> library(MASS) > > > > > > library("ISLR") Error in library("ISLR") : there is no package called ‘ISLR’ > > install.packages("MASS") Installing package into ‘C:/Users/201706401/Documents/R/win-library/3.4’ (as ‘lib’ is unspecified) There is a binary version available but the source version is later: binary source needs_compilation MASS 7.3-51.4 7.3-51.5 TRUE Binaries will be installed Warning in install.packages : package ‘MASS’ is in use and will not be installed > library(MASS) > library("ISLR") Error in library("ISLR") : there is no package called ‘ISLR’ > install.packages("ISLR") Installing package into ‘C:/Users/201706401/Documents/R/win-library/3.4’ (as ‘lib’ is unspecified) trying URL 'https://cran.rstudio.com/bin/windows/contrib/3.4/ISLR_1.2.zip' Content type 'application/zip' length 2923604 bytes (2.8 MB) downloaded 2.8 MB package ‘ISLR’ successfully unpacked and MD5 sums checked The downloaded binary packages are in C:\Users\201706401\AppData\Local\Temp\RtmpOAMblL\downloaded_packages > library("ISLR") Warning message: package ‘ISLR’ was built under R version 3.4.4 > fix(Boston) Warning messages: 1: In doTryCatch(return(expr), name, parentenv, handler) : restarting interrupted promise evaluation 2: In doTryCatch(return(expr), name, parentenv, handler) : restarting interrupted promise evaluation > names(Boston) [1] "crim" "zn" "indus" "chas" "nox" "rm" "age" "dis" "rad" "tax" [11] "ptratio" "black" "lstat" "medv" > attach(Boston) > > lm.fit = lm(medv-lstat) Error in formula.default(object, env = baseenv()) : invalid formula > lm.fit = lm(medv-lstat) Error in formula.default(object, env = baseenv()) : invalid formula > lm.fit = lm(medv~lstat) > summary(lm.fit) Call: lm(formula = medv ~ lstat) Residuals: Min 1Q Median 3Q Max -15.168 -3.990 -1.318 2.034 24.500 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 34.55384 0.56263 61.41 <2e-16 *** lstat -0.95005 0.03873 -24.53 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 6.216 on 504 degrees of freedom Multiple R-squared: 0.5441, Adjusted R-squared: 0.5432 F-statistic: 601.6 on 1 and 504 DF, p-value: < 2.2e-16 > names(lm.fit) [1] "coefficients" "residuals" "effects" "rank" "fitted.values" "assign" [7] "qr" "df.residual" "xlevels" "call" "terms" "model" > coefficients(lm.fit) (Intercept) lstat 34.5538409 -0.9500494 > confint(lm.fit) 2.5 % 97.5 % (Intercept) 33.448457 35.6592247 lstat -1.026148 -0.8739505 > predict(lm.fit,data.frame(lstat-c(5,10,15))) 1 2 3 4 5 6 7 8 9 29.8225951 25.8703898 30.7251420 31.7606958 29.4900778 29.6040837 22.7447274 16.3603958 6.1188637 10 11 12 13 14 15 16 17 18 18.3079969 15.1253316 21.9466860 19.6285655 26.7064332 24.8063345 26.5069229 28.3025161 20.6166169 19 20 21 22 23 24 25 26 27 23.4477639 23.8372842 14.5838035 21.4146583 16.7689170 15.6668597 19.0680364 18.8685260 20.4836100 28 29 30 31 32 33 34 35 36 18.1369880 22.3932092 23.1722496 13.0827255 22.1651973 8.2279733 17.1204352 15.2298370 25.3573631 37 38 39 40 41 42 43 44 45 23.7137778 26.2219080 24.9298409 30.4496277 32.6727432 29.9556020 29.0340541 27.4854737 25.4808696 46 47 48 49 50 51 52 53 54 24.8538370 21.1106425 16.6929130 5.2828203 19.1630413 21.7756771 25.5948755 29.5375803 26.5449248 55 56 57 58 59 60 61 62 63 20.4931104 29.9841035 29.0720561 30.8011459 28.0365023 25.7943858 22.0606919 20.8351282 28.1600087 64 65 66 67 68 69 70 71 72 25.5283720 26.9059436 30.1171104 24.8253355 26.8584411 22.1176948 26.2029071 28.1695092 25.1673533 73 74 75 76 77 78 79 80 81 29.3095684 27.3904688 28.1125063 26.0603997 23.1817501 24.7968340 22.8302319 25.9083918 29.5280798 82 83 84 85 86 87 88 89 90 27.6944845 28.1695092 27.4189702 25.4143661 28.3500186 22.3362062 26.5354243 29.3285694 29.1385596 91 92 93 94 95 96 97 98 99 26.1839061 26.7634362 26.8014382 28.6540344 24.4928182 28.2360127 23.7802812 30.5541331 31.1621647 100 101 102 103 104 105 106 107 108 28.6730354 25.6043760 27.2669623 24.4548162 21.7851776 22.8397323 18.9065280 16.8259199 21.1676455 109 110 111 112 113 114 115 116 117 22.8967353 19.7805734 22.2031993 24.9013394 19.1535409 18.3174974 24.6258251 19.5810631 23.1152467 118 119 120 121 122 123 124 125 126 24.7683325 19.9515823 21.6236692 20.9016317 20.9966366 17.5194560 10.4130868 17.8519732 20.4836100 127 128 129 130 131 132 133 134 135 8.6554955 18.2224925 19.9325813 17.1299357 22.5832190 22.9062358 23.9892921 20.2745991 18.1084866 136 137 138 139 140 141 142 143 144 18.4410038 18.4980068 20.6926208 14.2987887 17.0159298 11.6006485 1.8626426 9.0735172 9.4535370 145 146 147 148 149 150 151 152 153 6.7268953 8.1424688 18.7355191 6.4988835 7.6484432 14.1752822 21.1581450 21.9371855 23.0392427 