diff --git a/Lecture 13/P-value optimization.ipynb b/Lecture 13/P-value optimization.ipynb new file mode 100644 index 0000000..10b98b6 --- /dev/null +++ b/Lecture 13/P-value optimization.ipynb @@ -0,0 +1,468 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here we will trick a statistical analysis with what is known as a p-value optimization.\n", + "\n", + "We will test if clapping your boots three time before leave your house in the morning will increase your average salary. Try it, it doesnt." + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": {}, + "outputs": [], + "source": [ + "# set up some random data.\n", + "\n", + "\n", + "\n", + "income = rnorm(1000,mean=5000,sd=200) # your sample. Everybody must clap boots every day for 3 years, then you record salary. You know that average salary of this population is 5000.\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "31" + ], + "text/latex": [ + "31" + ], + "text/markdown": [ + "31" + ], + "text/plain": [ + "[1] 31" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "2.86334177746778" + ], + "text/latex": [ + "2.86334177746778" + ], + "text/markdown": [ + "2.86334177746778" + ], + "text/plain": [ + "[1] 2.863342" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "#now lets do a T test. Your hypothesis is that the average salary is larger than the known average 5000.\n", + "\n", + "for (i in seq(1,989,by=1))\n", + " \n", + " {\n", + " partialdata=income[i:(i+9)]\n", + " T=(mean(partialdata)-5000)/sd(partialdata)*sqrt(10)\n", + " if (T>qt(p=0.95,df=9))\n", + " {break} \n", + " }\n", + "\n", + "i\n", + "T" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "There you go, clap your boots to become rich! Where do we cheat? Incomplete data! We have performed i+10 experiments, but only analyze 10 datapoints, dropping those before i. Note that we did not have to \"pick\" specific values, like low incomes, that did not suit us. We did blindly, without bias, take 10 random datapoints. We let statistical fluctuations do the dirty work.\n", + "\n", + "Typical setting to look out for: \"Student A does not show effect\" \"Probably student A was not skilled\" \"Let student B have a go\". \"Student B is also not skilled\" \"Lets see if student C can do it\"..." + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "989" + ], + "text/latex": [ + "989" + ], + "text/markdown": [ + "989" + ], + "text/plain": [ + "[1] 989" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "0.00240801927382591" + ], + "text/latex": [ + "0.00240801927382591" + ], + "text/markdown": [ + "0.00240801927382591" + ], + "text/plain": [ + "[1] 0.002408019" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "#now lets do a proper T test. Your hypothesis is that the average salary is larger than the known average 5000.