 29 18.4
## 30 21.0
## 31 12.7
## 32 14.5
## 33 13.2
## 34 13.1
## 35 13.5
## 36 18.9
## 37 20.0
## 38 21.0
## 39 24.7
## 40 30.8
## 41 34.9
## 42 26.6
## 43 25.3
## 44 24.7
## 45 21.2
## 46 19.3
## 47 20.0
## 48 16.6
## 49 14.4
## 50 19.4
## 51 19.7
## 52 20.5
## 53 25.0
## 54 23.4
## 55 18.9
## 56 35.4
## 57 24.7
## 58 31.6
## 59 23.3
## 60 19.6
## 61 18.7
## 62 16.0
## 63 22.2
## 64 25.0
## 65 33.0
## 66 23.5
## 67 19.4
## 68 22.0
## 69 17.4
## 70 20.9
## 71 24.2
## 72 21.7
## 73 22.8
## 74 23.4
## 75 24.1
## 76 21.4
## 77 20.0
## 78 20.8
## 79 21.2
## 80 20.3
## 81 28.0
## 82 23.9
## 83 24.8
## 84 22.9
## 85 23.9
## 86 26.6
## 87 22.5
## 88 22.2
## 89 23.6
## 90 28.7
## 91 22.6
## 92 22.0
## 93 22.9
## 94 25.0
## 95 20.6
## 96 28.4
## 97 21.4
## 98 38.7
## 99 43.8
## 100 33.2
## 101 27.5
## 102 26.5
## 103 18.6
## 104 19.3
## 105 20.1
## 106 19.5
## 107 19.5
## 108 20.4
## 109 19.8
## 110 19.4
## 111 21.7
## 112 22.8
## 113 18.8
## 114 18.7
## 115 18.5
## 116 18.3
## 117 21.2
## 118 19.2
## 119 20.4
## 120 19.3
## 121 22.0
## 122 20.3
## 123 20.5
## 124 17.3
## 125 18.8
## 126 21.4
## 127 15.7
## 128 16.2
## 129 18.0
## 130 14.3
## 131 19.2
## 132 19.6
## 133 23.0
## 134 18.4
## 135 15.6
## 136 18.1
## 137 17.4
## 138 17.1
## 139 13.3
## 140 17.8
## 141 14.0
## 142 14.4
## 143 13.4
## 144 15.6
## 145 11.8
## 146 13.8
## 147 15.6
## 148 14.6
## 149 17.8
## 150 15.4
## 151 21.5
## 152 19.6
## 153 15.3
## 154 19.4
## 155 17.0
## 156 15.6
## 157 13.1
## 158 41.3
## 159 24.3
## 160 23.3
## 161 27.0
## 162 50.0
## 163 50.0
## 164 50.0
## 165 22.7
## 166 25.0
## 167 50.0
## 168 23.8
## 169 23.8
## 170 22.3
## 171 17.4
## 172 19.1
## 173 23.1
## 174 23.6
## 175 22.6
## 176 29.4
## 177 23.2
## 178 24.6
## 179 29.9
## 180 37.2
## 181 39.8
## 182 36.2
## 183 37.9
## 184 32.5
## 185 26.4
## 186 29.6
## 187 50.0
## 188 32.0
## 189 29.8
## 190 34.9
## 191 37.0
## 192 30.5
## 193 36.4
## 194 31.1
## 195 29.1
## 196 50.0
## 197 33.3
## 198 30.3
## 199 34.6
## 200 34.9
## 201 32.9
## 202 24.1
## 203 42.3
## 204 48.5
## 205 50.0
## 206 22.6
## 207 24.4
## 208 22.5
## 209 24.4
## 210 20.0
## 211 21.7
## 212 19.3
## 213 22.4
## 214 28.1
## 215 23.7
## 216 25.0
## 217 23.3
## 218 28.7
## 219 21.5
## 220 23.0
## 221 26.7
## 222 21.7
## 223 27.5
## 224 30.1
## 225 44.8
## 226 50.0
## 227 37.6
## 228 31.6
## 229 46.7
## 230 31.5
## 231 24.3
## 232 31.7
## 233 41.7
## 234 48.3
## 235 29.0
## 236 24.0
## 237 25.1
## 238 31.5
## 239 23.7
## 240 23.3
## 241 22.0
## 242 20.1
## 243 22.2
## 244 23.7
## 245 17.6
## 246 18.5
## 247 24.3
## 248 20.5
## 249 24.5
## 250 26.2
## 251 24.4
## 252 24.8
## 253 29.6
## 254 42.8
## 255 21.9
## 256 20.9
## 257 44.0
## 258 50.0
## 259 36.0
## 260 30.1
## 261 33.8
## 262 43.1
## 263 48.8
## 264 31.0
## 265 36.5
## 266 22.8
## 267 30.7
## 268 50.0
## 269 43.5
## 270 20.7
## 271 21.1
## 272 25.2
## 273 24.4
## 274 35.2
## 275 32.4
## 276 32.0