154 155 156 157 158 159 160 161 162 19.5525616 20.1890947 20.2840996 19.2200443 30.1931143 28.4450235 27.5329762 29.3285694 32.9102555 163 164 165 166 167 168 169 170 171 32.7297461 31.3996770 23.4952664 25.2338567 31.0386583 23.0202417 24.0082931 23.7992822 20.8446287 172 173 174 175 176 177 178 179 180 23.1247472 20.5976159 25.9653947 25.3953651 29.4900778 24.9488419 28.5780304 27.9794994 29.7655921 181 182 183 184 185 186 187 188 189 27.3714678 25.5758745 29.9746030 29.1575606 21.2721509 22.0606919 30.3261213 28.2075112 30.2216158 190 191 192 193 194 195 196 197 198 29.4330749 29.7085892 30.0981094 31.8271992 29.7750926 30.3926247 31.7321943 30.6776395 26.3739159 199 200 201 202 203 204 205 206 207 28.2645142 30.2216158 30.3261213 27.4949742 31.5991874 30.9341528 31.8176987 24.2268044 24.1317995 208 209 210 211 212 213 214 215 216 17.3959496 20.6261174 12.6172013 18.1464885 11.7716574 19.3245497 25.6423779 6.4798825 25.5568735 217 218 219 220 221 222 223 224 225 21.7186741 25.3478626 17.5289565 24.5783227 25.3288617 14.1657817 25.1198508 27.3334658 30.6206366 226 227 228 229 230 231 232 233 234 30.1551124 31.5801864 28.5115270 30.8296474 30.9816553 23.4857659 29.5660818 32.2072190 30.8011459 235 236 237 238 239 240 241 242 243 26.9059436 24.2173039 25.4903700 30.0601074 28.5115270 27.5519771 23.7422792 22.7732289 23.8942871 244 245 246 247 248 249 250 251 252 29.6230847 22.6782240 17.0159298 25.8513888 24.9108399 25.5093710 28.3215171 28.9485497 31.1431637 253 254 255 256 257 258 259 260 261 31.2001667 31.1906662 28.3120166 25.7658844 31.5991874 29.6895882 27.1529564 27.9985003 25.4428676 262 263 264 265 266 267 268 269 270 27.6564826 28.9390492 23.8657856 26.8584411 24.6258251 20.5026109 27.4854737 31.5516849 21.5856672 271 272 273 274 275 276 277 278 279 22.2031993 28.2930156 27.2099594 28.3025161 31.2001667 31.7226938 28.8060423 30.6016356 27.7229860 280 281 282 283 284 285 286 287 288 29.9461015 30.9816553 30.1931143 31.6941923 31.5516849 27.0959535 26.7349347 22.2697027 27.7704885 289 290 291 292 293 294 295 296 297 27.3334658 25.5188715 31.3901765 31.1716652 30.0886089 26.4024174 24.6733276 28.5970314 27.5329762 298 299 300 301 302 303 304 305 306 19.5050591 29.8320956 30.0506069 28.7870413 25.5283720 26.3169130 29.9366010 27.9699989 26.0699002 307 308 309 310 311 312 313 314 315 28.4070216 27.3999692 30.2406168 25.0818488 22.5452170 28.8725457 23.4192625 27.0484510 25.7373829 316 317 318 319 320 321 322 323 324 23.6282733 17.1394362 19.4100542 24.7113296 22.4597126 27.7134855 28.0270018 27.2384609 23.4002615 325 326 327 328 329 330 331 332 333 28.7395388 29.7275902 28.7110374 22.4027096 25.0818488 27.5804786 25.9178923 22.7447274 27.1149544 334 335 336 337 338 339 340 341 342 29.1575606 28.1410077 26.9439456 25.2433572 24.5213197 26.4689209 25.3003602 25.7278824 29.3380699 343 344 345 346 347 348 349 350 351 26.3359140 27.7324865 30.1741134 24.5498212 22.5167156 28.5115270 28.8630453 28.9580502 28.8725457 352 353 354 355 356 357 358 359 360 29.3380699 27.1529564 30.2786188 26.9059436 29.2620660 17.8329723 21.9466860 23.6472743 22.5167156 361 362 363 364 365 366 367 368 369 27.1529564 21.0726405 24.8728380 20.6451183 29.5280798 27.7894895 21.2531499 21.8896830 31.4566800 370 371 372 373 374 375 376 377 378 31.0101568 31.7416948 25.4998705 26.1174026 1.5206248 -1.5195331 21.7851776 12.4746939 14.3747926 379 380 381 382 383 384 385 386 387 12.0471717 13.8617660 18.2034915 14.5268005 12.1326761 11.2206288 5.4538292 5.2828203 7.6864452 388 389 390 391 392 393 394 395 396 4.1617621 5.4633297 14.7453119 18.2984964 16.7309150 10.1565735 20.1415922 19.0205339 18.2889959 397 398 399 400 401 402 403 404 405 16.1513849 15.6288578 5.4918311 6.0808617 9.1210197 15.2488380 15.2583385 15.7713652 8.5414896 406 407 408 409 410 411 412 413 414 12.7217067 12.3796890 23.0297422 9.4725379 15.7618647 24.9488419 14.3937936 1.9006446 15.4768499 415 416 417 418 419 420 421 422 423 -0.5789842 6.9549072 10.0520680 9.2445261 14.9638232 12.9497186 20.2840996 19.6380660 21.1581450 424 425 426 427 428 429 430 431 432 12.4271914 18.2509940 11.3821371 19.6475665 20.7591243 14.1087788 11.6766524 17.7949703 15.8473691 433 434 435 436 437 438 439 440 441 23.1247472 19.1440404 20.1415922 12.4461924 17.4054500 9.4250355 2.2331619 12.8167117 13.5482497 442 443 444 445 446 447 448 449 450 16.0088775 18.7925221 16.6454106 11.9521668 11.7716574 17.6524629 18.9350295 17.3294461 16.2083879 451 452 453 454 455 456 457 458 459 17.9849801 17.7094658 18.1464885 18.6500147 16.7784175 17.3294461 16.4934027 18.4600048 19.1345399 460 461 462 463 464 465 466 467 468 20.5881154 18.9540305 