\n", + "\n", + "for (i in seq(1,989,by=1))\n", + " \n", + " {\n", + " partialdata=income[1:(i+9)]\n", + " T=(mean(partialdata)-5000)/sd(partialdata)*sqrt(10)\n", + " if (T>qt(p=0.95,df=9))\n", + " {break} \n", + " }\n", + "\n", + "i\n", + "T" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The dirty variation: The Chocolate Diet\n", + "\n", + "It is obvious that omitting datapoints from a study, irrespective of whether you pick specific points or choose some at random, is dubious. How about dropping factors?" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We take 1000 people from the same population, sort them into 100 groups a 10 people each, and subject them to totally nonesense diets. \"Liquid butter diet\", \"Travel to the moon diet\", \"Chicken nuggets diet\", etc. None of these actually work." + ] + }, + { + "cell_type": "code", + "execution_count": 65, + "metadata": {}, + "outputs": [], + "source": [ + "# set up some random data. The value denotes the weight loss/gain after 6 weeks of diet.\n", + "\n", + "n=10 #group size\n", + "m=100 # number of groups\n", + "\n", + "weightdata = matrix( rnorm(n*m,mean=0,sd=3), n, m) # weightloss of participants. mean=0, so all 100 diets are useless." + ] + }, + { + "cell_type": "code", + "execution_count": 66, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + " V1 V2 V3 V4 \n", + " Min. :-2.7387 Min. :-6.2774 Min. :-4.5216 Min. :-2.4288 \n", + " 1st Qu.:-1.7837 1st Qu.:-0.2651 1st Qu.:-0.6638 1st Qu.:-0.1889 \n", + " Median : 0.2215 Median : 2.0853 Median : 1.6549 Median : 0.9696 \n", + " Mean : 0.2599 Mean : 1.1211 Mean : 1.1930 Mean : 1.0904 \n", + " 3rd Qu.: 1.2789 3rd Qu.: 3.4290 3rd Qu.: 2.6253 3rd Qu.: 2.1775 \n", + " Max. : 6.0157 Max. : 4.5204 Max. : 6.3859 Max. : 4.4411 \n", + " V5 V6 V7 V8 \n", + " Min. :-3.6839 Min. :-3.6010 Min. :-5.257 Min. :-3.9968 \n", + " 1st Qu.:-0.8926 1st Qu.:-2.9950 1st Qu.:-3.309 1st Qu.:-2.4232 \n", + " Median : 0.2585 Median : 0.7410 Median :-2.115 Median :-0.2503 \n", + " Mean :-0.2065 Mean :-0.2975 Mean :-1.351 Mean : 0.1509 \n", + " 3rd Qu.: 1.1194 3rd Qu.: 1.5913 3rd Qu.: 1.387 3rd Qu.: 2.6697 \n", + " Max. : 1.8276 Max. : 3.1081 Max. : 2.876 Max. : 4.7618 \n", + " V9 V10 V11 V12 \n", + " Min. :-3.6981 Min. :-8.5235 Min. :-6.127 Min. :-7.3124 \n", + " 1st Qu.:-1.6022 1st Qu.:-2.8714 1st Qu.:-2.587 1st Qu.:-1.4340 \n", + " Median :-0.9391 Median : 0.6387 Median : 1.322 Median : 1.2521 \n", + " Mean :-0.5562 Mean : 0.6502 Mean :-0.144 Mean : 0.6589 \n", + " 3rd Qu.: 0.5766 3rd Qu.: 4.3568 3rd Qu.: 2.635 3rd Qu.: 3.5662 \n", + " Max. : 4.0115 Max. : 7.9055 Max. : 3.583 Max. : 4.9850 \n", + " V13 V14 V15 V16 \n", + " Min. :-5.2386 Min. :-3.60503 Min. :-4.5492 Min. :-3.1401 \n", + " 1st Qu.:-3.2376 1st