## 277 33.2
## 278 33.1
## 279 29.1
## 280 35.1
## 281 45.4
## 282 35.4
## 283 46.0
## 284 50.0
## 285 32.2
## 286 22.0
## 287 20.1
## 288 23.2
## 289 22.3
## 290 24.8
## 291 28.5
## 292 37.3
## 293 27.9
## 294 23.9
## 295 21.7
## 296 28.6
## 297 27.1
## 298 20.3
## 299 22.5
## 300 29.0
## 301 24.8
## 302 22.0
## 303 26.4
## 304 33.1
## 305 36.1
## 306 28.4
## 307 33.4
## 308 28.2
## 309 22.8
## 310 20.3
## 311 16.1
## 312 22.1
## 313 19.4
## 314 21.6
## 315 23.8
## 316 16.2
## 317 17.8
## 318 19.8
## 319 23.1
## 320 21.0
## 321 23.8
## 322 23.1
## 323 20.4
## 324 18.5
## 325 25.0
## 326 24.6
## 327 23.0
## 328 22.2
## 329 19.3
## 330 22.6
## 331 19.8
## 332 17.1
## 333 19.4
## 334 22.2
## 335 20.7
## 336 21.1
## 337 19.5
## 338 18.5
## 339 20.6
## 340 19.0
## 341 18.7
## 342 32.7
## 343 16.5
## 344 23.9
## 345 31.2
## 346 17.5
## 347 17.2
## 348 23.1
## 349 24.5
## 350 26.6
## 351 22.9
## 352 24.1
## 353 18.6
## 354 30.1
## 355 18.2
## 356 20.6
## 357 17.8
## 358 21.7
## 359 22.7
## 360 22.6
## 361 25.0
## 362 19.9
## 363 20.8
## 364 16.8
## 365 21.9
## 366 27.5
## 367 21.9
## 368 23.1
## 369 50.0
## 370 50.0
## 371 50.0
## 372 50.0
## 373 50.0
## 374 13.8
## 375 13.8
## 376 15.0
## 377 13.9
## 378 13.3
## 379 13.1
## 380 10.2
## 381 10.4
## 382 10.9
## 383 11.3
## 384 12.3
## 385 8.8
## 386 7.2
## 387 10.5
## 388 7.4
## 389 10.2
## 390 11.5
## 391 15.1
## 392 23.2
## 393 9.7
## 394 13.8
## 395 12.7
## 396 13.1
## 397 12.5
## 398 8.5
## 399 5.0
## 400 6.3
## 401 5.6
## 402 7.2
## 403 12.1
## 404 8.3
## 405 8.5
## 406 5.0
## 407 11.9
## 408 27.9
## 409 17.2
## 410 27.5
## 411 15.0
## 412 17.2
## 413 17.9
## 414 16.3
## 415 7.0
## 416 7.2
## 417 7.5
## 418 10.4
## 419 8.8
## 420 8.4
## 421 16.7
## 422 14.2
## 423 20.8
## 424 13.4
## 425 11.7
## 426 8.3
## 427 10.2
## 428 10.9
## 429 11.0
## 430 9.5
## 431 14.5
## 432 14.1
## 433 16.1
## 434 14.3
## 435 11.7
## 436 13.4
## 437 9.6
## 438 8.7
## 439 8.4
## 440 12.8
## 441 10.5
## 442 17.1
## 443 18.4
## 444 15.4
## 445 10.8
## 446 11.8
## 447 14.9
## 448 12.6
## 449 14.1
## 450 13.0
## 451 13.4
## 452 15.2
## 453 16.1
## 454 17.8
## 455 14.9
## 456 14.1
## 457 12.7
## 458 13.5
## 459 14.9
## 460 20.0
## 461 16.4
## 462 17.7
## 463 19.5
## 464 20.2
## 465 21.4
## 466 19.9
## 467 19.0
## 468 19.1
## 469 19.1
## 470 20.1
## 471 19.9
## 472 19.6
## 473 23.2
## 474 29.8
## 475 13.8
## 476 13.3
## 477 16.7
## 478 12.0
## 479 14.6
## 480 21.4
## 481 23.0
## 482 23.7
## 483 25.0
## 484 21.8
## 485 20.6
## 486 21.2
## 487 19.1
## 488 20.6
## 489 15.2
## 490 7.0
## 491 8.1
## 492 13.6
## 493 20.1
## 494 21.8
## 495 24.5
## 496 23.1
## 497 19.7
## 498 18.3
## 499 21.2
## 500 17.5
## 501 16.8
## 502 22.4
## 503 20.6
## 504 23.9
## 505 22.0
## 506 11.9# Number of rows and columns
dim(Boston) # returns c(rows, columns)## [1] 506 13# Description of the dataset
?BostonAnswer: The dataset has 506 rows and 14 columns.