20.6356178 21.2626504 24.7778330 21.9941884 21.1296435 18.2604945 14.2987887 469 470 471 472 473 474 475 476 477 17.3294461 20.5311124 19.0775369 22.3267057 20.9111322 23.4762654 17.3199456 11.6576515 16.8069190 478 479 480 481 482 483 484 485 486 10.8881115 17.4244510 22.0986939 24.3503108 27.2004589 27.8939949 24.6543266 21.8801825 24.5023187 487 488 489 490 491 492 493 494 495 20.3221016 23.6757758 17.3959496 11.7811579 6.3563761 17.3864491 21.8706820 23.1437481 21.6426702 496 497 498 499 500 501 502 503 504 17.8329723 14.4697975 21.1581450 22.2792032 20.2080956 20.9396336 25.3668636 25.9273927 29.1955625 505 506 28.3975211 27.0674520 Warning message: In lstat - c(5, 10, 15) : longer object length is not a multiple of shorter object length > predict(lm.fit,data.frame(lstat-c(5,10,15)), interval = "confidence") fit lwr upr 1 29.8225951 29.0252990 30.6198912 2 25.8703898 25.2652456 26.4755340 3 30.7251420 29.8734766 31.5768074 4 31.7606958 30.8435939 32.6777976 5 29.4900778 28.7120765 30.2680791 6 29.6040837 28.8195155 30.3886520 7 22.7447274 22.2015728 23.2878820 8 16.3603958 15.6261142 17.0946773 9 6.1188637 4.6964329 7.5412945 10 18.3079969 17.6682721 18.9477218 11 15.1253316 14.3211061 15.9295571 12 21.9466860 21.4017704 22.4916015 13 19.6285655 19.0379343 20.2191967 14 26.7064332 26.0688676 27.3439989 15 24.8063345 24.2337154 25.3789536 16 26.5069229 25.8775896 27.1362562 17 28.3025161 27.5895545 29.0154778 18 20.6166169 20.0524476 21.1807861 19 23.4477639 22.8999502 23.9955777 20 23.8372842 23.2844310 24.3901373 21 14.5838035 13.7470635 15.4205434 22 21.4146583 20.8644307 21.9648859 23 16.7689170 16.0562575 17.4815765 24 15.6668597 14.8940813 16.4396381 25 19.0680364 18.4583230 19.6777498 26 18.8685260 18.2513745 19.4856776 27 20.4836100 19.9164496 21.0507703 28 18.1369880 17.4899128 18.7840633 29 22.3932092 21.8502047 22.9362136 30 23.1722496 22.6269495 23.7175497 31 13.0827255 12.1512199 14.0142311 32 22.1651973 21.6215101 22.7088845 33 8.2279733 6.9600530 9.4958936 34 17.1204352 16.4256854 17.8151851 35 15.2298370 14.4317674 16.0279067 36 25.3573631 24.7692166 25.9455097 37 23.7137778 23.1627087 24.2648468 38 26.2219080 25.6038085 26.8400076 39 24.9298409 24.3539993 25.5056826 40 30.4496277 29.6148475 31.2844078 41 32.6727432 31.6958039 33.6496824 42 29.9556020 29.1504707 30.7607333 43 29.0340541 28.2817852 29.7863231 44 27.4854737 26.8130853 28.1578620 45 25.4808696 24.8888470 26.0728921 46 24.8538370 24.2799965 25.4276775 47 21.1106425 20.5559305 21.6653546 48 16.6929130 15.9762946 17.4095315 49 5.2828203 3.7982718 6.7673688 50 19.1630413 18.5567540 19.7693287 51 21.7756771 21.2294109 22.3219432 52 25.5948755 24.9991517 26.1905993 53 29.5375803 28.7568490 30.3183116 54 26.5449248 25.9140464 27.1758033 55 20.4931104 19.9261699 21.0600510 56 29.9841035 29.1772848 30.7909222 57 29.0720561 28.3176769 29.8264354 58 30.8011459 29.9447810 31.6575109 59 28.0365023 27.3371644 28.7358402 60 25.7943858 25.1919064 26.3968653 61 22.0606919 21.5164871 22.6048967 62 20.8351282 20.2754673 21.3947892 63 28.1600087 27.4543928 28.8656246 64 25.5283720 24.9348216 26.1219225 65 26.9059436 26.2598551 27.5520321 66 30.1171104 29.3023789 30.9318418 67 24.8253355 24.2522306 25.3984404 68 26.8584411 26.2144076 27.5024746 69 22.1176948 21.5737883 22.6616014 70 26.2029071 25.5855337 26.8202804 71 28.1695092 27.4634070 28.8756115 72 25.1673533 24.5848920 25.7498145 73 29.3095684 28.5418575 30.0772794 74 27.3904688 26.7225420 28.0583955 75 28.1125063 27.4093148 28.8156977 76 26.0603997 25.4483794 26.6724199 77 23.1817501 22.6363780 23.7271222 78 24.7968340 24.2244565 25.3692115 79 22.8302319 22.2868201 23.3736436 80 25.9083918 25.3018967 26.5148869 81 29.5280798 28.7478952 30.3082644 82 27.6944845 27.0120847 28.3768844 83 28.1695092 27.4634070 28.8756115 84 27.4189702 26.7497109 28.0882295 85 25.4143661 24.8244481 26.0042841 86 28.3500186 27.6345847 29.0654525 87 22.3362062 21.7930886 22.8793238 88 26.5354243 25.9049332 27.1659155 89 29.3285694 28.5597816 30.0973572 90 29.1385596 28.3804719 29.8966472 91 26.1839061 25.5672560 26.8005562 92 26.7634362 26.1234646 27.4034078 93 26.8014382 26.1598495 27.4430268 94 28.6540344 27.9225070 29.3855618 95 24.4928182 23.9276839 25.0579525 96 28.2360127 27.5264925 28.9455329 97 23.7802812 23.2282728 24.3322896 98 30.5541331 29.7129765 31.3952897 99 31.1621647 30.2832444 32.0410850 100 28.6730354 27.9404870 29.4055838 101 25.6043760 25.0083385 26.2004135 102 27.2669623 26.6047498 27.9291749 103 24.4548162 23.8905200 25.0191125 104 21.7851776 21.2389954 22.3313598 105 22.8397323 22.2962867 23.3831780 106 18.9065280 18.2908184 19.5222377 