Qu.:-1.29229 1st Qu.:-1.0744 1st Qu.:-1.8784 \n", + " Median : 0.2321 Median :-0.02803 Median : 0.3035 Median : 1.2445 \n", + " Mean :-0.8010 Mean : 0.52617 Mean : 0.8219 Mean : 0.5374 \n", + " 3rd Qu.: 1.3161 3rd Qu.: 2.60564 3rd Qu.: 3.1928 3rd Qu.: 2.6841 \n", + " Max. : 2.6894 Max. : 4.36617 Max. : 6.4502 Max. : 3.7110 \n", + " V17 V18 V19 V20 \n", + " Min. :-4.7366 Min. :-2.82933 Min. :-6.6116 Min. :-4.27837 \n", + " 1st Qu.:-1.9053 1st Qu.:-1.33012 1st Qu.:-2.2258 1st Qu.:-2.26918 \n", + " Median :-0.9012 Median :-0.09478 Median :-1.0821 Median :-0.02806 \n", + " Mean :-0.3886 Mean : 1.01402 Mean :-1.2373 Mean : 0.32870 \n", + " 3rd Qu.: 1.1414 3rd Qu.: 0.69461 3rd Qu.: 0.6779 3rd Qu.: 3.48848 \n", + " Max. : 5.8054 Max. : 9.85899 Max. : 1.2416 Max. : 4.79003 \n", + " V21 V22 V23 V24 \n", + " Min. :-3.889556 Min. :-8.0861 Min. :-7.4525 Min. :-6.7814 \n", + " 1st Qu.:-2.506045 1st Qu.:-3.1595 1st Qu.:-2.7016 1st Qu.:-2.9653 \n", + " Median :-0.627048 Median :-1.6304 Median :-2.0253 Median : 0.2354 \n", + " Mean :-0.005771 Mean :-1.9243 Mean :-1.6573 Mean :-0.1857 \n", + " 3rd Qu.: 1.044636 3rd Qu.:-0.6009 3rd Qu.: 0.6433 3rd Qu.: 1.7496 \n", + " Max. : 6.382749 Max. : 3.3835 Max. : 3.4973 Max. : 6.6333 \n", + " V25 V26 V27 V28 \n", + " Min. :-5.2645 Min. :-4.1320 Min. :-5.4829 Min. :-7.2746 \n", + " 1st Qu.:-2.9971 1st Qu.:-2.4753 1st Qu.:-4.3276 1st Qu.:-3.9064 \n", + " Median :-0.8133 Median :-0.8886 Median :-3.2116 Median :-2.0478 \n", + " Mean :-1.0575 Mean :-0.2999 Mean :-2.3287 Mean :-1.7122 \n", + " 3rd Qu.: 0.5644 3rd Qu.: 1.4955 3rd Qu.:-0.4598 3rd Qu.: 0.8398 \n", + " Max. : 4.8862 Max. : 5.8128 Max. : 2.9291 Max. : 3.9566 \n", + " V29 V30 V31 V32 \n", + " Min. :-4.0293 Min. :-6.13637 Min. :-5.4326 Min. :-4.7377 \n", + " 1st Qu.:-1.3605 1st Qu.:-1.23132 1st Qu.:-2.0664 1st Qu.:-0.5838 \n", + " Median : 0.2867 Median :-1.04713 Median :-0.9520 Median : 0.3090 \n", + " Mean : 0.1865 Mean : 0.08589 Mean :-0.4567 Mean : 0.4762 \n", + " 3rd Qu.: 2.0280 3rd Qu.: 3.42855 3rd Qu.: 0.8119 3rd Qu.: 2.5849 \n", + " Max. : 3.9142 Max. : 4.48216 Max. : 5.0003 Max. : 3.6401 \n", + " V33 V34 V35 V36 \n", + " Min. :-1.3908 Min. :-6.11141 Min. :-1.2187 Min. :-3.5014 \n", + " 1st Qu.:-0.3587 1st Qu.:-2.26360 1st Qu.:-0.6871 1st Qu.:-2.4399 \n", + " Median : 0.6135 Median :-0.09133 Median : 0.4042 Median :-0.1805 \n", + " Mean : 1.5321 Mean :-0.28936 Mean : 0.8160 Mean :-0.4881 \n", + " 3rd Qu.: 3.1194 3rd Qu.: 2.21535 3rd Qu.: 2.1187 3rd Qu.: 1.0160 \n", + " Max. : 7.3491 Max. : 5.03239 Max. : 3.6831 Max. : 3.2615 \n", + " V37 V38 V39 V40 \n", + " Min. :-3.8037 Min. :-5.8418 Min. :-4.8735 Min. :-4.59629 \n", + " 1st Qu.:-2.6220 1st Qu.:-2.5992 1st Qu.:-3.2764 1st Qu.:-2.06800 \n", + " Median :-0.6491 Median : 0.4256 Median :-1.1163 