Each row represents a census tract in the Boston area.
Each column is a predictor or response variable describing attributes like crime rate, property tax rate, number of rooms, etc.
b)
# Scatterplot matrix of the first few columns (to avoid overwhelming the plot)
pairs(Boston[, 1:6]) # You can adjust the range to see more/less
Answer: You’ll see linear and non-linear relationships between
predictors.
For instance, rm (average number of rooms) may be positively correlated with medv (median value of homes), while lstat (lower status) is negatively correlated with medv.
c)
# Check correlations
cor(Boston$crim, Boston[, -1]) # -1 to exclude crim itself## zn indus chas nox rm age dis
## [1,] -0.2004692 0.4065834 -0.05589158 0.4209717 -0.2192467 0.3527343 -0.3796701
## rad tax ptratio lstat medv
## [1,] 0.6255051 0.5827643 0.2899456 0.4556215 -0.3883046# Plotting for stronger correlations
plot(Boston$crim, Boston$rad) # Example with high correlation
Answer: crim is positively associated with rad (access to radial
highways) and tax, and negatively associated with ptratio. Higher crime
rates tend to occur in areas with more highways and higher taxes.
d)
# Summary statistics
summary(Boston$crim)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.00632 0.08204 0.25651 3.61352 3.67708 88.97620summary(Boston$tax)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 187.0 279.0 330.0 408.2 666.0 711.0summary(Boston$ptratio)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 12.60 17.40 19.05 18.46 20.20 22.00# Find rows with high values
Boston[Boston$crim == max(Boston$crim), ]## crim zn indus chas nox rm age dis rad tax ptratio lstat medv
## 381 88.9762 0 18.1 0 0.671 6.968 91.9 1.4165 24 666 20.2 17.21 10.4Boston[Boston$tax == max(Boston$tax), ]## crim zn indus chas nox rm age dis rad tax ptratio lstat medv
## 489 0.15086 0 27.74 0 0.609 5.454 92.7 1.8209 4 711 20.1 18.06 15.2
## 490 0.18337 0 27.74 0 0.609 5.414 98.3 1.7554 4 711 20.1 23.97 7.0
## 491 0.20746 0 27.74 0 0.609 5.093 98.0 1.8226 4 711 20.1 29.68 8.1
## 492 0.10574 0 27.74 0 0.609 5.983 98.8 1.8681 4 711 20.1 18.07 13.6
## 493 0.11132 0 27.74 0 0.609 5.983 83.5 2.1099 4 711 20.1 13.35 20.1Boston[Boston$ptratio == max(Boston$ptratio), ]## crim zn indus chas nox rm age dis rad tax ptratio lstat medv
## 355 0.04301 80 1.91 0 0.413 5.663 21.9 10.5857 4 334 22 8.05 18.2
## 356 0.10659 80 1.91 0 0.413 5.936 19.5 10.5857 4 334 22 5.57 20.6Answer:
The crime rate goes up to 88.98 — indicating outliers.
The tax rate ranges from 187 to 711.
Pupil-teacher ratio ranges from 12.6 to 22.0.
Some census tracts clearly stand out as extreme.
e)
# `chas` = 1 means tract bounds river
sum(Boston$chas == 1)## [1] 35Answer: 35 census tracts bound the Charles River.
f)
median(Boston$ptratio)## [1] 19.05Answer: The median pupil-teacher ratio is 19.05.
g)
# Find index of min medv
which.min(Boston$medv)## [1] 399# Display the row
Boston[which.min(Boston$medv), ]## crim zn indus chas nox rm age dis rad tax ptratio lstat medv
## 399 38.3518 0 18.1 0 0.693 5.453 100 1.4896 24 666 20.2 30.59 5# Compare with summary
summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio lstat
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 1.73
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.: 6.95
## Median : 5.000 Median :330.0 Median :19.05 Median :11.36
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :12.65
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:16.95
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :37.97
## medv
## Min. : 5.00
## 1st Qu.:17.02
## Median :21.20
## Mean :22.53
## 3rd Qu.:25.00
## Max. :50.00Answer: The tract with the lowest median value has a medv of $5,000. It has high crime, high lstat, and low rm. These values are at the extremes compared to the dataset, indicating a lower socioeconomic area.