107 16.8259199 16.1162099 17.5356300 108 21.1676455 20.6138531 21.7214379 109 22.8967353 22.3530640 23.4404067 110 19.7805734 19.1946310 20.3665159 111 22.2031993 21.6596684 22.7467302 112 24.9013394 24.3262550 25.4764239 113 19.1535409 18.5469143 19.7601674 114 18.3174974 17.6781748 18.9568201 115 24.6258251 24.0576390 25.1940113 116 19.5810631 18.9889229 20.1732032 117 23.1152467 22.5703565 23.6601369 118 24.7683325 24.1966742 25.3399908 119 19.9515823 19.3706548 20.5325098 120 21.6236692 21.0759176 22.1714208 121 20.9016317 20.3432411 21.4600222 122 20.9966366 20.4399775 21.5532957 123 17.5194560 16.8441884 18.1947235 124 10.4130868 9.3008106 11.5253630 125 17.8519732 17.1921972 18.5117493 126 20.4836100 19.9164496 21.0507703 127 8.6554955 7.4184349 9.8925561 128 18.2224925 17.5791184 18.8658666 129 19.9325813 19.3511104 20.5140522 130 17.1299357 16.4356605 17.8242109 131 22.5832190 22.0403147 23.1261233 132 22.9062358 22.3625231 23.4499485 133 23.9892921 23.4340089 24.5445752 134 20.2745991 19.7023691 20.8468291 135 18.1084866 17.4601662 18.7568069 136 18.4410038 17.8068503 19.0751574 137 18.4980068 17.8662010 19.1298126 138 20.6926208 20.1300773 21.2551643 139 14.2987887 13.4445481 15.1530292 140 17.0159298 16.3159259 17.7159337 141 11.6006485 10.5703489 12.6309481 142 1.8626426 0.1202312 3.6050541 143 9.0735172 7.8664540 10.2805804 144 9.4535370 8.2735814 10.6334925 145 6.7268953 5.3493556 8.1044350 146 8.1424688 6.8683558 9.4165819 147 18.7355191 18.1132302 19.3578080 148 6.4988835 5.1045396 7.8932273 149 7.6484432 6.3384222 8.9584642 150 14.1752822 13.3133807 15.0371838 151 21.1581450 20.6042018 21.7120882 152 21.9371855 21.3922039 22.4821671 153 23.0392427 22.4948400 23.5836454 154 19.5525616 18.9595062 20.1456170 155 20.1890947 19.6146630 20.7635263 156 20.2840996 19.7121097 20.8560895 157 19.2200443 18.6157758 19.8243128 158 30.1931143 29.3738334 31.0123953 159 28.4450235 27.7246102 29.1654369 160 27.5329762 26.8583359 28.2076164 161 29.3285694 28.5597816 30.0973572 162 32.9102555 31.9174432 33.9030678 163 32.7297461 31.7490076 33.7104847 164 31.3996770 30.5057178 32.2936362 165 23.4952664 22.9469306 24.0436022 166 25.2338567 24.6494444 25.8182690 167 31.0386583 30.1674966 31.9098200 168 23.0202417 22.4759503 23.5645332 169 24.0082931 23.4526882 24.5638979 170 23.7992822 23.2469962 24.3515682 171 20.8446287 20.2851522 21.4041052 172 23.1247472 22.5797913 23.6697030 173 20.5976159 20.0330307 21.1622011 174 25.9653947 25.3568503 26.5739391 175 25.3953651 24.8060409 25.9846893 176 29.4900778 28.7120765 30.2680791 177 24.9488419 24.3724910 25.5251928 178 28.5780304 27.8505694 29.3054914 179 27.9794994 27.2830307 28.6759680 180 29.7655921 28.9716338 30.5595504 181 27.3714678 26.7044264 28.0385091 182 25.5758745 24.9807757 26.1709733 183 29.9746030 29.1683471 30.7808589 184 29.1575606 28.3984099 29.9167112 185 21.2721509 20.7199504 21.8243514 186 22.0606919 21.5164871 22.6048967 187 30.3261213 29.4988312 31.1534114 188 28.2075112 27.4994587 28.9155637 189 30.2216158 29.4006237 31.0426079 190 29.4330749 28.6583375 30.2078122 191 29.7085892 28.9179564 30.4992219 192 30.0981094 29.2845122 30.9117066 193 31.8271992 30.9057987 32.7485998 194 29.7750926 28.9805789 30.5696064 195 30.3926247 29.5613078 31.2239417 196 31.7321943 30.8169314 32.6474571 197 30.6776395 29.8289023 31.5263767 198 26.3739159 25.7499038 26.9979281 199 28.2645142 27.5535219 28.9755064 200 30.2216158 29.4006237 31.0426079 201 30.3261213 29.4988312 31.1534114 202 27.4949742 26.8221366 28.1678118 203 31.5991874 30.6924798 32.5058950 204 30.9341528 30.0695221 31.7987836 205 31.8176987 30.8969129 32.7384845 206 24.2268044 23.6672150 24.7863938 207 24.1317995 23.5740064 24.6895925 208 17.3959496 16.7147525 18.0771466 209 20.6261174 20.0621547 21.1900800 210 12.6172013 11.6551493 13.5792533 211 18.1464885 17.4998271 18.7931500 212 11.7716574 10.7529741 12.7903406 213 19.3245497 18.7239101 19.9251894 214 25.6423779 25.0450775 26.2396784 215 6.4798825 5.0841366 7.8756284 216 25.5568735 24.9623965 26.1513505 217 21.7186741 21.1718822 22.2654660 218 25.3478626 24.7600084 25.9357169 219 17.5289565 16.8541412 18.2037718 220 24.5783227 24.0112476 25.1453978 221 25.3288617 24.7415895 25.9161338 222 14.1657817 13.3032890 15.0282745 223 25.1198508 24.5387572 25.7009444 224 27.3334658 26.6681885 27.9987431 225 30.6206366 29.7754040 31.4658691 226 30.1551124 29.3381087 30.9721161 227 31.5801864 30.6746974 32.4856754 228 28.5115270 27.7876008 29.2354531 229 30.8296474 29.9715157 31.6877792 230 30.9816553 30.1140599 31.8492507 231 23.4857659 22.9375366 24.0339952 232 29.5660818 28.7837082 