12 Confidence and Prediction Intervals
12.1 What are they?
Essentially these are two different things:
Confidence interval: “Error bar” on the location of the regression line at any specific value of x, or “what is the uncertainty on the population mean of \(y\), for a specific value of \(x\)?”
Prediction interval: Predicting the range of likely y-values for a new data-point, or “if you had to put a new dot on the scatterplot, what range of y values would it likely be found in
These are described in detail here:
- https://online.stat.psu.edu/stat462/node/125/
- https://online.stat.psu.edu/stat462/node/126/
- https://online.stat.psu.edu/stat462/node/127/
- https://online.stat.psu.edu/stat462/node/128/
In R, lets look at the pirate dataset
#For our pirate weight/height dataset
lm.pirate <- lm(weight~height,data=pirates)
summary(lm.pirate)##
## Call:
## lm(formula = weight ~ height, data = pirates)
##
## Residuals:
## Min 1Q Median 3Q Max
## -12.3592 -2.6564 -0.0708 2.7275 11.1451
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -68.87722 1.71250 -40.22 <2e-16 ***
## height 0.81434 0.01003 81.16 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.928 on 998 degrees of freedom
## Multiple R-squared: 0.8684, Adjusted R-squared: 0.8683
## F-statistic: 6587 on 1 and 998 DF, p-value: < 2.2e-1612.2 Calculating a confidence interval
What are the AVERAGE WEIGHTS (and uncertainty on our estimate) of pirates whose heights are 150cm and 170cm?
# So we have the MODEL
lm.pirate <- lm(weight~height,data=pirates)
# a mini table of our new X-Values - you need the column name to be identical to your predictor
new.pirates <- data.frame(height=c(150,170))
# and the command
# predict (MODEL_NAME , NAME_OF_NEW_DATA, TYPE_OF_INTERVAL, SIG LEVEL)
predict(lm.pirate,newdata=new.pirates,interval="confidence",level=0.95)## fit lwr upr
## 1 53.27345 52.80652 53.74037
## 2 69.56020 69.31640 69.80400We are 95% certain that on average, the AVERAGE weight of pirates who are 150cm tall falls between 52.8 Kg and 53.74 Kg.
We are 95% certain that on average, the AVERAGE weight of pirates who are 170cm tall falls between 69.32 Kg and 69.80 Kg.
12.3 Calculating a prediction interval
A new pirate joins and her height is 160cm. What range of weights is she likely to have? with a significance level of 99%
# So we have the model
# MAKE SURE THAT YOU ONLY MENTION THE TABLE-NAME AT THE END
lm.pirate <- lm(weight~height,data=pirates)
# a mini table of our new X-Values - you need the column name to be identical to your predictor
new.pirates <- data.frame(height=c(160))
# and the command
# predict (MODEL_NAME , NAME_OF_NEW_DATA, TYPE_OF_INTERVAL, SIG LEVEL)
predict(lm.pirate,newdata=new.pirates,interval="predict",level=0.99)## fit lwr upr
## 1 61.41682 51.27067 71.56298Given our model, her weight is likely to be somewhere between 51.27 Kg and 71.56 Kg with 99% certainty.
12.4 HELP! MY CODE DOESN’T WORK
12.4.1 Predictor/model name spelt wrong
If you don’t spell your column name EXACTLY (case sensitive) the same as the predictor in your model, the code will break. Here I put HEIGHT not height for my new predictor:
# THIS WILL LOOK FINE BUT BREAK PREDICT.
# ONLY MENTION THE WORD PIRATES AFTER THE DATA BIT
lm.pirate <- lm(weight~height,data=pirates)
new.pirates <- data.frame(HEIGHT=c(160))
predict(lm.pirate,newdata=new.pirates,interval="predict",level=0.99)## Error in eval(predvars, data, env): object 'height' not found12.4.2 Your original lm command
The predict command is fussy and depends on your original lm command being exactly correct, where you only type the name of the data.table once at the end. If you type it at each stage, then the model doesn’t understand it all comes from one table, which means that predict will run, but give the wrong answer.