Median :-0.16480 \n", + " Mean :-0.4337 Mean :-0.7668 Mean :-1.3830 Mean :-0.04249 \n", + " 3rd Qu.: 0.7986 3rd Qu.: 1.0016 3rd Qu.: 0.6823 3rd Qu.: 1.27648 \n", + " Max. : 3.8737 Max. : 1.8474 Max. : 1.4900 Max. : 6.46894 \n", + " V41 V42 V43 V44 \n", + " Min. :-3.0495 Min. :-5.2623 Min. :-2.8562 Min. :-2.85473 \n", + " 1st Qu.:-0.6384 1st Qu.:-1.4195 1st Qu.:-1.1437 1st Qu.:-0.99668 \n", + " Median : 0.8009 Median :-0.4979 Median : 0.8957 Median : 0.04019 \n", + " Mean : 0.6714 Mean : 0.7669 Mean : 0.7331 Mean : 0.20297 \n", + " 3rd Qu.: 2.3859 3rd Qu.: 3.8956 3rd Qu.: 2.2222 3rd Qu.: 1.17033 \n", + " Max. : 3.4813 Max. : 7.0075 Max. : 4.6657 Max. : 4.07790 \n", + " V45 V46 V47 V48 \n", + " Min. :-5.38055 Min. :-3.4603 Min. :-6.7713 Min. :-2.2898 \n", + " 1st Qu.:-0.92784 1st Qu.:-0.6142 1st Qu.:-3.3013 1st Qu.:-1.4995 \n", + " Median :-0.21939 Median : 0.5084 Median :-1.9644 Median : 0.3583 \n", + " Mean :-0.07288 Mean : 0.0765 Mean :-1.8958 Mean : 0.7756 \n", + " 3rd Qu.: 1.70221 3rd Qu.: 1.3192 3rd Qu.:-0.1547 3rd Qu.: 0.8926 \n", + " Max. : 5.52106 Max. : 2.8123 Max. : 2.9755 Max. : 8.4801 \n", + " V49 V50 V51 V52 \n", + " Min. :-3.6087 Min. :-3.8625 Min. :-8.3220 Min. :-3.2985 \n", + " 1st Qu.:-0.8305 1st Qu.:-3.3829 1st Qu.:-1.9049 1st Qu.:-2.9366 \n", + " Median : 1.3502 Median :-2.9301 Median : 0.4876 Median :-0.5306 \n", + " Mean : 0.6117 Mean :-1.6878 Mean :-0.3437 Mean :-0.5468 \n", + " 3rd Qu.: 2.1119 3rd Qu.:-0.2984 3rd Qu.: 2.5113 3rd Qu.: 0.8550 \n", + " Max. : 2.7773 Max. : 2.9689 Max. : 4.4144 Max. : 4.4288 \n", + " V53 V54 V55 V56 \n", + " Min. :-3.4745 Min. :-4.96658 Min. :-4.2060 Min. :-5.6279 \n", + " 1st Qu.:-1.0855 1st Qu.:-1.97070 1st Qu.:-2.1646 1st Qu.:-1.6898 \n", + " Median : 0.5801 Median : 0.04154 Median :-0.8205 Median : 0.6932 \n", + " Mean : 0.8464 Mean :-0.10137 Mean :-0.7876 Mean : 0.3396 \n", + " 3rd Qu.: 2.8242 3rd Qu.: 1.82526 3rd Qu.:-0.2133 3rd Qu.: 2.4876 \n", + " Max. : 5.4924 Max. : 3.20107 Max. : 4.3878 Max. : 5.9250 \n", + " V57 V58 V59 V60 \n", + " Min. :-3.4964 Min. :-5.2207 Min. :-3.9131 Min. :-2.0504 \n", + " 1st Qu.:-0.3855 1st Qu.:-2.1553 1st Qu.: 0.3406 1st Qu.:-0.5297 \n", + " Median : 0.8869 Median : 0.1165 Median : 1.5290 Median : 1.0673 \n", + " Mean : 0.6363 Mean :-0.3584 Mean : 1.4126 Mean : 0.6702 \n", + " 3rd Qu.: 1.7703 3rd Qu.: 0.6783 3rd Qu.: 2.8064 3rd Qu.: 1.8851 \n", + " Max. : 3.7388 Max. : 4.0330 Max. : 5.9108 Max. : 2.9805 \n", + " V61 V62 V63 V64 \n", + " Min. :-7.7404 Min. :-4.999 Min. :-4.5805 Min. :-4.0908 \n", + " 1st Qu.:-1.7868 1st Qu.:-1.169 1st Qu.:-3.7782 1st Qu.:-0.3189 \n", + " Median :-0.1115 Median : 1.212 Median :-1.2213 Median : 1.5848 \n", + " Mean :-0.2392 Mean : 1.210 Mean :-0.7975 Mean : 1.5143 \n", + " 3rd Qu.: 2.4393 3rd Qu.: 