30.3484554 233 32.2072190 31.2610537 33.1533843 234 30.8011459 29.9447810 31.6575109 235 26.9059436 26.2598551 27.5520321 236 24.2173039 23.6578985 24.7767093 237 25.4903700 24.8980436 26.0826965 238 30.0601074 29.2487749 30.8714400 239 28.5115270 27.7876008 29.2354531 240 27.5519771 26.8764322 28.2275221 241 23.7422792 23.1908137 24.2937448 242 22.7732289 22.2299981 23.3164596 243 23.8942871 23.3405528 24.4480214 244 29.6230847 28.8374170 30.4087525 245 22.6782240 22.1352097 23.2212382 246 17.0159298 16.3159259 17.7159337 247 25.8513888 25.2469154 26.4558622 248 24.9108399 24.3355040 25.4861759 249 25.5093710 24.9164342 26.1023078 250 28.3215171 27.6075680 29.0354662 251 28.9485497 28.2010055 29.6960939 252 31.1431637 30.2654398 32.0208876 253 31.2001667 30.3188505 32.0814828 254 31.1906662 30.3099493 32.0713830 255 28.3120166 27.5985615 29.0254718 256 25.7658844 25.1643914 26.3673773 257 31.5991874 30.6924798 32.5058950 258 29.6895882 28.9000612 30.4791152 259 27.1529564 26.4959304 27.8099824 260 27.9985003 27.3010773 28.6959234 261 25.4428676 24.8520526 26.0336825 262 27.6564826 26.9759227 28.3370425 263 28.9390492 28.1920279 29.6860705 264 23.8657856 23.3124965 24.4190748 265 26.8584411 26.2144076 27.5024746 266 24.6258251 24.0576390 25.1940113 267 20.5026109 19.9358892 21.0693327 268 27.4854737 26.8130853 28.1578620 269 31.5516849 30.6480220 32.4553478 270 21.5856672 21.0375026 22.1338318 271 22.2031993 21.6596684 22.7467302 272 28.2930156 27.5805471 29.0054842 273 27.2099594 26.5503508 27.8695680 274 28.3025161 27.5895545 29.0154778 275 31.2001667 30.3188505 32.0814828 276 31.7226938 30.8080435 32.6373441 277 28.8060423 28.0662988 29.5457858 278 30.6016356 29.7575690 31.4457021 279 27.7229860 27.0392006 28.4067715 280 29.9461015 29.1415320 30.7506710 281 30.9816553 30.1140599 31.8492507 282 30.1931143 29.3738334 31.0123953 283 31.6941923 30.7813783 32.6070063 284 31.5516849 30.6480220 32.4553478 285 27.0959535 26.4414884 27.7504186 286 26.7349347 26.0961690 27.3737004 287 22.2697027 21.7264045 22.8130009 288 27.7704885 27.0843830 28.4565940 289 27.3334658 26.6681885 27.9987431 290 25.5188715 24.9256283 26.1121147 291 31.3901765 30.4968218 32.2835312 292 31.1716652 30.2921463 32.0511841 293 30.0886089 29.2755783 30.9016395 294 26.4024174 25.7772765 27.0275583 295 24.6733276 24.1040072 25.2426480 296 28.5970314 27.8685565 29.3255064 297 27.5329762 26.8583359 28.2076164 298 19.5050591 18.9104619 20.0996563 299 29.8320956 29.0342420 30.6299492 300 30.0506069 29.2398397 30.8613742 301 28.7870413 28.0483308 29.5257518 302 25.5283720 24.9348216 26.1219225 303 26.3169130 25.6951394 26.9386865 304 29.9366010 29.1325930 30.7406091 305 27.9699989 27.2740067 28.6659911 306 26.0699002 25.4575282 26.6822721 307 28.4070216 27.6886055 29.1254376 308 27.3999692 26.7315989 28.0683396 309 30.2406168 29.4184824 31.0627513 310 25.0818488 24.5018338 25.6618639 311 22.5452170 22.0023268 23.0881072 312 28.8725457 28.1291735 29.6159179 313 23.4192625 22.8717494 23.9667755 314 27.0484510 26.3961032 27.7007987 315 25.7373829 25.1368693 26.3378965 316 23.6282733 23.0783385 24.1782081 317 17.1394362 16.4456351 17.8332373 318 19.4100542 18.8123127 20.0077956 319 24.7113296 24.1410850 25.2815741 320 22.4597126 21.9167917 23.0026335 321 27.7134855 27.0301625 28.3968086 322 28.0270018 27.3281434 28.7258602 323 27.2384609 26.5775529 27.8993688 324 23.4002615 22.8529438 23.9475792 325 28.7395388 28.0034034 29.4756743 326 29.7275902 28.9358503 30.5193301 327 28.7110374 27.9764418 29.4456329 328 22.4027096 21.8597204 22.9456989 329 25.0818488 24.5018338 25.6618639 330 27.5804786 26.9035726 28.2573847 331 25.9178923 25.3110575 26.5247270 332 22.7447274 22.2015728 23.2878820 333 27.1149544 26.4596381 27.7702708 [ reached getOption("max.print") -- omitted 173 rows ] Warning message: In lstat - c(5, 10, 15) : longer object length is not a multiple of shorter object length > predict(lm.fit,data.frame(lstat-c(5,10,15)), interval = "prediction") fit lwr upr 1 29.8225951 17.5846032 42.06059 2 25.8703898 13.6434129 38.09737 3 30.7251420 18.4834878 42.96680 4 31.7606958 19.5143151 44.00708 5 29.4900778 17.2533279 41.72683 6 29.6040837 17.3669145 41.84125 7 22.7447274 10.5206618 34.96879 8 16.3603958 4.1263477 28.59444 9 6.1188637 -6.1756909 18.41342 10 18.3079969 6.0792598 30.53673 11 15.1253316 2.8868863 27.36378 12 21.9466860 9.7225420 34.17083 13 19.6285655 7.4022984 31.85483 14 26.7064332 14.4778089 38.93506 15 24.8063345 12.5809243 37.03174 16 26.5069229 14.2787250 38.73512 17 28.3025161 16.0697291 40.53530 18 20.6166169 8.3915995 32.84163 19 23.4477639 11.2234904 35.67204 20 23.8372842 11.6127838 36.06178 21 14.5838035 2.3431786 26.82443 22 21.4146583 9.1902764 33.63904 23 16.7689170 4.5361476 29.00169 24 15.6668597 3.4304407 27.90328 25 19.0680364 6.8408326 31.29524 26 18.8685260 6.6409490 31.09610 27 20.4836100 8.2584542 32.70877 28 18.1369880 5.9078642 30.36611 29 22.3932092 10.1691502 34.61727 30 23.1722496 10.9480885 35.39641 31 13.0827255 0.8352577 25.33019 32 22.1651973 9.9411080 34.38929 33 8.2279733 -4.0496643 20.50561 34 17.1204352 4.8886962 29.35217 35 15.2298370 2.9917948 27.46788 36 25.3573631 13.1312158 37.58351 37 23.7137778 11.4893579 35.93820 38 26.2219080 13.9942832 38.44953 39 24.9298409 12.7042793 37.15540 40 30.4496277 18.2091366 42.69012 41 32.6727432 20.4217361 44.92375 42 29.9556020 17.7170972 42.19411 43 29.0340541 16.7989133 41.26919 44 27.4854737 15.2549843 39.71596 45 25.4808696 13.2545351 37.70720 46 24.8538370 12.6283694 37.07930 47 21.1106425 8.8860579 33.33523 48 16.6929130 4.4599124 28.92591 49 5.2828203 -7.0190758 17.58472 50 19.1630413 6.9360079 31.39007 51 21.7756771 9.5514728 33.99988 52 25.5948755 13.3683612 37.82139 53 29.5375803 17.3006565 41.77450 54 26.5449248 14.3166473 38.77320 55 20.4931104 8.2679648 32.71826 56 29.9841035 17.7454876 42.22272 57 29.0720561 16.8367853 41.30733 58 30.8011459 18.5591639 43.04313 59 28.0365023 15.8045018 40.26850 60 25.7943858 13.5675406 38.02123 61 22.0606919 9.8365795 34.28480 62 20.8351282 8.6103180 33.05994 63 28.1600087 15.9276477 40.39237 64 25.5283720 13.3019635 37.75478 65 26.9059436 14.6768719 39.13502 66 30.1171104 17.8779703 42.35625 67 24.8253355 12.5999025 37.05077 68 26.8584411 14.6294778 39.08740 69 22.1176948 9.8935958 34.34179 70 26.2029071 13.9753189 38.43050 71 28.1695092 15.9371201 40.40190 72 25.1673533 12.9414781 37.39323 73 29.3095684 17.0734684 41.54567 74 27.3904688 15.1602239 39.62071 75 28.1125063 15.8802848 40.34473 76 26.0603997 13.8330806 38.28772 77 23.1817501 10.9575858 35.40591 78 24.7968340 12.5714351 37.02223 79 22.8302319 10.6061548 35.05431 80 25.9083918 13.6813480 38.13544 81 29.5280798 17.2911908 41.76497 82 27.6944845 15.4634407 39.92553 83 28.1695092 15.9371201 40.40190 84 27.4189702 15.1886525 39.64929 85 25.4143661 13.1881334 37.64060 86 28.3500186 16.1170873 40.58295 87 22.3362062 10.1121422 34.56027 88 26.5354243 14.3071668 38.76368 89 29.3285694 17.0924018 41.56474 90 29.1385596 16.9030596 41.37406 91 26.1839061 13.9563544 38.41146 92 26.7634362 14.5346862 38.99219 93 26.8014382 14.5726034 39.03027 94 28.6540344 16.4201513 40.88792 95 24.4928182 12.2677563 36.71788 96 28.2360127 16.0034258 40.46860 97 23.7802812 11.5558190 36.00474 98 30.5541331 18.3132055 42.79506 99 31.1621647 18.9185842 43.40575 100 28.6730354 16.4390912 40.90698 101 25.6043760 13.3778464 37.83091 102 27.2669623 15.0370282 39.49690 103 24.4548162 12.2297930 36.67984 104 21.7851776 9.5609770 34.00938 105 22.8397323 10.6156538 35.06381 106 18.9065280 6.6790237 31.13403 107 16.8259199 4.5933220 29.05852 108 21.1676455 8.9431026 33.39219 109 22.8967353 10.6726467 35.12082 110 19.7805734 7.5545319 32.00661 111 22.2031993 9.9791169 34.42728 112 24.9013394 12.6758135 37.12687 113 19.1535409 6.9264905 31.38059 114 18.3174974 6.0887813 30.54621 115 24.6258251 12.4006217 36.85103 116 19.5810631 7.3547229 31.80740 117 23.1152467 10.8911038 35.33939 118 24.7683325 12.5429672 36.99370 119 19.9515823 7.7257801 32.17738 120 21.6236692 9.3993984 33.84794 121 20.9016317 8.6768796 33.12638 122 20.9966366 8.7719635 33.22131 123 17.5194560 5.2888080 29.75010 124 10.4130868 -1.8494546 22.67563 125 17.8519732 5.6221708 30.08178 126 20.4836100 8.2584542 32.70877 127 8.6554955 -3.6189936 20.92998 128 18.2224925 5.9935639 30.45142 129 19.9325813 7.7067533 32.15841 130 17.1299357 4.8982236 29.36165 131 22.5832190 10.3591645 34.80727 132 22.9062358 10.6821454 35.13033 133 23.9892921 11.7646815 36.21390 134 20.2745991 8.0492071 32.49999 135 18.1084866 5.8792968 30.33768 136 18.4410038 6.2125569 30.66945 137 18.4980068 6.2696814 30.72633 138 20.6926208 8.4676783 32.91756 139 14.2987887 2.0569550 26.54062 140 17.0159298 4.7838912 29.24797 141 11.6006485 -0.6547291 23.85603 142 1.8626426 -10.4730277 14.19831 143 9.0735172 -3.1979849 21.34502 144 9.4535370 -2.8153285 21.72240 145 6.7268953 -5.5625465 19.01634 146 8.1424688 -4.1358098 20.42075 147 18.7355191 6.5076818 30.96336 148 6.4988835 -5.7924533 18.79022 149 7.6484432 -4.6336136 19.93050 150 14.1752822 1.9329116 26.41765 151 21.1581450 8.9335952 33.38269 152 21.9371855 9.7130385 34.16133 153 23.0392427 10.8151216 35.26336 154 19.5525616 7.3261771 31.77895 155 20.1890947 7.9635994 32.41459 156 20.2840996 8.0587188 32.50948 157 19.2200443 6.9931108 31.44698 158 30.1931143 17.9536705 42.43256 159 28.4450235 16.2118000 40.67825 160 27.5329762 15.3023628 39.76359 161 29.3285694 17.0924018 41.56474 162 32.9102555 20.6579725 45.16254 163 32.7297461 20.4784355 44.98106 164 31.3996770 19.1550077 43.64435 165 23.4952664 11.2709695 35.71956 166 25.2338567 13.0078884 37.45983 167 31.0386583 18.7956323 43.28168 168 23.0202417 10.7961255 35.24436 169 24.0082931 11.7836679 36.23292 170 23.7992822 11.5748074 36.02376 171 20.8446287 8.6198270 33.06943 172 23.1247472 10.9006014 35.34889 173 20.5976159 8.3725793 32.82265 174 25.9653947 13.7382491 38.19254 175 25.3953651 13.1691610 37.62157 176 29.4900778 17.2533279 41.72683 177 24.9488419 12.7232563 37.17443 178 28.5780304 16.3443898 40.81167 179 27.9794994 15.7476625 40.21134 180 29.7655921 17.5278173 42.00337 181 27.3714678 15.1412712 39.60166 182 25.5758745 13.3493907 37.80236 183 29.9746030 17.7360242 42.21318 184 29.1575606 16.9219946 41.39313 185 21.2721509 9.0476800 33.49662 186 22.0606919 9.8365795 34.28480 187 30.3261213 18.0861387 42.56610 188 28.2075112 15.9750093 40.44001 189 30.2216158 17.9820574 42.46117 190 29.4330749 17.1965320 41.66962 191 29.7085892 17.4710296 41.94615 192 30.0981094 17.8590447 42.33717 193 31.8271992 19.5804959 44.07390 194 29.7750926 17.5372817 42.01290 195 30.3926247 18.1523693 42.63288 196 31.7321943 19.4859512 43.97844 197 30.6776395 18.4361887 42.91909 198 26.3739159 14.1459908 38.60184 199 28.2645142 16.0318418 40.49719 200 30.2216158 17.9820574 42.46117 201 30.3261213 18.0861387 42.56610 202 27.4949742 15.2644601 39.72549 203 31.5991874 19.3535807 43.84479 204 30.9341528 18.6915898 43.17672 205 31.8176987 19.5710417 44.06436 206 24.2268044 12.0019975 36.45161 207 24.1317995 11.9070747 36.35652 208 17.3959496 5.1649728 29.62693 209 20.6261174 8.4011095 32.85113 210 12.6172013 0.3673724 24.86703 211 18.1464885 5.9173866 30.37559 212 11.7716574 -0.4827491 24.02606 213 19.3245497 7.0977950 31.55130 214 25.6423779 13.4157868 37.86897 215 6.4798825 -5.8116134 18.77138 216 25.5568735 13.3304200 37.78333 217 21.7186741 9.4944463 33.94290 218 25.3478626 13.1217293 37.57400 219 17.5289565 5.2983335 29.75958 220 24.5783227 12.3531708 36.80347 221 25.3288617 13.1027563 37.55497 222 14.1657817 1.9233695 26.40819 223 25.1198508 12.8940407 37.34566 224 27.3334658 15.1033653 39.56357 225 30.6206366 18.3794282 42.86184 226 30.1551124 17.9158208 42.39440 227 31.5801864 19.3346699 43.82570 228 28.5115270 16.2780961 40.74496 229 30.8296474 18.5875417 43.07175 230 30.9816553 18.7388825 43.22443 231 23.4857659 11.2614738 35.71006 232 29.5660818 17.3290531 41.80311 233 32.2072190 19.9586275 44.45581 234 30.8011459 18.5591639 43.04313 235 26.9059436 14.6768719 39.13502 236 24.2173039 11.9925054 36.44210 237 25.4903700 13.2640209 37.71672 238 30.0601074 17.8211931 42.29902 239 28.5115270 16.2780961 40.74496 240 27.5519771 15.3213139 39.78264 241 23.7422792 11.5178415 35.96672 242 22.7732289 10.5491599 34.99730 243 23.8942871 11.6697469 36.11883 244 29.6230847 17.3858450 41.86032 245 22.6782240 10.4541646 34.90228 246 17.0159298 4.7838912 29.24797 247 25.8513888 13.6244451 38.07833 248 24.9108399 12.6853021 37.13638 249 25.5093710 13.2829923 37.73575 250 28.3215171 16.0886725 40.55436 251 28.9485497 16.7136984 41.18340 252 31.1431637 18.8996690 43.38666 253 31.2001667 18.9564139 43.44392 254 31.1906662 18.9469565 43.43438 255 28.3120166 16.0792009 40.54483 256 25.7658844 13.5390877 37.99268 257 31.5991874 19.3535807 43.84479 258 29.6895882 17.4521000 41.92708 259 27.1529564 14.9233020 39.38261 260 27.9985003 15.7666091 40.23039 261 25.4428676 13.2165915 37.66914 262 27.6564826 15.4255413 39.88742 263 28.9390492 16.7042299 41.17387 264 23.8657856 11.6412655 36.09031 265 26.8584411 14.6294778 39.08740 266 24.6258251 12.4006217 36.85103 267 20.5026109 8.2774755 32.72775 268 27.4854737 15.2549843 39.71596 269 31.5516849 19.3063033 43.79707 270 21.5856672 9.3613779 33.80996 271 22.2031993 9.9791169 34.42728 272 28.2930156 16.0602574 40.52577 273 27.2099594 14.9801660 39.43975 274 28.3025161 16.0697291 40.53530 275 31.2001667 18.9564139 43.44392 276 31.7226938 19.4764965 43.96889 277 28.8060423 16.5716652 41.04042 278 30.6016356 18.3605077 42.84276 279 27.7229860 15.4918648 39.95411 280 29.9461015 17.7076337 42.18457 281 30.9816553 18.7388825 43.22443 282 30.1931143 17.9536705 42.43256 283 31.6941923 19.4481320 43.94025 284 31.5516849 19.3063033 43.79707 285 27.0959535 14.8664364 39.32547 286 26.7349347 14.5062477 38.96362 287 22.2697027 10.0456307 34.49377 288 27.7704885 15.5392374 40.00174 289 27.3334658 15.1033653 39.56357 290 25.5188715 13.2924779 37.74527 291 31.3901765 19.1455513 43.63480 292 31.1716652 18.9280417 43.41529 293 30.0886089 17.8495819 42.32764 294 26.4024174 14.1744346 38.63040 295 24.6733276 12.4480714 36.89858 296 28.5970314 16.3633305 40.83073 297 27.5329762 15.3023628 39.76359 298 19.5050591 7.2785997 31.73152 299 29.8320956 17.5940674 42.07012 300 30.0506069 17.8117301 42.28948 301 28.7870413 16.5527266 41.02136 302 25.5283720 13.3019635 37.75478 303 26.3169130 14.0891018 38.54472 304 29.9366010 17.6981701 42.17503 305 27.9699989 15.7381892 40.20181 306 26.0699002 13.8425634 38.29724 307 28.4070216 16.1739155 40.64013 308 27.3999692 15.1697001 39.63024 309 30.2406168 18.0009817 42.48025 310 25.0818488 12.8560899 37.30761 311 22.5452170 10.3211632 34.76927 312 28.8725457 16.6379487 41.10714 313 23.4192625 11.1950024 35.64352 314 27.0484510 14.8190470 39.27785 315 25.7373829 13.5106343 37.96413 316 23.6282733 11.4039045 35.85264 317 17.1394362 4.9077510 29.37112 318 19.4100542 7.1834415 31.63667 319 24.7113296 12.4860303 36.93663 320 22.4597126 10.2356574 34.68377 321 27.7134855 15.4823902 39.94458 322 28.0270018 15.7950287 40.25897 323 27.2384609 15.0085973 39.46832 324 23.4002615 11.1760102 35.62451 325 28.7395388 16.5053793 40.97370 326 29.7275902 17.4899590 41.96522 327 28.7110374 16.4769704 40.94510 328 22.4027096 10.1786514 34.62677 329 25.0818488 12.8560899 37.30761 330 27.5804786 15.3497401 39.81122 331 25.9178923 13.6908316 38.14495 332 22.7447274 10.5206618 34.96879 333 27.1149544 14.8853918 39.34452 [ reached getOption("max.print") -- omitted 173 rows ] Warning message: In lstat - c(5, 10, 15) : longer object length is not a multiple of shorter object length > plot(lstat,medv) > abline(lm.fit) > abline(lm.fit, lwd=5) > abline(lm.fit, lwd=5, col="blue") > plot(lstat, medv, pch="+") > plot(lstat, medv, pch=20) > dev.off() null device 1 > par(mfrow=c(2,2)) > plot(lm.fit) > dev.off() null device 1 > plot(predict(lm.fit),residuals(lm.fit)) > plot(predict(lm.fit),rstudent(lm.fit)) > plot(hatvalues(lm.fit)) > mlm.fit = lm(medv~lstat+age, data=Boston) > summary(mlm.fit) Call: lm(formula = medv ~ lstat + age, data = Boston) Residuals: Min 1Q Median 3Q Max -15.981 -3.978 -1.283 1.968 23.158 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 33.22276 0.73085 45.458 < 2e-16 *** lstat -1.03207 0.04819 -21.416 < 2e-16 *** age 0.03454 0.01223 2.826 0.00491 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 6.173 on 503 degrees of freedom Multiple R-squared: 0.5513, Adjusted R-squared: 0.5495 F-statistic: 309 on 2 and 503 DF, p-value: < 2.2e-16 > summary(mlm.fit) Call: lm(formula = medv ~ lstat + age, data = Boston) Residuals: Min 1Q Median 3Q Max -15.981 -3.978 -1.283 1.968 23.158 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 33.22276 0.73085 45.458 < 2e-16 *** lstat -1.03207 0.04819 -21.416 < 2e-16 *** age 0.03454 0.01223 2.826 0.00491 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 6.173 on 503 degrees of freedom Multiple R-squared: 0.5513, Adjusted R-squared: 0.5495 F-statistic: 309 on 2 and 503 DF, p-value: < 2.2e-16 > summary(mlm.fit)$r.sq [1] 0.5512689 > mlm.fit.inter=lm(medv~lstat*age,data=Boston) > summary(mlm.fit.inter) Call: lm(formula = medv ~ lstat * age, data = Boston) Residuals: Min 1Q Median 3Q Max -15.806 -4.045 -1.333 2.085 27.552 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 36.0885359 1.4698355 24.553 < 2e-16 *** lstat -1.3921168 0.1674555 -8.313 8.78e-16 *** age -0.0007209 0.0198792 -0.036 0.9711 lstat:age 0.0041560 0.0018518 2.244 0.0252 * --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 6.149 on 502 degrees of freedom Multiple R-squared: 0.5557, Adjusted R-squared: 0.5531 F-statistic: 209.3 on 3 and 502 DF, p-value: < 2.2e-16 > summary(mlm.fit.inter)$r.sq [1] 0.5557265 > lm.fit.poly=lm(medv~lstat+I(lstat^2)) > summary(lm.fit.poly) Call: lm(formula = medv ~ lstat + I(lstat^2)) Residuals: Min 1Q Median 3Q Max -15.2834 -3.8313 -0.5295 2.3095 25.4148 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 42.862007 0.872084 49.15 <2e-16 *** lstat -2.332821 0.123803 -18.84 <2e-16 *** I(lstat^2) 0.043547 0.003745 11.63 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 5.524 on 503 degrees of freedom Multiple R-squared: 0.6407, Adjusted R-squared: 0.6393 F-statistic: 448.5 on 2 and 503 DF, p-value: < 2.2e-16 > plot(lstat,medv) > lm.fit = lm(medv~lstat) > anova(lm.fit,lm.fit.poly) Analysis of Variance Table Model 1: medv ~ lstat Model 2: medv ~ lstat + I(lstat^2) Res.Df RSS Df Sum of Sq F Pr(>F) 1 504 19472 2 503 15347 1 4125.1 135.2 < 2.2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1