diff --git a/Lecture 8/2-sample t-test.ipynb b/Lecture 8/2-sample t-test.ipynb
new file mode 100644
index 0000000..0ab649c
--- /dev/null
+++ b/Lecture 8/2-sample t-test.ipynb
@@ -0,0 +1,172 @@
+{
+ "cells": [
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "'Pooled Variance: 10.0085189219356 True Variance: 16'"
+ ],
+ "text/latex": [
+ "'Pooled Variance: 10.0085189219356 True Variance: 16'"
+ ],
+ "text/markdown": [
+ "'Pooled Variance: 10.0085189219356 True Variance: 16'"
+ ],
+ "text/plain": [
+ "[1] \"Pooled Variance: 10.0085189219356 True Variance: 16\""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "text/html": [
+ "'T : -1.31136522266901 Critical T: -1.8595480375309'"
+ ],
+ "text/latex": [
+ "'T : -1.31136522266901 Critical T: -1.8595480375309'"
+ ],
+ "text/markdown": [
+ "'T : -1.31136522266901 Critical T: -1.8595480375309'"
+ ],
+ "text/plain": [
+ "[1] \"T : -1.31136522266901 Critical T: -1.8595480375309\""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "text/html": [
+ "'accept H0'"
+ ],
+ "text/latex": [
+ "'accept H0'"
+ ],
+ "text/markdown": [
+ "'accept H0'"
+ ],
+ "text/plain": [
+ "[1] \"accept H0\""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "image/png": 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4ZV9lqZmZm9e/cuf02DBg0ObyQAgOrk5AkAgEAIOwCAQAg7AIBABBV2o0ePbt26\ndaqnAABIjaDCbtu2bRs2bEj1FAAAqRFU2AEAfJOlzdudXHnllRWuycvLq4ZJAABqprQJu5kz\nZ6Z6BACAGi1twu7YY49t2bLlhAkTylnz4IMPzps3r9pGAgCoUdIm7Dp27Pj+++9ffPHF5Xy0\n16xZs6pzJACAGiVtTp7Izc3dtWvX2rVrUz0IAEANlTaP2PXp02fRokWbNm1q06ZNWWsGDBjQ\nqlWr6pwKAKDmSJuwGzRo0KBBg458DQBAqNLmqVgAAMon7AAAAiHsAAACIewAAAIh7AAAAiHs\nAAACIewAAAIh7AAAAiHsAAACIewAAAIh7AAAAiHsAAACIewAAAIh7AAAAiHsAAACIewAAAIh\n7AAAAiHsAAACIewAAAIh7AAAAiHsAAACIewAAAIh7AAAAiHsAAACIewAAAIh7AAAAiHsAAAC\nIewAAAIh7AAAAiHsAAACIewAAAIh7AAAAiHsAAACIewAAAIh7AAAAiHsAAACIewAAAIh7AAA\nAiHsAAACIewAAAIh7AAAAiHsAAACIewAAAIh7AAAAiHsAAACIewAAAIh7AAAAiHsAAACIewA\nAAIh7AAAAiHsAAACIewAAAIh7AAAAiHsAAACIewAAAIh7AAAAiHsAAACIewAAAIh7AAAAiHs\nAAACIewAAAIh7AAAAiHsAAACIewAAAIh7AAAAiHsAAACIewAAAIh7AAAAiHsAAACIewAAAIh\n7AAAAiHsAAACIewAAAIh7AAAAiHsAAACIewAAAJRcdh99dVX1TAHAABHqOKwa9my5bBhwxYu\nXFgN0wAAcNgqDrtWrVpNnTr13HPPPeuss37/+9/v2rWrGsYCAKCyKg671atXz5s3b/DgwR99\n9NFNN93UokWLa6+9dsmSJdUwHAAAyas47DIyMvr06TNz5syNGzf+9re/bdas2eTJk7t3796l\nS5dHH310z5491TAlAAAVqsRZsU2bNv3Zz3728ccfv/rqq//0T/+0cuXKESNGtGjR4sYbb1y1\nalXVjQgAQDIq/XYnGRkZ7dq1O/3003NycqIo2r1796RJkzp27DhkyJD8/PwqmBAAgKRUIuwK\nCwtffPHFiy+++NRTTx03bly9evXuvvvuTZs2vfzyy9/97ndnzJhx0003Vd2gAACUr3YyizZu\n3Dh58uTHH3988+bNGRkZ/fr1+8lPfnLppZfWqlUriqKWLVv279//sssue/nllwAeO1oAACAA\nSURBVKt4WgAAylRx2F166aVz584tLCw87rjjbr/99htvvPG0004rsSYjI6NHjx6zZ8+umiEB\nAKhYxWH30ksvdevW7Sc/+cmVV155zDHHlLWsf//+jRs3PqqzAQBQCRWH3TvvvNOlS5cKl+Xm\n5ubm5h6NkQAAOBwVnzyRTNUBAJByFYfd008/fcEFF2zatKnE8U2bNvXu3fvZZ5+tmsEAAKic\nisPuscce2717d6tWrUocb9Wq1c6dOx977LGqGQwAgMqpOOxWrlzZtWvXQ17UtWvXlStXHu2R\nAAA4HBWH3Y4dO5o0aXLIi5o2bbpt27ajPRIAAIej4rBr0qTJxx9/fMiLPvnkk+zs7KM9EgAA\nh6PisDvvvPNefPHFjz76qMTxDz/88MUXX+zZs2fVDAYAQOVUHHa33377gQMHevbsOXHixE8+\n+aSgoOCTTz6ZOHHieeedd+DAgdGjR1fDlAAAVKjiNyg+55xzHnnkkZ/+9Kc333xz4vFatWo9\n8sgj5557bpXNBgBAJVQcdlEU3XDDDeeee+7vf//7vLy8nTt3Zmdn9+jR4yc/+cmZZ55Z1fMB\nAJCkpMIuiqKOHTtOmjSpSkcBAOBIVPwaOwAA0oKwAwAIRFJh98YbbwwYMKBZs2b16tWrXUpV\njwgAQDIqzrKXXnrpsssuKyoqysrKatu2rZIDAKiZKq60u+66KyMj409/+tOQIUMyMjKqYSYA\nAA5DxWG3atWqgQMH/uhHP6qGaQAAOGwVv8bu2GOPbdq0aTWMAgDAkag47Pr165eXl1cNowAA\ncCQqDrt7771306ZNY8eOLSwsrIaBAAA4PBW/xu7Xv/71GWeccddddz3xxBOdOnXKzs4usWDK\nlClVMhoAAJVRcdhNnTq1+IsNGzZs2LCh9AJhBwBQE1QcdsuWLauGOQAAOEIVh12nTp2qYQ4A\nAI5QJT4rdsOGDQsXLszPz6+6aQAAOGxJhd2iRYvOOuus1q1bn3vuuUuWLCk+OGPGjA4dOrzx\nxhtVOd4hxGKx1atXz549+8knn/zjH/84e/bs1atXx2Kxah4DAKCmqfip2A8//LBfv34ZGRmX\nXXbZCy+8ED9+ySWXXHvttc8888x3v/vdqpzw/ykoKJgwYcKkSZM2b95c4qJWrVqNGDFi1KhR\n9evXr55hAABqmorDbty4cQcOHHjnnXeaN2+eGHYNGza84IILFixYUJXj/T979+7t27dvXl5e\nZmZm586d27Ztm5WVlZGRsXPnzjVr1qxYsWLMmDFz5syZN29egwYNqmckAIAapeKwmzdv3sCB\nA88888xt27aVuOjb3/72woULq2awksaPH5+Xlzd06NB77723RYsWJS7dvHnzHXfcMX369PHj\nx48bN656RgIAqFEqfo3d9u3bW7dufciLatWqtXv37qM8URlmzJjRpUuXadOmla66KIpatmz5\n5JNP5ubmzpw5s3rmAQCoaSoOu5ycnK1btx7yomXLljVv3vxoj3RomzZt6tWrV2ZmmQNnZmb2\n6tVr48aN1TMPAEBNU3HY9ezZc86cOfv27StxfP78+a+99lrv3r2rZK5SsrKy1q1bV/6atWvX\nlv7EMwCAb4iKw2706NFbt24dOHDgBx98EEVRQUHBkiVLRo0a1b9//9q1a99+++1VP2QURVG/\nfv1mz549bdq0shZMmTLlpZde6tu3b/XMAwBQ01R88