h)
# > 7 rooms
sum(Boston$rm > 7)## [1] 64# > 8 rooms
sum(Boston$rm > 8)## [1] 13# Show those tracts
Boston[Boston$rm > 8, ]## crim zn indus chas nox rm age dis rad tax ptratio lstat medv
## 98 0.12083 0 2.89 0 0.4450 8.069 76.0 3.4952 2 276 18.0 4.21 38.7
## 164 1.51902 0 19.58 1 0.6050 8.375 93.9 2.1620 5 403 14.7 3.32 50.0
## 205 0.02009 95 2.68 0 0.4161 8.034 31.9 5.1180 4 224 14.7 2.88 50.0
## 225 0.31533 0 6.20 0 0.5040 8.266 78.3 2.8944 8 307 17.4 4.14 44.8
## 226 0.52693 0 6.20 0 0.5040 8.725 83.0 2.8944 8 307 17.4 4.63 50.0
## 227 0.38214 0 6.20 0 0.5040 8.040 86.5 3.2157 8 307 17.4 3.13 37.6
## 233 0.57529 0 6.20 0 0.5070 8.337 73.3 3.8384 8 307 17.4 2.47 41.7
## 234 0.33147 0 6.20 0 0.5070 8.247 70.4 3.6519 8 307 17.4 3.95 48.3
## 254 0.36894 22 5.86 0 0.4310 8.259 8.4 8.9067 7 330 19.1 3.54 42.8
## 258 0.61154 20 3.97 0 0.6470 8.704 86.9 1.8010 5 264 13.0 5.12 50.0
## 263 0.52014 20 3.97 0 0.6470 8.398 91.5 2.2885 5 264 13.0 5.91 48.8
## 268 0.57834 20 3.97 0 0.5750 8.297 67.0 2.4216 5 264 13.0 7.44 50.0
## 365 3.47428 0 18.10 1 0.7180 8.780 82.9 1.9047 24 666 20.2 5.29 21.9Answer:
64 tracts average more than 7 rooms.
13 tracts average more than 8 rooms.
These tracts tend to also have high medv, low crim, and favorable socio-economic indicators.
Chapter 3 Exercise 2
# Load libraries
library(class)
library(datasets)
# Prepare data
data(iris)
set.seed(123)
# Split into training and test sets
train_index <- sample(1:nrow(iris), 100)
train_data <- iris[train_index, 1:4]
train_labels <- iris[train_index, 5]
test_data <- iris[-train_index, 1:4]
test_labels <- iris[-train_index, 5]
# Run KNN Classifier
knn_pred <- knn(train = train_data, test = test_data, cl = train_labels, k = 3)
# Accuracy
mean(knn_pred == test_labels)## [1] 0.96This predicts species (a class label) using majority vote from K=3 nearest neighbors.
# Load required packages
library(FNN)##
## Attaching package: 'FNN'## The following objects are masked from 'package:class':
##
## knn, knn.cvlibrary(ISLR2)
# Load the Boston dataset
data("Boston")
# Set seed for reproducibility
set.seed(123)
# Split into training and test sets
train_index <- sample(1:nrow(Boston), 400)
train_data <- Boston[train_index, -14] # All predictors
train_target <- Boston[train_index, "medv"] # Target variable
test_data <- Boston[-train_index, -14]
test_target <- Boston[-train_index, "medv"]
# Run KNN Regression
knn_reg <- knn.reg(train = train_data, test = test_data, y = train_target, k = 3)
# View first few predictions
head(knn_reg$pred)## [1] 22.06667 24.46667 19.36667 27.53333 24.76667 15.23333# Calculate Mean Squared Error
mse <- mean((knn_reg$pred - test_target)^2)
print(mse)## [1] 19.35238This predicts medv (median house value, a numeric value) using average of neighbors.
Chapter 3 Exercise 10 a)
# Load packages and data
library(ISLR2)
data(Carseats)
# Fit model with Price, Urban, and US as predictors
model_a <- lm(Sales ~ Price + Urban + US, data = Carseats)
# View summary
summary(model_a)##
## 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-16b)
Term | Estimate | Interpretation Intercept | 13.0435 | Expected Sales when Price = 0, Urban = “No”, and US = “No”. While not realistic (since Price = 0 is not practical), it’s the baseline for the model. Price | -0.0545 | For each $1 increase in price, Sales decrease by 0.0545 units, holding other variables constant. UrbanYes | -0.0219 | Stores in urban areas sell 0.0219 units fewer than non-urban stores, on average, holding Price and US constant — but this effect is not statistically significant (p = 0.936). USYes | +1.2006 | Stores located in the US sell 1.2 more units than non-US stores, on average, holding Price and Urban constant. Statistically significant.
c) The model is: Sales=13.0435−0.0545⋅Price−0.0219⋅UrbanYes+1.2006⋅USYes+ϵ Where: UrbanYes = 1 if Urban = “Yes”, otherwise 0 USYes = 1 if US = “Yes”, otherwise 0
d)
Variable | p-value | Decision
Price | < 2e-16 | Reject H₀ → Significant
UrbanYes | 0.936 | Fail to reject H₀ → Not significant
USYes | 4.86e-06 | Reject H₀ → Significant
e)
model_e <- lm(Sales ~ Price + US, data = Carseats)
summary(model_e)##
## Call:
## lm(formula = Sales ~ Price + US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9269 -1.6286 -0.0574 1.5766 7.0515
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.03079 0.63098 20.652 < 2e-16 ***
## Price -0.05448 0.00523 -10.416 < 2e-16 ***
## USYes 1.19964 0.25846 4.641 4.71e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.469 on 397 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2354
## F-statistic: 62.43 on 2 and 397 DF, p-value: < 2.2e-16f)
summary(model_a)$adj.r.squared## [1] 0.2335123summary(model_e)$adj.r.squared## [1] 0.2354305Compare Adjusted R-squared: If they are very similar, the simpler model is preferable (fewer predictors, similar performance).
g)
confint(model_e, level = 0.95)## 2.5 % 97.5 %
## (Intercept) 11.79032020 14.27126531
## Price -0.06475984 -0.04419543
## USYes 0.69151957 1.70776632This gives 95% CIs for each coefficient in the smaller model.
h)
# Plot diagnostic plots
par(mfrow = c(2, 2))
plot(model_e)
Chapter 3 Exersice 12
a) log(Pr(Y=orange∣x)/Pr(Y=apple∣x))=β0+β1x
b) log(Pr(Y=orange)/Pr(Y=apple))=(αorange0−αapple0)+(αorange1−αapple1)x
c) Your model: 𝛽0=2, 𝛽1=−1
Let’s find friend’s coefficients assuming softmax uses apple as the reference (arbitrary shift possible because softmax is invariant to adding same constant to both linear terms):
One way to convert:
Let: 𝛼orange0=2, 𝛼orange1=−1 𝛼apple0=0, 𝛼apple1=0
Then: log(exp(2−𝑥)/exp(2−𝑥)+exp(0)) matches 𝛽0+𝛽x=2−x So: 𝛼orange0=2 𝛼orange1=−1 𝛼apple0=0 𝛼apple1=0
d) 𝛼orange0=1.2, 𝛼orange1=−2 𝛼apple0=3, 𝛼apple1=0.6
Convert to the model form (log odds): log(Pr(𝑌=apple)/Pr(𝑌=orange))=(𝛼orange0−𝛼apple0)+(𝛼orange1−𝛼apple1)𝑥=(1.2−3)+(−2−0.6)𝑥=−1.8−2.6𝑥
So the model estimates:
𝛽0=−1.8 𝛽1=−2.6
e)
# Generate a sequence of x values
set.seed(123)
x_vals <- seq(-5, 5, length.out = 1000)
# Friend's softmax model probabilities
alpha_orange0 <- 1.2; alpha_orange1 <- -2
alpha_apple0 <- 3; alpha_apple1 <- 0.6
logit_orange <- alpha_orange0 + alpha_orange1 * x_vals
logit_apple <- alpha_apple0 + alpha_apple1 * x_vals
p_orange_softmax <- exp(logit_orange) / (exp(logit_orange) + exp(logit_apple))
p_apple_softmax <- 1 - p_orange_softmax
# Your model (binary logistic)
beta0 <- -1.8; beta1 <- -2.6
p_orange_logit <- exp(beta0 + beta1 * x_vals) / (1 + exp(beta0 + beta1 * x_vals))
# Predicted labels
labels_softmax <- ifelse(p_orange_softmax > 0.5, "orange", "apple")
labels_logit <- ifelse(p_orange_logit > 0.5, "orange", "apple")
# Agreement rate
mean(labels_softmax == labels_logit)## [1] 1This code calculates the fraction of predictions that match between your model and your friend’s model.