3.943 3rd Qu.: 1.1804 3rd Qu.: 3.9967 \n", + " Max. : 4.1072 Max. : 8.360 Max. : 5.4101 Max. : 6.0471 \n", + " V65 V66 V67 V68 \n", + " Min. :-3.593 Min. :-5.5280 Min. :-6.6670 Min. :-4.4258 \n", + " 1st Qu.:-2.959 1st Qu.:-2.7325 1st Qu.:-3.5371 1st Qu.:-3.6897 \n", + " Median :-2.030 Median : 0.9414 Median : 0.8520 Median :-2.8846 \n", + " Mean :-1.296 Mean : 0.1612 Mean :-0.4244 Mean :-1.5048 \n", + " 3rd Qu.:-1.004 3rd Qu.: 2.9022 3rd Qu.: 1.7432 3rd Qu.: 0.9953 \n", + " Max. : 2.962 Max. : 4.4893 Max. : 4.6650 Max. : 3.1263 \n", + " V69 V70 V71 V72 \n", + " Min. :-4.1867 Min. :-3.5920 Min. :-1.5778 Min. :-6.2671 \n", + " 1st Qu.:-2.9221 1st Qu.:-1.7924 1st Qu.:-0.6966 1st Qu.:-1.4800 \n", + " Median : 0.4474 Median :-0.2807 Median : 0.4413 Median :-0.4585 \n", + " Mean :-0.2254 Mean :-0.1259 Mean : 0.9255 Mean :-0.5278 \n", + " 3rd Qu.: 2.1030 3rd Qu.: 1.8812 3rd Qu.: 1.9306 3rd Qu.: 1.2729 \n", + " Max. : 3.2509 Max. : 3.1321 Max. : 5.0731 Max. : 3.7592 \n", + " V73 V74 V75 V76 \n", + " Min. :-2.8741 Min. :-6.063 Min. :-2.437 Min. :-6.36781 \n", + " 1st Qu.:-0.2621 1st Qu.:-2.462 1st Qu.:-1.362 1st Qu.:-1.44698 \n", + " Median : 0.3135 Median :-1.479 Median : 1.118 Median :-0.22778 \n", + " Mean : 0.7916 Mean :-1.101 Mean : 1.030 Mean : 0.08612 \n", + " 3rd Qu.: 1.1646 3rd Qu.: 1.749 3rd Qu.: 3.551 3rd Qu.: 2.51234 \n", + " Max. : 7.2964 Max. : 2.853 Max. : 4.281 Max. : 4.38210 \n", + " V77 V78 V79 V80 \n", + " Min. :-2.9275 Min. :-5.9778 Min. :-3.52059 Min. :-5.1360 \n", + " 1st Qu.:-1.8053 1st Qu.:-3.1982 1st Qu.:-1.73778 1st Qu.:-3.3350 \n", + " Median : 0.7098 Median :-1.9262 Median :-0.26206 Median :-1.0066 \n", + " Mean : 0.3157 Mean :-1.5544 Mean :-0.04367 Mean :-1.4568 \n", + " 3rd Qu.: 1.7679 3rd Qu.: 0.3854 3rd Qu.: 1.56273 3rd Qu.: 0.5765 \n", + " Max. : 4.0622 Max. : 3.3637 Max. : 4.30285 Max. : 2.0140 \n", + " V81 V82 V83 V84 \n", + " Min. :-4.1083 Min. :-1.7479 Min. :-6.2640 Min. :0.4621 \n", + " 1st Qu.:-2.5350 1st Qu.:-0.9795 1st Qu.:-1.5113 1st Qu.:1.0495 \n", + " Median :-1.2026 Median : 1.3971 Median : 0.7994 Median :1.4419 \n", + " Mean :-0.9683 Mean : 1.4375 Mean : 0.3501 Mean :1.8670 \n", + " 3rd Qu.: 0.4388 3rd Qu.: 3.6563 3rd Qu.: 3.1489 3rd Qu.:2.8223 \n", + " Max. : 3.5350 Max. : 5.4576 Max. : 5.7092 Max. :3.8786 \n", + " V85 V86 V87 V88 \n", + " Min. :-2.8525 Min. :-1.2197 Min. :-5.00529 Min. :-5.4783 \n", + " 1st Qu.:-0.9390 1st Qu.:-0.1384 1st Qu.:-3.15741 1st Qu.:-0.4943 \n", + " Median :-0.4581 Median : 1.1440 Median :-0.08258 Median : 1.4565 \n", + " Mean : 0.8771 Mean : 0.8451 Mean :-0.65594 Mean : 0.8884 \n", + " 3rd Qu.: 3.2803 3rd Qu.: 1.8053 3rd Qu.: 1.74454 3rd Qu.: 2.6112 \n", + " Max. : 5.3565 Max. : 2.6130 Max. : 2.57776 Max. : 6.0365 \n", + " V89 V90 V91 V92 \n", + " Min. :-3.59884 