kTPnj0feeSRkSNHzp07N4qiAQMGFB+v\nU6fO448/3rFjx6od8H/cc889L7/88tVXX/3ggw/279+/ffv2WVlZURTl5+evXr167ty5y5cv\nz87Ovvvuu6tnHgCAmqbisIui6IYbbujVq9ekSZMWLly4ffv2rKysHj16jBw58owzzqjq+eLa\ntGmzYMGC4cOHL168+JAfX9u9e/fJkye3adOm2kYCAKhRkgq7KIrOOOOMiRMnVukoFerQoUNe\nXt7SpUvnz5+/evXq4g83y8rKat++fZ8+fXJzc1M7HgBAaiUbdjVHbm6uhgMAKC2pz4oFAKDm\nq/gRu9NOO638BZ988slRGuaIbNmy5dNPP42iqGvXrqmeBQAgBSoOu9KfJLZ3796DBw9GUdS4\nceOMjIwqmavynnrqqdtuuy2Kolgslvy1vvrqqzvvvLP4P6csX3755ZEOBykyZ86cF198sVJX\nGTBgwMUXX5ym82zatCl/9eoJEyYkuf7v77zzdb16I0aMSHL9u+++u2XLloYNGya5fs+ePU2b\nNu3SpUuS66Mqvv2rWk37e4NvoIrDbufOnSWOHDhwYNmyZbfeeuvxxx//7LPPVs1glZadne2U\nWChh1qxZr7/+dqdOvZNcv3z53/bv3191d7RVPc/69esPbt2xfXuy83z+1c7MjCj67LMk17//\n3nu1CgubND0pyfWfbtn0+edbmzdPNuyq+vavarNmzXr79dd7d+qU5Pq/LV+e1v+9UAMdzskT\nderU6d69+5w5c84444zx48f/+te/PupjHYZhw4YNGzasstfKycl55JFHyl/z9ttvv/DCC4c5\nFqRahw7njhr1f5Nc/B//cU2VDhNV/TwNG2YPHjwqycWTls3PyYz+76hk1z87f36jouiJf5ua\n5PqBP794S2b9GnX7V7VzO3RI/va85j/+o0qHgW+gwz95Iicnp1+/flOnJvt/cAAAVKkjOiu2\nXr16mzdvPlqjAABwJA4/7L744ovZs2e3bNnyKE4DAMBhq/g1dnfddVeJIwcPHty4cePzzz+/\na9euGvXZrKNHj541a9b69etTPQgAQApUHHZjx4495PH69euPHj36l7/85dEe6fBt27Ztw4YN\nqZ4CACA1Kg672bNnlziSmZmZk5Nz5plnJv9mTgAAVLWKw+6SSy6phjkqdOWVV1a4Ji8vrxom\nAQComQ7nfexSYubMmakeAQCgRkubsDv22GNbtmxZ/icFPfjgg/Pmzau2kQAAapSKw65169bJ\n/7iqOyO1Y8eO77///sUXX1zOp9POmjWrin47AEDNV3HY7dmzp7CwMP6Jsccee+zevXuLv87O\nzq5Vq1YVTpcgNzd34cKFa9eu9YGwAACHVPEbFK9fv75Dhw65ublz5szZvXv3nj17du/ePWfO\nnM6dO3fo0GH9+vXbElTdoH369OnSpcumTZvKWTNgwIAa9fYrAADVqeJH7MaMGfPZZ5+tXLmy\nQYMGxUcaNmx40UUX9e7d+8wzzxwzZswDDzxQxUNGURQNGjRo0KBBR74GACBUFT9i98wzzwwa\nNChedXENGjQYNGiQl7UBANQQFYfd1q1bY7HYIS+KxWJbt2492iMBAHA4Kg671q1bP/vss/ET\nJuL27t07a9asU045pWoGAwCgcioOuxtuuGH9+vU9e/Z8/vnnd+zYEUXRjh07nn/++Z49e27Y\nsGHEiBFVPyQAABWr+OSJW2655cMPP3zssccGDhwYRVHt2rUPHjxYfNH1119/8803V+2AAAAk\np+Kwy8zMfPTRR4cMGTJ16tRly5bl5+dnZWV17tx52LBhvXv3rvoJAQBISrIfKXbBBRdccMEF\nVToKAABHouLX2MVt2LBh4cKF+fn5VTcNAACHLamwW7Ro0VlnndW6detzzz13yZIlxQdnzJjR\noUOHN954oyrHAwAgWRWH3YcfftivX7+1a9dedtlliccvueSS9evXP/PMM1U2GwAAlVDxa+zG\njRt34MCBd955p3nz5i+88EL8eMOGDS+44IIFCxZU5XgAACSr4kfs5s2bN3DgwDPPPLP0Rd/+\n9rc3bdpUBVMBAFBpFYfd9u3bW7dufciLatWqtXv37qM8EQAAh6XisMvJySnrA2GXLVvWvHnz\noz0SAACHo+Kw69mz55w5c/bt21fi+Pz581977TXvUQwAUENUHHajR4/eunXrwIEDP/jggyiK\nCgoKlixZMmrUqP79+9euXfv222+v+iEBAKhYxWfF9uzZ85FHHhk5cuTcuXOjKBowYEDx8Tp1\n6jz++OMdO3as2gEBAEhOUh8pdsMNN/Tq1WvSpEkLFy7cvn17VlZWjx49Ro4cecYZZ1T1fAAA\nJKnisFu0aNExxxzTqVOniRMnVsNAAAAcnopfY3fuueeOGzeuGkYBAOBIVBx2TZo0adCgQTWM\nAgDAkag47Hr37r148eLCwsJqmAYAgMNWcdiNHz9+27Ztt95669dff10NAwEAcHgqPnniN7/5\nTceOHR9++OEZM2Z06tSpRYsWGRkZiQumTJlSVdMBAJC0isNu6tSpxV9s27bt9ddfL71A2AEA\n1AQVh92yZcuqYQ4AAI5QxWHXqVOnapgDAIAjVObJEzNmzMjLy6vOUQAAOBJlht2QIUP+67/+\nK/7thAkT+vfvXy0jAQBwOCp+u5NiK1eu/Mtf/lKlowAAcCSSDTsAAGo4YQcAEAhhBwAQCGEH\nABCI8t7H7qmnnnr++eeLvy7+oNjs7OzSy3bu3FkVkwEAUCnlhd2BAwfy8/MTj5T4FgCAmqPM\nsCsoKKjOOQAAOEJlht0xxxxTnXMAAHCEnDwBABAIYQcAEAhhBwAQCGEHABAIYQcAEAhhBwAQ\nCGEHABAIYQcAEAhhBwAQCGEHABAIYQcAEAhhBwAQCGEHABAIYQcAEAhhBwAQCGEHABAIYQcA\nEAhhBwAQCGEHABAIYQcAEAhhBwAQCGEHABAIYQcAEAhhBwAQCGEHABAIYQcAEAhhBwAQCGEH\nABAIYQcAEAhhBwAQCGEHABAIYQcAEAhhBwAQCGEHABAIYQcAEAhhBwAQCGEHABAIYQcAEAhh\nBwAQCGEHABAIYQcAEAhhBwAQCGEHABAIYQcAEAhhBwAQCGEHABAIYQcAEAhhBwAQCGEHABAI\nYQcAEAhhBwAQCGEHABAIYQcAEAhhBwAQCGEHABAIYQcAEAhhBwAQCGEHABAIYQcAEAhhBwAQ\nCGEHABAIYQcAEAhhBwAQCGEHABAIYQcAEAhhBwAQCGEHABAIYQcAEAhhBwAQCGEHABAIYQcA\nEAhhBwAQCGEHABAIYQcAEAhhBwAQCGEHABCINA67t95666KLLjr++OMbNWrUqVOnCRMmHDx4\nMNVDAQCkTNqEXbNmzW655Zb4t9OnT7/gggvmzp27ffv2PXv2vPfee6NHj77iiitisVgKhwQA\nSKG0Cbsvv/wyPz+/+Ovt27dff/31sVjszjvvXLt27Y4dO5577rnmzZu/8MILTz31VGrnBABI\nlbQJu0SzZs3as2fPzTfffM8995xyyik5OTkDBw7885//HEXR1KlTUz0dAEBqpGXYrVixIoqi\n6667LvHg2Wef3alTp+XLl6doKACAFEvLsCsoKIii6JRTTilx/NRTT925c2cqJgIASL20DLvT\nTjstiqJdu3aVOP7VV19lZWWlYiIAgNSrneoBKuGPf/zjjBkzoigqKiqKomjVqlUnnnhi4oJ1\n69addNJJqRkOACDV0ibs2rdvX+LI4sWL+/btG/926dKl69ev79+/f/XOBQBQU6RN2H300Ufl\nLygsLLzvvvsSUw8A4BslbcKuQt26devWrVuqpwAASJm0PHkCAIDSwnnEbsuWLZ9++mkURV27\ndk31LAAAKRBO2D311FO33XZbFEWV+rjYoqKiN9988+DBg+Wsef/99490OACAqhdO2GVnZ7dp\n06ay19qwYcPgwYPLD7viSyvViwAA1S+csBs2bNiwYcMqe61TTjlly5Yt5a95++23e/bsmZGR\ncZiTAQBUCydPAAAEQtgBAAQi/Z6KjcVia9asWbNmTX5+fiwWy87ObteuXbt27TxVCgB8w6VT\n2BUUFEyYMGHSpEmbN28ucVGrVq1GjBgxatSo+vXrp2Q2AICUS5uw27t3b9++ffPy8jIzMzt3\n7ty2bdusrKyMjIydO3euWbNmxYoVY8aMmTNnzrx58xo0aJDqYQEAUiBtwm78+PF5eXlDhw69\n9957W7RoUeLSzZs333HHHdOnTx8/fvy4ceNSMiEAQGqlzckTM2bM6NKly7Rp00pXXRRFLVu2\nfPLJJ3Nzc2fOnFn9swEA1ARpE3abNm3q1atXZmaZA2dmZvbq1Wvjxo3VORUAQM2RNmGXlZW1\nbt268tesXbs2Ozu7euYBAKhp0ibs+vXrN3v27GnTppW1YMqUKS+99FLfvn2rcyoAgJojbU6e\nuOeee15++eWrr776wQcf7N+/f/v27bOysqIoys/PX7169dy5c5cvX56dnX333XenelIAgNRI\nm7Br06bNggULhg8fvnjx4mXLlpVe0L1798mTJ7dp06b6ZwMAqAnSJuyiKOrQoUNeXt7SpUvn\nz5+/evXq/Pz8KIqysrLat2/fp0+f3NzcVA8IAJBK6RR2xXJzczUcAEBpaXPyBAAA5RN2AACB\nEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAA\ngRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYA\nAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2\nAACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQ\ndgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACB\nEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAA\ngRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYA\nAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2\nAACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQ\ndgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACBEHYAAIEQdgAAgRB2AACB\nEHYAAIEQdgAAgRB2AACBqJ3qASotFoutWbNmzZo1+fn5sVgsOzu7Xbt27dq1y8jISPVoAACp\nlE5hV1BQMGHChEmTJm3evLnERa1atRoxYsSoUaPq16+fktkAAFIubcJu7969ffv2zcvLy8zM\n7Ny5c9u2bbOysjIyMnbu3LlmzZoVK1aMGTNmzpw58+bNa9CgQaqHBQBIgbQJu/Hjx+fl5Q0d\nOvTee+9t0aJFiUs3b958xx13TJ8+ffz48ePGjUvJhAAAqZU2J0/MmDGjS5cu06ZNK111URS1\nbNnyySefzM3NnTlzZvXPBgBQE6RN2G3atKlXr16ZmWUOnJmZ2atXr40bN1bnVAAANUfahF1W\nVta6devKX7N27drs7OzqmQcAoKZJm7Dr16/f7Nmzp02bVtaCKVOmvPTSS3379q3OqQAAao60\nOXninnvuefnll6+++uoHH3ywf//+7du3z8rKiqIoPz9/9erVc+fOXb58eXZ29t13353qSQEA\nUiNtwq5NmzYLFiwYPnz44sWLly1bVnpB9+7dJ0+e3KZNm+qfDQCgJkibsIuiqEOHDnl5eUuX\nLp0/f/7q1avz8/OjKMrKymrfvn2fPn1yc3NTPSAAQCqlU9gVy83N1XAAAKWlzckTAACUL/0e\nsSvLli1bPv300yiKunbtmupZAABSICMWi6V6hqPjwQcfvO2226IoqtR/0bp1684+++yDBw+W\ns+bgwYO7d+/ev39/nTp1jnTKKOrXr9/8+fMrdZVYLJaRkWG99Yexvvh/DlW3vqbNE4vFMqOo\nVtLrD8ZiGVW8PlaTbv+qXn8Yt3+lbp/KzmO99SX06dPn9ddfT359OgrnEbvs7OzDOCX25JNP\nfvrpp8sPu1gstmXLlqNSdVEU3X///a+++mry6z///PNYLHbID1Kz3nrrrbfeeuuTXB9F0fe/\n//3kF6epcB6xAwD4hnPyBABAIIQdAEAg0u81drFYbM2aNWvWrMnPz4/FYtnZ2e3atWvXrl2l\nXj4JABCedAq7goKCCRMmTJo0afPmzSUuatWq1YgRI0aNGlW/fv2UzAYAkHJpc/LE3r17+/bt\nm5eXl5mZedZZZ7Vt2zYrKysjI2Pnzp1r1qxZsWJFUVFRjx495s2b16BBg1QPCwCQAmnziN34\n8ePz8vKGDh167733lj63efPmzXfcccf06dPHjx8/bty4lEwIAJBaafOIXZs2bXJychYvXpyZ\neegTPoqKirp167Zr166PP/64mmcDAKgJ0uas2E2bNvXq1ausqouiKDMzs1evXhs3bqzOqQAA\nao60CbusrKx169aVv2bt2rXZ2dnVMw8AQE2TNmHXr1+/2bNnT5s2rawFU6ZMeemll/r27Vud\nUwEA1Bxp8xq7f/zjH12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+ "text/plain": [
+ "Plot with title “Compare groups A and B”"
+ ]
+ },
+ "metadata": {
+ "image/png": {
+ "height": 420,
+ "width": 420
+ },
+ "text/plain": {
+ "height": 420,
+ "width": 420
+ }
+ },
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "# 2 independent sample test for the mean (2-sample t-test)\n",
+ "# \n",
+ "# Situation: Want to see if a new drug X is more effective than the existing drug Y.\n",
+ "# We take 2 random groups of patients willing to participate in the study.\n",
+ "# n1 patients in group X receive the new treatment, n2 patients in group Y receive the old one.\n",
+ "# Let X_i be the observed time in days it takes patient X_i to heal, same for Y_j.\n",
+ "# We assume the same variance for both treatments (not guaranteed!)\n",
+ "\n",
+ "# 2 groups: n1 samples X_1..X_n1, n2 samples Y_1..Y_n2\n",
+ "\n",
+ "# Generate some data to test on\n",
+ "\n",
+ "n1=5\n",
+ "n2=5\n",
+ "sigma= 4\n",
+ "\n",
+ "X=rnorm(n = n1,mean = 18,sd = sigma) #new drug on average reduces healing time, but variance is large\n",
+ "Y=rnorm(n = n2,mean = 21,sd = sigma) #sd are equal, as assumed\n",
+ "\n",
+ "histx <- hist(X, breaks = seq(0,40,by=1),plot=\"FALSE\") # centered at 4\n",
+ "histy <- hist(Y, breaks = seq(0,40,by=1),plot=\"FALSE\") # centered at 6\n",
+ "plot( histx, col=rgb(0,0,1,1/4), xlim=c(0,40), ylim=c(0,max(c(histx$counts,histy$counts))), xlab = \"Days until cured\", main = \"Compare groups A and B\") # first histogram\n",
+ "plot( histy, col=rgb(1,0,0,1/4), xlim=c(0,40), add=T) # second\n",
+ "\n",
+ "#2) Research hypothesis: new drug is better than old drug (which we know is true)\n",
+ "#H1: mu_x < mu_y -->> H0: mu_x >= mu_y\n",
+ "\n",
+ "#3) significance\n",
+ "alpha=0.05\n",
+ "\n",
+ "#4) Compute test statistic\n",
+ "S= sqrt( ((n1-1)*sd(X)^2+(n2-1)*sd(Y)^2)/(n1+n2-2) )\n",
+ "T= (mean(X)-mean(Y))/S * sqrt( (n1*n2)/(n1+n2))\n",
+ "\n",
+ "#5) Critical region\n",
+ "\n",
+ "#H0 will have to be rejected if mu_x K, reject H0, accept H1, your drug is better than the old at the significance level alpha.\n",
+ "\n",
+ "# Show all output:\n",
+ "\n",
+ "paste('Pooled Variance: ',S^2, ' True Variance: ',sigma^2 )\n",
+ "paste('T :', T, ' Critical T: ', TK)\n",
+ "if (T'measured chi^2: 12.8 critical chi^2: 11.0704976935164'"
+ ],
+ "text/latex": [
+ "'measured chi\\textasciicircum{}2: 12.8 critical chi\\textasciicircum{}2: 11.0704976935164'"
+ ],
+ "text/markdown": [
+ "'measured chi^2: 12.8 critical chi^2: 11.0704976935164'"
+ ],
+ "text/plain": [
+ "[1] \"measured chi^2: 12.8 critical chi^2: 11.0704976935164\""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "image/png": 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LWHQuA54DinqtwbpPL1vWB5T/iWdv8l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\nQbAAWINgAbAGwQJgDYIFwBoEC4A1/g+VntJOtE3iCQAAAABJRU5ErkJggg==",
+ "text/plain": [
+ "Plot with title “Histogram of sample(1:6, n, replace = T, prob = probabilities)”"
+ ]
+ },
+ "metadata": {
+ "image/png": {
+ "height": 390,
+ "width": 600
+ },
+ "text/plain": {
+ "height": 390,
+ "width": 600
+ }
+ },
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "# chi^2 testing\n",
+ "\n",
+ "# Example: The unfair dice\n",
+ "\n",
+ "# Generate some data for testing:\n",
+ "n = 10 # number of dice throw\n",
+ "\n",
+ "# Probabilities : Vector of p_i. i-th number: probability to find i on dice\n",
+ "#probabilities <- c(1/6,1/6,1/6,1/6,1/6,1/6) # a fair die\n",
+ "probabilities <- c(1/10,1/10,1/10,1/10,3/10,3/10) # an unfair die, lots of 5 and 6\n",
+ "\n",
+ "\n",
+ "\n",
+ "mydata=hist(sample(1:6,n,replace=T,prob=probabilities), breaks = seq(0.5,6.5,by=1)) # role die n times, count observations.\n",
+ "# 1) Hypothesis\n",
+ "\n",
+ "#H_1: This dice is unfair! H_0: the dice is fair. Under H_0, all p_i = 1/6.\n",
+ "# Is there significant evidence that the observed numbers deviate from what we expect if H_0 was valid?\n",
+ "\n",
+ "# 2) Significance\n",
+ "alpha=0.05\n",
+ "\n",
+ "# 3) Compute test statistic\n",
+ "\n",
+ "chisquare = 0\n",
+ "\n",
+ "for (i in seq(1,6,by=1)) { chisquare = chisquare + (mydata$counts[i]-n*1/6)^2/(n*1/6) }\n",
+ "\n",
+ "# 4) Critical region\n",
+ "# Reject H_0 if chisquare > chi_5(1-alpha)\n",
+ "\n",
+ "K= qchisq(1-alpha,5)\n",
+ "\n",
+ "# 5) decide\n",
+ "\n",
+ "paste('measured chi^2: ',chisquare, ' critical chi^2: ',K )\n",
+ "\n",
+ "\n",
+ "# Repeat in R\n",
+ "\n",
+ "#chisq.test(x=mydata$counts,p = c(1/6,1/6,1/6,1/6,1/6,1/6))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 30,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "\n",
+ "\tChi-squared test for given probabilities\n",
+ "\n",
+ "data: mydata$counts\n",
+ "X-squared = 51.44, df = 5, p-value = 7.027e-10\n"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "# Repeat in R\n",
+ "\n",
+ "chisq.test(x=mydata$counts,p = c(1/6,1/6,1/6,1/6,1/6,1/6))"
+ ]
+ },
+ {
+ "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
+}
diff --git a/Lecture 8/Demonstrate 2sampleT.ipynb b/Lecture 8/Demonstrate 2sampleT.ipynb
new file mode 100644
index 0000000..b4901bb
--- /dev/null
+++ b/Lecture 8/Demonstrate 2sampleT.ipynb
@@ -0,0 +1,75 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "options(repr.plot.width=10, repr.plot.height=6.5) # this command just formats the size of the figures. Adapt to view them nicely\n",
+ " # in your browser."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Demonstrate the distribution of the 2-sample T-test"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "n1=5\n",
+ "n2=5\n",
+ "sigma= 4\n",
+ "n=10000\n",
+ "\n",
+ "for (i in seq(1,n,by=1)) \n",
+ " {\n",
+ "X=rnorm(n = n1,mean = 18,sd = sigma) \n",
+ "Y=rnorm(n = n2,mean = 18,sd = sigma) \n",
+ "S= sqrt( ((n1-1)*sd(X)^2+(n2-1)*sd(Y)^2)/(n1+n2-2) )\n",
+ "T[i] = (mean(X)-mean(Y))/S * sqrt( (n1*n2)/(n1+n2))\n",
+ " }"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "Trange=seq(min(T),max(T),length.out=100)\n",
+ "\n",
+ "hist(T, probability=\"TRUE\", xlim=c(-10,10), breaks=Trange)\n",
+ "lines(seq(-10,10,by=0.05),dt(seq(-10,10,by=0.05),n1+n2-2),col=\"red\")"
+ ]
+ },
+ {
+ "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
+}