For example
# this is correct
# lm_correct <- lm(weight~height,data=pirates)
# If you add the data name at each step, the predict behaves wrongly
# THIS WILL WORK FINE INITIALY BUT CAUSE ISSUES FOR PREDICT
lm_WRONG <- lm(pirates$weight~pirates$height,data=pirates)
new.pirates <- data.frame(height=c(160))
predict(lm_WRONG,newdata=new.pirates,interval="predict",level=0.99)## fit lwr upr
## 1 72.09279 61.94981 82.23577
## 2 101.52296 91.33004 111.71587
## 3 69.51948 59.37678 79.66219
## 4 48.62358 38.45866 58.78849
## 5 59.66600 49.51823 69.81376
## 6 86.00982 75.85394 96.16570
## 7 59.82887 49.68126 69.97647
## 8 71.61233 61.46946 81.75521
## 9 65.13835 54.99459 75.28211
## 10 80.57005 70.42151 90.71859
## 11 63.44452 53.29984 73.58921
## 12 64.38916 54.24503 74.53329
## 13 70.40711 60.26439 80.54984
## 14 76.06676 65.92207 86.21145
## 15 67.64651 57.50359 77.78943
## 16 65.56995 55.42638 75.71352
## 17 75.68402 65.53956 85.82848
## 18 60.03245 49.88505 70.17985
## 19 72.06022 61.91725 82.20319
## 20 73.99834 63.85474 84.14195
## 21 84.07984 73.92690 94.23278
## 22 76.93810 66.79282 87.08338
## 23 60.32561 50.17849 70.47273
## 24 73.86805 63.72450 84.01160
## 25 83.79483 73.64229 93.94736
## 26 56.15620 46.00430 66.30810
## 27 70.76542 60.62267 80.90818
## 28 75.29314 65.14890 85.43738
## 29 49.99981 39.83769 60.16192
## 30 40.95252 30.76858 51.13645
## 31 91.29488 81.12905 101.46070
## 32 79.29154 69.14429 89.43878
## 33 85.86324 75.70760 96.01889
## 34 63.53410 53.38948 73.67873
## 35 76.27034 66.12552 86.41517
## 36 63.93313 53.78874 74.07751
## 37 62.39403 52.24863 72.53942
## 38 59.81258 49.66496 69.96020
## 39 36.84825 26.65175 47.04475
## 40 56.27021 46.11846 66.42196
## 41 60.50477 50.35781 70.65172
## 42 74.92669 64.78265 85.07073
## 43 60.27675 50.12958 70.42392
## 44 76.61237 66.46731 86.75742
## 45 67.63022 57.48730 77.77315
## 46 74.63353 64.48963 84.77742
## 47 59.54385 49.39596 69.69173
## 48 66.80774 56.66461 76.95087
## 49 61.93800 51.79226 72.08374
## 50 54.58453 44.43038 64.73868
## 51 69.76379 59.62108 79.90649
## 52 76.93810 66.79282 87.08338
## 53 86.66944 76.51247 96.82641
## 54 62.15787 52.01230 72.30344
## 55 62.56504 52.41977 72.71031
## 56 83.81926 73.66668 93.97183
## 57 79.75571 69.60801 89.90341
## 58 58.46892 48.31988 68.61796
## 59 69.97551 59.83281 80.11822
## 60 54.76368 44.60980 64.91757
## 61 51.49819 41.33891 61.65747
## 62 73.58303 63.43959 83.72647
## 63 76.19705 66.05228 86.34183
## 64 59.71486 49.56714 69.86257
## 65 68.14325 58.00042 78.28608
## 66 82.82576 72.67454 92.97699
## 67 73.33059 63.18724 83.47393
## 68 66.02598 55.88258 76.16937
## 69 77.40227 67.25665 87.54790
## 70 77.54071 67.39498 87.68644
## 71 65.00805 54.86423 75.15187
## 72 55.07313 44.91971 65.22656
## 73 82.75247 72.60134 92.90361
## 74 62.65462 52.50941 72.79982
## 75 82.08472 71.93443 92.23501
## 76 57.78488 47.63505 67.93471
## 77 64.82890 54.68499 74.97281
## 78 68.77029 58.62754 78.91304
## 79 70.63513 60.49239 80.77787
## 80 61.88100 51.73521 72.02678
## 81 55.52916 45.37639 65.68193
## 82 83.47723 73.32514 93.62933
## 83 61.19695 51.05061 71.34330
## 84 77.63029 67.48449 87.77609
## 85 52.33696 42.17916 62.49475
## 86 61.11552 50.96911 71.26193
## 87 65.93640 55.79297 76.07982
## 88 68.79472 58.65198 78.93747
## 89 72.02764 61.88468 82.17061
## 90 69.21004 59.06732 79.35275
## 91 77.51628 67.37057 87.66199
## 92 78.25733 68.11102 88.40364
## 93 69.47062 59.32792 79.61333
## 94 76.20520 66.06042 86.34998
## 95 51.80764 41.64891 61.96636
## 96 57.90703 47.75734 68.05671
## 97 83.53424 73.38206 93.68641
## 98 67.02761 56.88454 77.17068
## 99 70.24425 60.10153 80.38696
## 100 62.19044 52.04490 72.33599
## 101 68.59928 58.45652 78.74205
## 102 63.24908 53.10428 73.39389
## 103 65.26864 55.12494 75.41234
## 104 62.95592 52.81092 73.10092
## 105 62.91521 52.77018 73.06023
## 106 84.43001 74.27656 94.58345
## 107 71.17259 61.02979 81.31539
## 108 63.67254 53.52800 73.81708
## 109 83.91698 73.76427 94.06968
## 110 83.34694 73.19502 93.49886
## 111 84.37301 74.21964 94.52637
## 112 72.75241 62.60925 82.89556
## 113 73.26544 63.12212 83.40876
## 114 58.68879 48.54000 68.83759
## 115 84.34858 74.19525 94.50190
## 116 65.63509 55.49155 75.77864
## 117 81.09122 70.94211 91.24034
## 118 74.78825 64.64428 84.93222
## 119 64.03899 53.89467 74.18332
## 120 62.34517 52.19974 72.49060
## 121 66.30285 56.15956 76.44614
## 122 53.97378 43.81868 64.12887
## 123 75.11398 64.96985 85.25812
## 124 74.11235 63.96870 84.25600
## 125 76.84038 66.69517 86.98559
## 126 64.50316 54.35909 74.64724
## 127 62.36146 52.21604 72.50687
## 128 67.21491 57.07189 77.35793
## 129 67.45107 57.30810 77.59403
## 130 77.91531 67.76928 88.06134
## 131 52.21481 42.05680 62.37281
## 132 66.76702 56.62388 76.91017
## 133 70.61884 60.47610 80.76158
## 134 86.79973 76.64254 96.95692
## 135 67.81752 57.67463 77.96041
## 136 82.36159 72.21096 92.51222
## 137 59.89401 49.74647 70.04155
## 138 69.49505 59.35235 79.63776
## 139 71.52276 61.37990 81.66562
## 140 69.63349 59.49079 79.77619
## 141 67.32892 57.18592 77.47191
## 142 59.05524 48.90685 69.20364
## 143 61.53897 51.39292 71.68503
## 144 69.78822 59.64551 79.93092
## 145 76.62865 66.48359 86.77372
## 146 62.64647 52.50126 72.79169
## 147 57.21484 47.06432 67.36537
## 148 86.51471 76.35800 96.67142
## 149 83.28179 73.12996 93.43362
## 150 74.66610 64.52219 84.81001
## 151 71.33546 61.19263 81.47829
## 152 70.76542 60.62267 80.90818
## 153 54.92655 44.77291 65.08019
## 154 60.73278 50.58603 70.87953
## 155 77.63029 67.48449 87.77609
## 156 59.93473 49.78723 70.08223
## 157 59.64971 49.50193 69.79749
## 158 80.94464 70.79569 91.09360
## 159 62.14973 52.00415 72.29530
## 160 67.14976 57.00672 77.29280
## 161 83.59124 73.43899 93.74350
## 162 83.39580 73.24381 93.54779
## 163 68.90059 58.75785 79.04333
## 164 87.19061 77.03275 97.34848
## 165 58.30605 48.15683 68.45528
## 166 78.84365 68.69682 88.99048
## 167 80.07330 69.92528 90.22132
## 168 59.28326 49.13510 69.43141
## 169 92.94798 82.77848 103.11748
## 170 56.44936 46.29785 66.60087
## 171 68.65629 58.51353 78.79905
## 172 64.00642 53.86207 74.15076
## 173 57.68716 47.53721 67.83710
## 174 51.09916 40.93915 61.25918
## 175 70.21981 60.07710 80.36253
## 176 47.66266 37.49568 57.82964
## 177 85.98539 75.82955 96.14123
## 178 51.94608 41.78760 62.10456
## 179 66.44129 56.29804 76.58453
## 180 85.17920 75.02463 95.33377
## 181 77.25569 67.11018 87.40121
## 182 63.03736 52.89241 73.18230
## 183 75.05698 64.91287 85.20109
## 184 76.65308 66.50800 86.79816
## 185 70.97715 60.83437 81.11993
## 186 50.61056 40.44963 60.77150
## 187 69.67421 59.53151 79.81691
## 188 78.38762 68.24120 88.53405
## 189 69.07160 58.92887 79.21432
## 190 53.35488 43.19880 63.51096
## 191 64.98362 54.83979 75.12746
## 192 67.14162 56.99858 77.28466
## 193 72.08465 61.94167 82.22762
## 194 83.49352 73.34140 93.64564
## 195 91.27045 81.10467 101.43622
## 196 63.78655 53.64208 73.93102
## 197 62.82563 52.68054 72.97072
## 198 67.30449 57.16149 77.44748
## 199 90.55383 80.38957 100.71809
## 200 74.85340 64.70939 84.99740
## 201 76.11562 65.97090 86.26035
## 202 58.00475 47.85518 68.15432
## 203 82.54889 72.39802 92.69976
## 204 76.48207 66.33711 86.62704
## 205 53.03729 42.88068 63.19389
## 206 62.50804 52.36272 72.65335
## 207 67.31263 57.16963 77.45563
## 208 70.58627 60.44353 80.72900
## 209 74.69053 64.54661 84.83445
## 210 57.41843 47.26815 67.56870
## 211 59.13668 48.98837 69.28499
## 212 58.43635 48.28727 68.58542
## 213 54.03078 43.87578 64.18578
## 214 80.31760 70.16933 90.46588
## 215 83.64824 73.49591 93.80058
## 216 66.64487 56.50169 76.78805
## 217 87.08475 76.92707 97.24243
## 218 76.88924 66.74400 87.03449
## 219 89.70692 79.54437 99.86946
## 220 79.18567 69.03853 89.33282
## 221 58.08618 47.93671 68.23566
## 222 52.23924 42.08127 62.39720
## 223 41.18053 30.99724 51.36382
## 224 66.80774 56.66461 76.95087
## 225 68.99831 58.85558 79.14104
## 226 75.30128 65.15704 85.44552
## 227 76.49836 66.35338 86.64333
## 228 67.48364 57.34068 77.62660
## 229 64.05528 53.91096 74.19959
## 230 72.81755 62.67438 82.96073
## 231 73.68889 63.54542 83.83237
## 232 62.53247 52.38717 72.67776
## 233 72.94785 62.80463 83.09106
## 234 69.64978 59.50708 79.79248
## 235 72.67097 62.52784 82.81410
## 236 74.81268 64.66870 84.95666
## 237 73.43645 63.29307 83.57983
## 238 59.95916 49.81169 70.10663
## 239 72.27195 62.12893 82.41497
## 240 68.56671 58.42394 78.70948
## 241 72.42667 62.28361 82.56973
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## 243 84.57659 74.42293 94.73025
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## 245 71.87292 61.72999 82.01585
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## 249 79.44626 69.29887 89.59365
## 250 69.17746 59.03474 79.32018
## 251 75.92832 65.78371 86.07293
## 252 75.99347 65.84882 86.13812
## 253 68.04553 57.90269 78.18838
## 254 77.92345 67.77741 88.06949
## 255 64.64974 54.50575 74.79374
## 256 90.58640 80.42207 100.75073
## 257 56.61223 46.46094 66.76353
## 258 70.78985 60.64709 80.93261
## 259 65.71653 55.57302 75.86004
## 260 79.69056 69.54293 89.83819
## 261 77.64658 67.50076 87.79239
## 262 76.81595 66.67076 86.96115
## 263 66.63673 56.49355 76.77991
## 264 73.66446 63.52100 83.80793
## 265 80.62705 70.47845 90.77565
## 266 68.76215 58.61940 78.90490
## 267 67.10090 56.95785 77.24395
## 268 55.94447 45.79228 66.09666
## 269 57.85817 47.70843 68.00791
## 270 79.91858 69.77072 90.06644
## 271 61.30282 51.15656 71.44907
## 272 91.34374 81.17781 101.50967
## 273 60.20346 50.05622 70.35070
## 274 62.48361 52.33828 72.62894
## 275 75.03255 64.88846 85.17665
## 276 67.54879 57.40584 77.69173
## 277 89.53591 79.37370 99.69811
## 278 58.28977 48.14053 68.43901
## 279 64.65789 54.51390 74.80188
## 280 91.89749 81.73035 102.06462
## 281 59.34841 49.20032 69.49649
## 282 65.26050 55.11679 75.40420
## 283 79.96744 69.81953 90.11535
## 284 84.43815 74.28469 94.59161
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## 286 62.80934 52.66424 72.95444
## 287 85.43979 75.28481 95.59476
## 288 72.50810 62.36502 82.65119
## 289 61.78328 51.63741 71.92914
## 290 80.05702 69.90901 90.20502
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## 292 94.49522 84.32203 104.66841
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## 294 63.42824 53.28355 73.57293
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## 299 80.28503 70.13679 90.43327
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## 301 65.93640 55.79297 76.07982
## 302 82.70361 72.55254 92.85468
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## 304 81.46582 71.31627 91.61537
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## 313 48.66429 38.49947 58.82912
## 314 73.34687 63.20352 83.49022
## 315 61.05852 50.91205 71.20498
## 316 76.42507 66.28014 86.57000
## 317 59.42984 49.28184 69.57784
## 318 58.63993 48.49109 68.78878
## 319 69.03903 58.89630 79.18175
## 320 62.69533 52.55016 72.84051
## 321 78.91694 68.77005 89.06384
## 322 60.07317 49.92580 70.22053
## 323 88.70528 78.54468 98.86589
## 324 51.27018 41.11048 61.42988
## 325 85.04891 74.89454 95.20328
## 326 92.34537 82.17724 102.51350
## 327 67.53250 57.38955 77.67545
## 328 74.14492 64.00126 84.28859
## 329 57.71973 47.56982 67.86964
## 330 58.59922 48.45032 68.74811
## 331 79.20196 69.05480 89.34912
## 332 68.43642 58.29363 78.57920
## 333 60.00802 49.86059 70.15545
## 334 77.07654 66.93116 87.22192
## 335 61.25396 51.10766 71.40025
## 336 60.50477 50.35781 70.65172
## 337 59.49499 49.34705 69.64292
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## 342 58.23276 48.08346 68.38207
## 343 75.96090 65.81627 86.10552
## 344 80.31760 70.16933 90.46588
## 345 72.06022 61.91725 82.20319
## 346 61.75885 51.61297 71.90473
## 347 69.83708 59.69437 79.97978
## 348 74.01463 63.87102 84.15824
## 349 91.00171 80.83651 101.16691
## 350 68.92502 58.78228 79.06775
## 351 43.03722 32.85903 53.21541
## 352 82.82576 72.67454 92.97699
## 353 73.45274 63.30935 83.59612
## 354 81.27852 71.12919 91.42785
## 355 71.18073 61.03793 81.32354
## 356 65.83053 55.68707 75.97400
## 357 90.03265 79.86945 100.19585
## 358 61.70999 51.56407 71.85590
## 359 67.24748 57.10447 77.39050
## 360 66.97875 56.83567 77.12183
## 361 75.47229 65.32796 85.61663
## 362 77.79316 67.64723 87.93909
## 363 77.71987 67.57399 87.86574
## 364 63.46895 53.32429 73.61362
## 365 75.52115 65.37679 85.66552
## 366 60.11388 49.96656 70.26121
## 367 63.33866 53.19391 73.48341
## 368 63.50153 53.35688 73.64617
## 369 65.06506 54.92126 75.20885
## 370 74.96740 64.82334 85.11147
## 371 76.71009 66.56497 86.85521
## 372 76.49022 66.34525 86.63519
## 373 75.52115 65.37679 85.66552
## 374 78.27362 68.12729 88.41994
## 375 76.35178 66.20690 86.49666
## 376 72.23123 62.08822 82.37424
## 377 77.02768 66.88233 87.17302
## 378 60.60249 50.45562 70.74935
## 379 76.75895 66.61379 86.90410
## 380 75.03255 64.88846 85.17665
## 381 88.14339 77.98383 98.30295
## 382 67.95596 57.81309 78.09882
## 383 61.31910 51.17286 71.46534
## 384 62.80934 52.66424 72.95444
## 385 72.01950 61.87654 82.16246
## 386 71.11559 60.97279 81.25838
## 387 75.18728 65.04310 85.33145
## 388 67.75237 57.60947 77.89527
## 389 75.52930 65.38493 85.67367
## 390 57.84188 47.69212 67.99164
## 391 55.87118 45.71889 66.02347
## 392 63.65625 53.51170 73.80080
## 393 71.08301 60.94022 81.22580
## 394 83.90883 73.75614 94.06153
## 395 79.14496 68.99785 89.29206
## 396 70.45597 60.31325 80.59870
## 397 62.01943 51.87376 72.16511
## 398 61.98686 51.84116 72.13256
## 399 71.58790 61.44503 81.73077
## 400 58.21648 48.06715 68.36580
## 401 64.00642 53.86207 74.15076
## 402 65.88754 55.74409 76.03098
## 403 71.06673 60.92394 81.20952
## 404 61.58783 51.44182 71.73385
## 405 67.01132 56.86825 77.15440
## 406 54.46238 44.30804 64.61672
## 407 62.91521 52.77018 73.06023
## 408 61.62855 51.48257 71.77454
## 409 94.45451 84.28142 104.62760
## 410 65.52923 55.38564 75.67282
## 411 69.29147 59.14876 79.43418
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## 900 67.85009 57.70721 77.99297
## 901 80.52119 70.37270 90.66968
## 902 53.77019 43.61478 63.92561
## 903 70.92829 60.78552 81.07106
## 904 49.59264 39.42972 59.75556
## 905 76.36807 66.22318 86.51295
## 906 68.11882 57.97599 78.26166
## 907 82.01143 71.86123 92.16163
## 908 71.75077 61.60787 81.89367
## 909 71.05044 60.90765 81.19323
## 910 83.91698 73.76427 94.06968
## 911 70.01623 59.87352 80.15894
## 912 51.66920 41.51023 61.82818
## 913 81.13194 70.98278 91.28110
## 914 71.63676 61.49388 81.77964
## 915 63.44452 53.29984 73.58921
## 916 66.91360 56.77050 77.05671
## 917 57.96403 47.81441 68.11365
## 918 81.42510 71.27560 91.57460
## 919 75.35014 65.20587 85.49441
## 920 72.15794 62.01495 82.30093
## 921 71.55533 61.41246 81.69819
## 922 79.19382 69.04667 89.34097
## 923 80.63519 70.48658 90.78381
## 924 69.59277 59.45007 79.73548
## 925 82.58961 72.43868 92.74053
## 926 73.58303 63.43959 83.72647
## 927 78.71336 68.56665 88.86007
## 928 52.12523 41.96707 62.28339
## 929 79.53584 69.38836 89.68332
## 930 73.05371 62.91046 83.19696
## 931 51.92979 41.77128 62.08830
## 932 70.86314 60.72038 81.00591
## 933 75.31757 65.17332 85.46182
## 934 73.99834 63.85474 84.14195
## 935 68.48528 58.34249 78.62806
## 936 85.38278 75.22790 95.53767
## 937 83.34694 73.19502 93.49886
## 938 74.62538 64.48149 84.76927
## 939 62.58133 52.43607 72.72659
## 940 81.88927 71.73922 92.03933
## 941 72.62211 62.47900 82.76523
## 942 62.13344 51.98785 72.27903
## 943 68.69700 58.55425 78.83976
## 944 72.78498 62.64182 82.92814
## 945 82.58146 72.43055 92.73237
## 946 58.81094 48.66228 68.95960
## 947 65.11392 54.97015 75.25769
## 948 60.92008 50.77349 71.06666
## 949 60.08945 49.94211 70.23680
## 950 56.34350 46.19185 66.49515
## 951 73.07000 62.92674 83.21325
## 952 80.61076 70.46218 90.75935
## 953 62.53247 52.38717 72.67776
## 954 86.58800 76.43117 96.74484
## 955 86.21341 76.05720 96.36962
## 956 60.80607 50.65939 70.95275
## 957 81.02608 70.87703 91.17512
## 958 75.73288 65.58839 85.87737
## 959 57.45914 47.30892 67.60936
## 960 70.10581 59.96310 80.24852
## 961 78.78665 68.63987 88.93342
## 962 54.52753 44.37329 64.68176
## 963 51.43304 41.27364 61.59244
## 964 71.67748 61.53459 81.82037
## 965 77.85016 67.70418 87.99614
## 966 75.92018 65.77558 86.06478
## 967 63.09436 52.94945 73.23927
## 968 74.67424 64.53033 84.81816
## 969 69.30776 59.16505 79.45047
## 970 82.75247 72.60134 92.90361
## 971 76.88924 66.74400 87.03449
## 972 75.64330 65.49887 85.78774
## 973 80.60262 70.45404 90.75120
## 974 84.52773 74.37414 94.68132
## 975 55.03242 44.87893 65.18590
## 976 75.21985 65.07565 85.36405
## 977 62.12530 51.97970 72.27089
## 978 74.71496 64.57103 84.85889
## 979 55.26043 45.10728 65.41358
## 980 62.72791 52.58275 72.87306
## 981 62.31260 52.16714 72.45805
## 982 70.14652 60.00381 80.28923
## 983 65.83053 55.68707 75.97400
## 984 82.83391 72.68267 92.98515
## 985 52.97214 42.81543 63.12886
## 986 77.12540 66.97998 87.27082
## 987 62.76048 52.61535 72.90561
## 988 61.86471 51.71891 72.01051
## 989 74.40551 64.26173 84.54930
## 990 65.31750 55.17382 75.46118
## 991 82.80948 72.65827 92.96068
## 992 72.08465 61.94167 82.22762
## 993 62.36960 52.22419 72.51501
## 994 56.49008 46.33863 66.64154
## 995 71.55533 61.41246 81.69819
## 996 76.75080 66.60566 86.89595
## 997 59.13668 48.98837 69.28499
## 998 74.23450 64.09079 84.37821
## 999 73.82733 63.68380 83.97086
## 1000 85.39907 75.24416 95.55398