Min. :-4.6714 Min. :-3.3265 Min. :-3.9736 \n", + " 1st Qu.:-1.09594 1st Qu.:-1.6123 1st Qu.:-1.4255 1st Qu.: 0.6773 \n", + " Median :-0.06314 Median : 0.2407 Median :-0.2765 Median : 2.2629 \n", + " Mean : 0.18418 Mean : 0.5685 Mean : 0.4017 Mean : 1.5662 \n", + " 3rd Qu.: 1.82922 3rd Qu.: 3.1508 3rd Qu.: 2.4910 3rd Qu.: 2.6754 \n", + " Max. : 3.65400 Max. : 6.1124 Max. : 5.8832 Max. : 6.1553 \n", + " V93 V94 V95 V96 \n", + " Min. :-2.47096 Min. :-6.3451 Min. :-4.3624 Min. :-3.5323 \n", + " 1st Qu.:-1.73569 1st Qu.:-1.6137 1st Qu.:-0.4261 1st Qu.:-1.1714 \n", + " Median :-0.71519 Median :-1.3399 Median : 1.3717 Median :-0.3008 \n", + " Mean :-0.08103 Mean :-0.9288 Mean : 0.4653 Mean : 0.4759 \n", + " 3rd Qu.: 0.66572 3rd Qu.: 0.4041 3rd Qu.: 1.9404 3rd Qu.: 2.2224 \n", + " Max. : 4.24817 Max. : 4.8719 Max. : 3.1000 Max. : 6.0299 \n", + " V97 V98 V99 V100 \n", + " Min. :-3.84499 Min. :-4.6715 Min. :-5.4650 Min. :-5.4981 \n", + " 1st Qu.:-1.29470 1st Qu.:-1.3319 1st Qu.:-2.6876 1st Qu.:-3.5600 \n", + " Median : 0.06828 Median :-0.9314 Median :-0.4928 Median :-2.5573 \n", + " Mean : 0.32726 Mean :-0.7450 Mean :-0.9623 Mean :-2.0001 \n", + " 3rd Qu.: 0.84174 3rd Qu.: 0.1483 3rd Qu.: 0.8145 3rd Qu.:-0.5725 \n", + " Max. : 5.92879 Max. : 3.0226 Max. : 2.5577 Max. : 2.2577 " + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "summary(weightdata)" + ] + }, + { + "cell_type": "code", + "execution_count": 81, + "metadata": {}, + "outputs": [], + "source": [ + "successfuldiets=vector()\n", + "\n", + "for (i in seq(1,100))\n", + " \n", + " {\n", + " mypvalue=t.test(weightdata[,i],mu=0,alternative=\"less\")$p.value\n", + " \n", + " if (mypvalue<0.05)\n", + " {successfuldiets=c(successfuldiets,i)}\n", + " \n", + " \n", + "}" + ] + }, + { + "cell_type": "code", + "execution_count": 82, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
    \n", + "\t
  1. 22
    2. \n", + "\t
  2. 27
    3. \n", + "\t
  3. 39
    4. \n", + "\t
  4. 47
    5. \n", + "\t
  5. 50
    6. \n", + "\t
  6. 100
    7. \n", + "
\n" + ], + "text/latex": [ + "\\begin{enumerate*}\n", + "\\item 22\n", + "\\item 27\n", + "\\item 39\n", + "\\item 47\n", + "\\item 50\n", + "\\item 100\n", + "\\end{enumerate*}\n" + ], + "text/markdown": [ + "1. 22\n", + "2. 27\n", + "3. 39\n", + "4. 47\n", + "5. 50\n", + "6. 100\n", + "\n", + "\n" + ], + "text/plain": [ + "[1] 22 27 39 47 50 100" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "successfuldiets" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "R", + "language": "R", + "name": "ir" + }, + "language_info": { + "codemirror_mode": "r", + "file_extension": ".r", + "mimetype": "text/x-r-source", + "name": "R", + "pygments_lexer": "r", + "version": "3.6.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +}