part3_Lesson08-09-10-Statistics_and_Probabilities.ipynb

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   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Put these at the top of every notebook, to get automatic reloading and inline plotting\n",
    "%reload_ext autoreload\n",
    "%autoreload 2\n",
    "%matplotlib inline\n",
    "from IPython.core.interactiveshell import InteractiveShell\n",
    "InteractiveShell.ast_node_interactivity = \"all\"\n",
    "def pv(variable):\n",
    "    print(variable, \":\", eval(variable))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "# Please note again that we are using 0 to represent heads, and 1 to represent tails."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## simulate 1 million tests of two fair coin flips"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.250494"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# simulate 1 million tests of two fair coin flips\n",
    "tests = np.random.randint(2, size=(int(1e6), 2))\n",
    "\n",
    "# sums of all tests\n",
    "test_sums = tests.sum(axis=1)\n",
    "\n",
    "# proportion of tests that produced exactly two heads\n",
    "(test_sums == 0).mean()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Mathematics verification"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.25"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_h = 0.5\n",
    "p_t = p_h\n",
    "flips = 2\n",
    "p_two_heads = p_h**flips\n",
    "p_two_heads"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 1, 1],\n",
       "       [0, 0, 0],\n",
       "       [1, 1, 0],\n",
       "       ...,\n",
       "       [0, 1, 0],\n",
       "       [0, 1, 0],\n",
       "       [0, 1, 1]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "array([2, 0, 2, ..., 1, 1, 2])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "array([False, False, False, ...,  True,  True, False])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "375337"
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     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       "375337"
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     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
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     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
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    {
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       "0.375337"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# simulate 1 million tests of three fair coin flips\n",
    "tests = np.random.randint(2, size=(int(1e6), 3))\n",
    "tests\n",
    "\n",
    "\n",
    "# sums of all tests\n",
    "test_sums = tests.sum(axis=1)\n",
    "test_sums\n",
    "\n",
    "\n",
    "# proportion of tests that produced exactly one head\n",
    "# One head on 3 flips = 1 + 0 + 0 = 1\n",
    "one_head = (test_sums == 1)\n",
    "one_head\n",
    "np.bincount(one_head)[1]\n",
    "(one_head == True).sum()\n",
    "np.count_nonzero(one_head == True)\n",
    "#len(test_sums == 1)\n",
    "p_one_head=np.bincount(one_head)[1]/(np.bincount(one_head)[0]+np.bincount(one_head)[1])\n",
    "p_one_head\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Mathematics verification - Binomial Distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.375"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "0.375"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import math\n",
    "p_h = 0.5\n",
    "p_t = p_h\n",
    "n_flips = 3\n",
    "k_heads = 1\n",
    "\n",
    "def binomial_distribution(p, n, k):\n",
    "    return math.factorial(n) / ( math.factorial(n - k) * math.factorial(k)) * p**k * (1-p)**(n-k)\n",
    "\n",
    "p_one_head = math.factorial(n_flips) / ( math.factorial(n_flips - k_heads) * math.factorial(k_heads)) * p_h**k_heads * (1-p_h)**(n_flips-k_heads)\n",
    "p_one_head\n",
    "binomial_distribution(p_h, n_flips,  k_heads)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Three biased coin flips with P(H) = 0.6 produce exactly one head"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 0, 1],\n",
       "       [1, 1, 0],\n",
       "       [1, 0, 0],\n",
       "       ...,\n",
       "       [0, 0, 0],\n",
       "       [1, 1, 1],\n",
       "       [1, 1, 1]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "array([1, 2, 1, ..., 0, 3, 3])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "0.28784"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# simulate 1 million tests of three biased coin flips\n",
    "# hint: use np.random.choice()\n",
    "tests = np.random.choice(a=2, size=(int(1e6), 3), p=[0.4, 0.6])\n",
    "tests\n",
    "# sums of all tests\n",
    "test_sums = tests.sum(axis=1)\n",
    "test_sums\n",
    "# proportion of tests that produced exactly one head\n",
    "one_head = (test_sums == 1)\n",
    "p_one_head=np.bincount(one_head)[1]/(np.bincount(one_head)[0]+np.bincount(one_head)[1])\n",
    "p_one_head"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Mathematics verification"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.28800000000000003"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "binomial_distribution(0.6, 3, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. A die rolls an even number"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1],\n",
       "       [4],\n",
       "       [5],\n",
       "       ...,\n",
       "       [4],\n",
       "       [6],\n",
       "       [6]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "3.498754"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
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    {
     "data": {
      "text/plain": [
       "array([1, 0, 1, ..., 0, 0, 0])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "array([499574, 500426])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "499574"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "500426"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "0.500426"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# simulate 1 million tests of one die roll\n",
    "#tests = np.random.choice(6, size=(int(1e6),1), p=[0.5, 0.1, 0.1, 0.1, 0.1, 0.1])+1\n",
    "tests = np.random.randint(6, size=(int(1e6),1))+1\n",
    "tests\n",
    "tests.min()\n",
    "tests.max()\n",
    "tests.mean()\n",
    "# proportion of tests that produced an even (pair) number\n",
    "odd = ( tests & 0x1)\n",
    "odd = odd.flatten()\n",
    "odd\n",
    "even_odd = np.bincount(odd)\n",
    "even_odd\n",
    "odd_number = even_odd[0]\n",
    "odd_number\n",
    "even_number = even_odd[1]\n",
    "even_number\n",
    "p_even = even_number / (even_number + odd_number)\n",
    "p_even"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Mathematics verification"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 1 , 2, 3, 4, 5, 6\n",
    "p_even = 3/6\n",
    "p_even"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5. Two dice roll a double"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([833078, 166922])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "0.166922"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# simulate the first million die rolls\n",
    "first = np.random.randint(6, size=(int(1e6),1))+1\n",
    "\n",
    "# simulate the second million die rolls\n",
    "second = np.random.randint(6, size=(int(1e6),1))+1\n",
    "\n",
    "# proportion of tests where the 1st and 2nd die rolled the same number\n",
    "tests = (first == second).flatten()\n",
    "np.bincount(tests)\n",
    "p_same = np.bincount(tests)[1] / (np.bincount(tests)[1] + np.bincount(tests)[0])\n",
    "p_same"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 1])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "array([2, 1])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "array([1, 2])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.bincount([True, False, False])\n",
    "np.bincount([False, True, False])\n",
    "np.bincount([True, True, False])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Mathematics verification"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.16666666666666666"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# It's the probability to roll a given number (the number of the first roll)\n",
    "p_same = 1/6\n",
    "p_same"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simulating Many Coin Flips"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# number of heads from 10 fair coin flips\n",
    "np.random.binomial(10, 0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([5, 4, 8, 7, 6, 6, 5, 4, 5, 4, 6, 6, 5, 6, 6, 7, 6, 4, 5, 6])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# results from 20 tests with 10 coin flips\n",
    "np.random.binomial(10, 0.5, 20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.8"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# mean number of heads from the 20 tests\n",
    "np.random.binomial(10, 0.5, 20).mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.998086"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# reflects the fairness of the coin more closely as # tests increases\n",
    "np.random.binomial(10, 0.5, 1000000).mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "plt.hist(np.random.binomial(10, 0.5, 1000000));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# gets more narrow as number of flips increase per test\n",
    "plt.hist(np.random.binomial(100, 0.5, 1000000));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Binomial Distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.499446"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# simulate 1 million tests of one fair coin flip\n",
    "# remember, the output of these tests are the # successes, or # heads\n",
    "tests = np.random.binomial(1, 0.5, int(1e6))\n",
    "\n",
    "# proportion of tests that produced heads\n",
    "(tests == 1).mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_h=1/2\n",
    "p_h"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 0, 1, ..., 2, 2, 3])"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "0.155633"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# simulate 1 million tests of five fair coin flips\n",
    "tests = np.random.binomial(5, 0.5, int(1e6))\n",
    "tests\n",
    "# proportion of tests that produced 1 head\n",
    "(tests == 1).mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.15625"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "binomial_distribution(0.5, 5, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.205427"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# simulate 1 million tests of ten fair coin flips\n",
    "tests = np.random.binomial(10, 0.5, int(1e6))\n",
    "\n",
    "# proportion of tests that produced 4 heads\n",
    "(tests == 4).mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.205078125"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "binomial_distribution(p=0.5, n=10, k=4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.327921"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# simulate 1 million tests of five biased coin flips\n",
    "tests = np.random.binomial(5, 0.8, int(1e6))\n",
    "\n",
    "# proportion of tests that produced 5 heads\n",
    "(tests == 5).mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.3276800000000001"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "binomial_distribution(0.8, 5, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.179355"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# simulate 1 million tests of ten biased coin flips\n",
    "tests = np.random.binomial(10, 0.15, int(1e6))\n",
    "\n",
    "# proportion of tests that produced at least 3 heads\n",
    "(tests >= 3).mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Prob of  3 : 0.1298337207539062  Total : 0.1298337207539062\n",
      "Prob of  4 : 0.0400957078798828  Total : 0.16992942863378901\n",
      "Prob of  5 : 0.008490855786328123  Total : 0.17842028442011715\n",
      "Prob of  6 : 0.001248655262695312  Total : 0.17966893968281247\n",
      "Prob of  7 : 0.00012591481640624995  Total : 0.17979485449921873\n",
      "Prob of  8 : 8.332598144531246e-06  Total : 0.17980318709736326\n",
      "Prob of  9 : 3.267685546874999e-07  Total : 0.17980351386591795\n",
      "Prob of  10 : 5.7665039062499975e-09  Total : 0.17980351963242186\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "0.17980351963242186"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_h = 0\n",
    "for heads in range(3,10+1):\n",
    "    p_heads = binomial_distribution(0.15, 10, heads)\n",
    "    p_h += p_heads\n",
    "    print(\"Prob of \", heads, \":\", p_heads, \" Total :\", p_h)\n",
    "p_h"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Cancer Test Results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>patient_id</th>\n",
       "      <th>test_result</th>\n",
       "      <th>has_cancer</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>79452</td>\n",
       "      <td>Negative</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>81667</td>\n",
       "      <td>Positive</td>\n",
       "      <td>True</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>76297</td>\n",
       "      <td>Negative</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   patient_id test_result  has_cancer\n",
       "0       79452    Negative       False\n",
       "1       81667    Positive        True\n",
       "2       76297    Negative       False"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "# number of patients 2914\n",
      "number of patients with cancer 306\n",
      "number of patients without cancer 2608\n",
      "proportion of patients with cancer 0.10501029512697323\n",
      "proportion of patients without cancer 0.8949897048730268\n",
      "proportion of patients with cancer who test positive 0.9052287581699346\n",
      "proportion of patients with cancer who test negative 0.09477124183006536\n",
      "proportion of patients without cancer who test positive 0.2036042944785276\n",
      "proportion of patients without cancer who test negative 0.7963957055214724\n"
     ]
    }
   ],
   "source": [
    "import pandas as pd\n",
    "# load dataset\n",
    "df = pd.read_csv(\"cancer_test_data.csv\")\n",
    "df.head(3)\n",
    "patients = len(df)\n",
    "print(\"# number of patients\",patients)\n",
    "\n",
    "with_cancer = len(df[(df['has_cancer'] == True)])\n",
    "print(\"number of patients with cancer\", with_cancer)\n",
    "no_cancer = len(df[(df['has_cancer'] == False)])\n",
    "print(\"number of patients without cancer\", no_cancer)\n",
    "print(\"proportion of patients with cancer\", with_cancer / patients)\n",
    "print(\"proportion of patients without cancer\", no_cancer / patients)\n",
    "pos_test_with_cancer = len(\n",
    "    df[(df['has_cancer'] == True)\n",
    "       & (df['test_result'] == 'Positive')]\n",
    "                  )\n",
    "print(\"proportion of patients with cancer who test positive\", pos_test_with_cancer / with_cancer)\n",
    "neg_test_with_cancer = len(    df[(df['has_cancer'] == True)   & (df['test_result'] == 'Negative')]         )\n",
    "print(\"proportion of patients with cancer who test negative\",neg_test_with_cancer / with_cancer)\n",
    "pos_test_no_cancer = len(    df[(df['has_cancer'] == False)   & (df['test_result'] == 'Positive')]         )\n",
    "print(\"proportion of patients without cancer who test positive\",pos_test_no_cancer / no_cancer)\n",
    "neg_test_no_cancer = len(    df[(df['has_cancer'] == False)   & (df['test_result'] == 'Negative')]         )\n",
    "print(\"proportion of patients without cancer who test negative\",neg_test_no_cancer / no_cancer)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Conditional Probability & Bayes Rule Quiz\n",
    "In the previous section, you found the following proportions from the cancer results dataset.\n",
    "\n",
    "    Patients with cancer: 0.105\n",
    "    Patients without cancer: 0.895\n",
    "    Patients with cancer who tested positive: 0.905\n",
    "    Patients with cancer who tested negative: 0.095\n",
    "    Patients without cancer who tested positive: 0.204\n",
    "    Patients without cancer who tested negative: 0.796\n",
    "\n",
    "    Based on the above proportions observed in the data, we can assume the following probabilities.\n",
    "    Probability \tMeaning\n",
    "    P(cancer) = 0.105 \tProbability a patient has cancer\n",
    "    P(~cancer) = 0.895 \tProbability a patient does not have cancer\n",
    "    P(positive|cancer) = 0.905 \tProbability a patient with cancer tests positive\n",
    "    P(negative|cancer) = 0.095 \tProbability a patient with cancer tests negative\n",
    "    P(positive|~cancer) = 0.204 \tProbability a patient without cancer tests positive\n",
    "    P(negative|~cancer) = 0.796 \tProbability a patient without cancer tests negative"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Probability a patient who tested positive has cancer, or P(cancer|positive) 0.34230291241151994\n",
      "Probability a patient who tested positive doesn't have cancer, or P(~cancer|positive) 0.65769708758848\n",
      "Probability a patient who tested negative has cancer, or P(cancer|negative) 0.013808235106832134\n",
      "Probability a patient who tested negative doesn't have cancer, or P(~cancer|negative) 0.986191764893168\n"
     ]
    }
   ],
   "source": [
    "p_cancer = 0.105\n",
    "p_no_cancer = 0.895\n",
    "p_pos_when_cancer = 0.905\n",
    "p_pos_whithout_cancer = 0.204\n",
    "p_pos_and_no_cancer = p_pos_whithout_cancer * p_no_cancer\n",
    "p_cancer_and_pos = p_pos_when_cancer * p_cancer\n",
    "p_pos = p_cancer * p_pos_when_cancer + p_no_cancer * p_pos_whithout_cancer\n",
    "p_cancer_when_pos = p_cancer_and_pos / p_pos\n",
    "print(\"Probability a patient who tested positive has cancer, or P(cancer|positive)\", p_cancer_when_pos)\n",
    "p_no_cancer_when_pos = p_pos_and_no_cancer / p_pos\n",
    "print(\"Probability a patient who tested positive doesn't have cancer, or P(~cancer|positive)\", p_no_cancer_when_pos)\n",
    "\n",
    "p_neg_given_cancert = 0.095\n",
    "p_neg_given_no_cancert = 0.796\n",
    "p_neg = p_cancer * p_neg_given_cancert + p_no_cancer * p_neg_given_no_cancert\n",
    "p_neg_and_cancer = p_cancer * p_neg_given_cancert\n",
    "p_cancer_when_neg = p_neg_and_cancer / p_neg\n",
    "print(\"Probability a patient who tested negative has cancer, or P(cancer|negative)\", p_cancer_when_neg)\n",
    "p_neg_and_no_cancer = p_neg_given_no_cancert * p_no_cancer\n",
    "p_no_cancer_when_neg = p_neg_and_no_cancer / p_neg\n",
    "print(\"Probability a patient who tested negative doesn't have cancer, or P(~cancer|negative)\", p_no_cancer_when_neg)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Or more simple"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.013770180436847104"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.query('test_result == \"Negative\"')['has_cancer'].mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Normal Distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Sampling Distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Find the proportion of students who drink coffee in the above array. Store this value in a variable p.\n",
      "Use numpy's random.choice to simulate 5 draws from the students array.     What is proportion of your sample drink coffee?\n",
      "Repeat the above to obtain 10,000 additional proportions, where each sample was of size 5.       Store these in a variable called sample_props.\n",
      "What is the mean proportion of all 10,000 of these proportions ?     This is often called the mean of the sampling distribution.\n",
      "What are the variance and standard deviation for the original 21 data values?\n",
      "mean 0.7142857142857143 var 0.20408163265306126 std 0.45175395145262565\n",
      "What are the variance and standard deviation for the 10,000 proportions you created?\n",
      "mean 0.714 var 0.20420400000000002 std 0.45188936699152377\n",
      "\n",
      "7.Compute p(1-p), which of your answers does this most closely match?\n",
      "p * (1-p) 0.20408163265306123\n",
      "p_sample * (1-p_sample) 0.24\n",
      "\n",
      "8. Compute p(1-p)/n, which of your answers does this most closely match?\n",
      "p * (1-p)/n_sample 0.04081632653061225\n",
      "9. Notice that your answer to 8. is commonly called the variance of the sampling distribution.\n",
      "\n",
      "If you were to change your first sample to be 20,     what would this do for the variance of the sampling distribution?     Simulate and calculate the new answers in 6. and 8.    to check that the consistency you found before still holds.\n",
      "p_sample (ie mean) 0.25\n",
      "p * (1-p)/n_sample 0.010204081632653062\n",
      "mean 0.25 var 0.1875 std 0.4330127018922193\n",
      "\n",
      "Repeat the above to obtain 10,000 additional proportions, where each sample was of size 5. Store these in a variable called sample_props.\n",
      "10. Finally, plot a histgram of the 10,000 draws from both the proportions with a sample size of 5 and the proportions with a sample size of 20. Each of these distributions is a sampling distribution. One is for the proportions of sample size 5 and the other a sampling distribution for proportions with sample size 20.\n",
      "\n",
      "students\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "np.random.seed(42)\n",
    "\n",
    "students = np.array([1,0,1,1,1,1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,0])\n",
    "n = students.shape[0]\n",
    "n\n",
    "print(\"Find the proportion of students who drink coffee in the above array. Store this value in a variable p.\")\n",
    "p = (students == 1).sum() / students.shape[0]\n",
    "# Or shorter :\n",
    "p = students.mean()\n",
    "p\n",
    "print(\"Use numpy's random.choice to simulate 5 draws from the students array. \\\n",
    "    What is proportion of your sample drink coffee?\")\n",
    "indexes = np.random.choice(students, size=5)\n",
    "sample = students[indexes]\n",
    "n_sample = sample.shape[0]\n",
    "p_sample = (sample == 1).sum() / sample.shape[0]\n",
    "p_sample\n",
    "print(\"Repeat the above to obtain 10,000 additional proportions, where each sample was of size 5.\\\n",
    "       Store these in a variable called sample_props.\")\n",
    "sample_props = np.random.choice(students, size=(10000,5))\n",
    "sample_props\n",
    "print(\"What is the mean proportion of all 10,000 of these proportions ? \\\n",
    "    This is often called the mean of the sampling distribution.\")\n",
    "sample_props.mean()\n",
    "print(\"What are the variance and standard deviation for the original 21 data values?\")\n",
    "print(\"mean\", students.mean(), \"var\", students.var(), \"std\", students.std())\n",
    "print(\"What are the variance and standard deviation for the 10,000 proportions you created?\")\n",
    "print(\"mean\", sample_props.mean(), \"var\", sample_props.var(), \"std\", sample_props.std())\n",
    "print(\"\\n7.Compute p(1-p), which of your answers does this most closely match?\")\n",
    "print(\"p * (1-p)\", p * (1-p) )\n",
    "print(\"p_sample * (1-p_sample)\", p_sample * (1-p_sample))\n",
    "\n",
    "print(\"\\n8. Compute p(1-p)/n, which of your answers does this most closely match?\")\n",
    "#print(\"p * (1-p)/n\", p * (1-p)/n)\n",
    "#print(\"p_sample * (1-p_sample)/n_sample\", p_sample * (1-p_sample)/n_sample)\n",
    "print(\"p * (1-p)/n_sample\", p * (1-p)/n_sample)\n",
    "\n",
    "print(\"9. Notice that your answer to 8. is commonly called the variance of the sampling distribution.\")\n",
    "\n",
    "\n",
    "\n",
    "print(\"\\nIf you were to change your first sample to be 20, \\\n",
    "    what would this do for the variance of the sampling distribution? \\\n",
    "    Simulate and calculate the new answers in 6. and 8.\\\n",
    "    to check that the consistency you found before still holds.\")\n",
    "sample_20 = students[np.random.choice(students, size=20)]\n",
    "n_sample = sample_20.shape[0]\n",
    "p_sample = sample_20.mean()\n",
    "print(\"p_sample (ie mean)\", p_sample)\n",
    "print(\"p * (1-p)/n_sample\", p * (1-p)/n_sample)\n",
    "\n",
    "print(\"mean\", sample_20.mean(), \"var\", sample_20.var(), \"std\", sample_20.std())\n",
    "print(\"\\nRepeat the above to obtain 10,000 additional proportions, where each sample was of size 5. Store these in a variable called sample_props.\")\n",
    "sample_props = np.random.choice(students, size=(10000,5))\n",
    "sample_props\n",
    "\n",
    "print(\"10. Finally, plot a histgram of the 10,000 draws from both the proportions with a sample size of 5 and the proportions with a sample size of 20. Each of these distributions is a sampling distribution. One is for the proportions of sample size 5 and the other a sampling distribution for proportions with sample size 20.\")\n",
    "print(\"\")\n",
    "\n",
    "print(\"students\")\n",
    "plt.hist(students);\n",
    "plt.show();\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sample_props\n"
     ]
    },
    {
     "data": {
      "image/png": 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mFrRzkqTvO9sAeLlN7dCej7b6JLB2Wrs1wJEz1E9TVXuqakNVbRgbGzvL7kmS5nK2AXAAOHknz3bg/mn1m9rdQBuBE22K6EFgc5JV7eLv5laTJC2RFXM1SPJ54MPAhUkmGdzN82lgf5IdwAvAja35A8B1wATwKnAzQFUdS3Ib8Fhrd2tVnXphWZI0QnMGQFV9dJZNm2ZoW8CuWY6zF9g7r95JkhaNnwSWpE4ZAJLUKQNAkjplAEhSpwwASeqUASBJnTIAJKlTBoAkdcoAkKROGQCS1CkDQJI6ZQBIUqcMAEnqlAEgSZ0yACSpUwaAJHXKAJCkThkAktQpA0CSOmUASFKnRh4ASbYkeTbJRJJbRv36kqSBkQZAkvOAzwDXApcDH01y+Sj7IEkaGPUZwJXARFU9V1V/DtwDbB1xHyRJjD4AVgMvTlufbDVJ0oilqkb3YsmNwDVV9Tfa+i8CV1bVx6e12QnsbKvvAZ6dx0tcCPzZAnV3uel17I67L457OD9SVWNzNVpx9v05K5PA2mnra4Aj0xtU1R5gz9kcPMl4VW04++4tX72O3XH3xXEvrFFPAT0GrE9yWZLzgW3AgRH3QZLEiM8Aquq1JH8HeBA4D9hbVU+Nsg+SpIFRTwFRVQ8ADyzS4c9q6uhNotexO+6+OO4FNNKLwJKkNw6/CkKSOrUsA2Cur5NI8rYk97btjyZZN/peLrwhxv33kzyd5Mkkh5L8yFL0c6EN+/UhST6SpJK8Ke4SGWbcSX6u/cyfSvJ7o+7jYhnid/3SJA8neaL9vl+3FP1cSEn2Jjma5OuzbE+SO9q/yZNJrjjnF62qZfVgcPH4G8C7gfOBPwIuP6XN3wY+25a3Afcudb9HNO6fAX6gLf9SL+Nu7d4BfBl4BNiw1P0e0c97PfAEsKqtX7TU/R7h2PcAv9SWLweeX+p+L8C4fxq4Avj6LNuvA/4TEGAj8Oi5vuZyPAMY5usktgL72vJ9wKYkGWEfF8Oc466qh6vq1bb6CIPPWSx3w359yG3AvwC+PcrOLaJhxv03gc9U1XGAqjo64j4ulmHGXsAPt+V3csrniZajqvoycOwMTbYCd9fAI8DKJJecy2suxwAY5uskvtemql4DTgDvGknvFs98v0ZjB4N3C8vdnONO8kFgbVV9cZQdW2TD/Lx/DPixJP8tySNJtoysd4trmLH/GvALSSYZ3FX4cd78FvyrdEZ+G+gCmOmd/Km3Mg3TZrkZekxJfgHYAPzlRe3RaJxx3EneAtwOfGxUHRqRYX7eKxhMA32Ywdnef0nyk1X1yiL3bbENM/aPAr9TVf8qyU8Bn2tj/4vF796SWfC/a8vxDGDOr5OY3ibJCganiGc6tVoOhhk3SX4W+MfA9VX1nRH1bTHNNe53AD8JfCnJ8wzmRg+8CS4ED/t7fn9V/b+q+iaD781aP6L+LaZhxr4D2A9QVf8deDuD78t5Mxvqb8B8LMcAGObrJA4A29vyR4CHql1FWcbmHHebCvk3DP74v1nmg8847qo6UVUXVtW6qlrH4NrH9VU1vjTdXTDD/J7/RwYX/klyIYMpoedG2svFMczYXwA2ASR5L4MAmBppL0fvAHBTuxtoI3Ciql46lwMuuymgmuXrJJLcCoxX1QHgLganhBMM3vlvW7oeL4whx/0vgR8Cfr9d836hqq5fsk4vgCHH/aYz5LgfBDYneRr4LvAPqupbS9frhTHk2H8F+LdJ/h6DaZCPLfc3eUk+z2A678J2bWM38FaAqvosg2sd1wETwKvAzef8msv830ySdJaW4xSQJGkBGACS1CkDQJI6ZQBIUqcMAEnqlAEgSZ0yACSpUwaAJHXq/wNEWMa3NHDSjgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(\"sample_props\")\n",
    "plt.hist(sample_props);\n",
    "plt.show();\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sample_20\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(\"sample_20\")\n",
    "plt.hist(sample_20);\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Notation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([1663.,  747.,  329.,  134.,   73.,   34.,   13.,    5.,    0.,\n",
       "           2.]),\n",
       " array([1.16348231e-03, 8.17255031e+01, 1.63449843e+02, 2.45174182e+02,\n",
       "        3.26898522e+02, 4.08622862e+02, 4.90347201e+02, 5.72071541e+02,\n",
       "        6.53795881e+02, 7.35520220e+02, 8.17244560e+02]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "pop_data.mean()\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "100.35978700795846"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sample_10.mean()\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "103.47556549464393"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " sample standard deviation of your 10 draws\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "130.00074035573317"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "population standard deviation\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "99.77860187968906"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "population variance\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "9955.76939306549"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sample variance\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "16900.192493038754"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.seed(42)\n",
    "pop_data = np.random.gamma(1,100, 3000)\n",
    "plt.hist(pop_data);\n",
    "plt.show();\n",
    "print(\"pop_data.mean()\")\n",
    "pop_data.mean()\n",
    "print(\"sample_10.mean()\")\n",
    "sample_10 = np.random.choice(pop_data, size=10)\n",
    "sample_10.mean()\n",
    "print(\" sample standard deviation of your 10 draws\")\n",
    "sample_10.std()\n",
    "print(\"population standard deviation\")\n",
    "pop_data.std()\n",
    "print(\"population variance\")\n",
    "pop_data.var()\n",
    "print(\"sample variance\")\n",
    "sample_10.var()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Central Limit Theorem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "%matplotlib inline\n",
    "np.random.seed(42)\n",
    "\n",
    "pop_data = np.random.gamma(1,100,3000)\n",
    "plt.hist(pop_data);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. In order to create the sampling distribution for the average of 3 draws of this distribution, follow these steps:\n",
    "\n",
    "a. Use numpy's random.choice to simulate 3 draws from the pop_data array.\n",
    "\n",
    "b. Compute the mean of these 3 draws.\n",
    "\n",
    "c. Write a loop to simulate this process 10,000 times, and store each mean into an array called means_size_3.\n",
    "\n",
    "d. Plot a histogram of your sample means.\n",
    "\n",
    "e. Use means_size_3 and pop_data to answer the quiz questions below.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([2.033e+03, 3.687e+03, 2.476e+03, 1.133e+03, 4.720e+02, 1.350e+02,\n",
       "        4.200e+01, 1.700e+01, 3.000e+00, 2.000e+00]),\n",
       " array([  3.33913192,  52.06020861, 100.78128531, 149.502362  ,\n",
       "        198.2234387 , 246.94451539, 295.66559209, 344.38666878,\n",
       "        393.10774548, 441.82882217, 490.54989887]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "pop_data.mean() : 100.35978700795846\n",
      "means_size_3.mean() : 100.87276259567305\n",
      "pop_data.std() : 99.77860187968906\n",
      "means_size_3.std() : 57.72348461511113\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "sample = np.random.choice(pop_data, size=3)\n",
    "#sample\n",
    "means_size_3 = np.array([])\n",
    "for i in range(0,10000):\n",
    "    means_size_3 = np.append(means_size_3, np.random.choice(pop_data, size=3).mean())\n",
    "#pv(\"means_size_3\")\n",
    "plt.hist(means_size_3);\n",
    "pv(\"pop_data.mean()\")\n",
    "pv(\"means_size_3.mean()\")\n",
    "pv(\"pop_data.std()\")\n",
    "pv(\"means_size_3.std()\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Central Limit Theorem - Part II"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([1663.,  747.,  329.,  134.,   73.,   34.,   13.,    5.,    0.,\n",
       "           2.]),\n",
       " array([1.16348231e-03, 8.17255031e+01, 1.63449843e+02, 2.45174182e+02,\n",
       "        3.26898522e+02, 4.08622862e+02, 4.90347201e+02, 5.72071541e+02,\n",
       "        6.53795881e+02, 7.35520220e+02, 8.17244560e+02]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sample.mean() : 115.89760480820178\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(array([  22.,  167.,  815., 2205., 2924., 2297., 1158.,  326.,   76.,\n",
       "          10.]), array([ 65.40255233,  72.93722432,  80.47189631,  88.00656829,\n",
       "         95.54124028, 103.07591227, 110.61058426, 118.14525625,\n",
       "        125.67992824, 133.21460023, 140.74927222]), <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "pop_data.mean() : 100.35978700795846\n",
      "means_size_100.mean() : 100.45707456518824\n",
      "pop_data.std() : 99.77860187968906\n",
      "means_size_100.std() : 9.971979588536067\n",
      "pop_data.var() : 9955.76939306549\n",
      "means_size_100.var() : 99.44037691417994\n"
     ]
    }
   ],
   "source": [
    "np.random.seed(42)\n",
    "\n",
    "pop_data = np.random.gamma(1,100,3000)\n",
    "plt.hist(pop_data);\n",
    "plt.show()\n",
    "\n",
    "sample = np.random.choice(pop_data, size=100)\n",
    "pv(\"sample.mean()\")\n",
    "\n",
    "means_size_100 = np.array([])\n",
    "for i in range(0,10000):\n",
    "    means_size_100 = np.append(means_size_100, np.random.choice(pop_data, size=100).mean())\n",
    "#pv(\"means_size_3\")\n",
    "plt.hist(means_size_100);\n",
    "plt.show()\n",
    "pv(\"pop_data.mean()\")\n",
    "pv(\"means_size_100.mean()\")\n",
    "pv(\"pop_data.std()\")\n",
    "pv(\"means_size_100.std()\")\n",
    "pv(\"pop_data.var()\")\n",
    "pv(\"means_size_100.var()\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Central Limit Theorem - Part III"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([1663.,  747.,  329.,  134.,   73.,   34.,   13.,    5.,    0.,\n",
       "           2.]),\n",
       " array([1.16348231e-03, 8.17255031e+01, 1.63449843e+02, 2.45174182e+02,\n",
       "        3.26898522e+02, 4.08622862e+02, 4.90347201e+02, 5.72071541e+02,\n",
       "        6.53795881e+02, 7.35520220e+02, 8.17244560e+02]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sample.var() : 12098.749344943082\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(array([ 176., 1338., 2857., 2728., 1695.,  758.,  315.,   91.,   32.,\n",
       "          10.]),\n",
       " array([ 3615.72815393,  5472.35978523,  7328.99141652,  9185.62304782,\n",
       "        11042.25467911, 12898.88631041, 14755.51794171, 16612.149573  ,\n",
       "        18468.7812043 , 20325.41283559, 22182.04446689]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "pop_data.var() : 9955.76939306549\n",
      "var_size_100.var() : 6508086.114509583\n",
      "pop_data.std() : 99.77860187968906\n",
      "var_size_100.mean() : 9874.361793621825\n",
      "pop_data.var() : 9955.76939306549\n",
      "var_size_100.var() : 6508086.114509583\n"
     ]
    }
   ],
   "source": [
    "np.random.seed(42)\n",
    "\n",
    "pop_data = np.random.gamma(1,100,3000)\n",
    "plt.hist(pop_data);\n",
    "plt.show()\n",
    "sample = np.random.choice(pop_data, size=100)\n",
    "pv(\"sample.var()\")\n",
    "\n",
    "var_size_100 = np.array([])\n",
    "for i in range(0,10000):\n",
    "    var_size_100 = np.append(var_size_100, np.random.choice(pop_data, size=100).var())\n",
    "#pv(\"means_size_3\")\n",
    "plt.hist(var_size_100);\n",
    "plt.show()\n",
    "pv(\"pop_data.var()\")\n",
    "pv(\"var_size_100.var()\")\n",
    "pv(\"pop_data.std()\")\n",
    "pv(\"var_size_100.mean()\")\n",
    "pv(\"pop_data.var()\")\n",
    "pv(\"var_size_100.var()\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bootstrap Sampling"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([4, 5, 3, 5, 5, 2, 3, 3, 3, 5, 4, 3, 6, 5, 2, 4, 6, 6, 2, 4])"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Confidence Intervals for the Population Mean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "nb_drinks_coffee/200 : 0.595\n",
      "1-nb_drinks_coffee/200 : 0.405\n",
      "drinks_coffee['height'].mean() : 68.11962990858618\n",
      "coffee_red[(coffee_red['drinks_coffee'] == False)]['height'].mean() : 66.78492279927877\n",
      "nb_drinks_coffee/200 : 0.605\n",
      "1-nb_drinks_coffee/200 : 0.395\n",
      "drinks_coffee['height'].mean() : 68.25918199231297\n",
      "coffee_new[(coffee_new['drinks_coffee'] == False)]['height'].mean() : 66.28588551349722\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f2172c83630>]"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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e3YhfJvp/GIxjXCe35XVKJ8S/L4lUGnbHq64YYyoGsKes2MxzXy/ztZ9YrW+7HpMx7t+dRD3nkk3OCb8TXm6Tl6Bu0YOAbnFbnvDCGjv2uu8hpDtA18MfL65io3Yi0wUqyPI53Zfvi0MD3B98f9DLafvesqQ0KmIROdfg1Smrq/jlz3SxwpoXvli0gcLhH1aZsPjYZ0s46u6x7NpXxoXPTuLBjxZ62m8QmpzIPjLB3JNVwj915WYKh3/I2q3xZ7Y63big7skH39tGn7Y5XmJHTLbr3oC/f1HhdlaBQ5Ef/XQxhcM/ZHq0Pd4mfaT75mUvTGbYS/EHruf79PDp/sDHvGNNmY+871XKmQDJauD94+NFtgOFU1du5gcXawAEWa6LIrxzIsecFlozcl/6blVwBwNemhTa35wfKt/316aGbOd+G1V+MSa6tx07bSaTVcIfjicSy5YcqUHJbPne+PLBafWrN+3GqQ2fjta3m8om8sH9u8vwBqMmhAa/zn96oqcwxt/EGPCMZJlPW/IWh+Bq5z890VXs+cjrleg8i1jZw8fZu/9gZf7450srzdR1zOux/eCn1Rnr+u/dX+6p1+mF8gMm7WsVh9ldGvLEWbohtpdZ9G1evjF07TKlQsgq4ffyUkYKbkwhtPmp/IDhvKe+de3G9vs3va9oFHku0V3d8fPWsy2NkSKdrlbk1Y+MERRILypi57e8Oovvi4Pxgb7gGfc+5sl+acPPZJBB9JyKHPS5nPPEBD6Z7y10RDw/+vA7cPc7czn+Tx8nPDny7ZkhN2834THsiKws/zNppe9y2L0Py0p20n/E52zYEd9aEQRZJfxuiLx53y4NxRTx0+qeuXork1c4x9W3ObLD1viy2PvBz6ps++en/pfh83O+z361LL5pIYnCGO2C93CSlyGMJPKZiR7fiT7lLbtLUxL9NGg78b6yxGb8Ll7vfZ7FGJsJZuUHTJX5N+GQ4okOmIYtAuHQIF6v4di5iS/d6XTMURNW8MPWPYwP4BhuyCrhj34svNjPw60LJ7OAEz9uC03iiDWwlYpJYEET/YD+bexCx99s8wdUjpXWxKTocYYaSbykpWUHKBz+IX8fX3nQMJbshK/J7tJyTn7IewC7ZA34xXNLDNPpnnGc5KPciWC3GtrfPlrACQ9+xiabVnmyelwlO/b5CqwYi1i9E7tfwulHjPU2UO2XrBJ+O4wxtouQuH6I4gzghO3To+PGK/f21G7fW8Z/Jq5MSuyeyIqotOwAw9/8Pm58GDc4uio6qNrAh7+k3V1VZz5C1Yoj0nQEoXGTeHZWv+y1Ii/+d2IoBPS0lVUr9ejWX+RtSkXM9ejGxOcLN/i+h98sCXYd4kT4fGEo8urWPft9dSDXbN7NQ+MWunpvwtfwmv9O4/Y3ZrsKdx5m7g/bfJuM7Ag/y7tKUzObN6uEP/peGwxPfbmMXn/9tCLmi50IZWKL/PHPlvDH9+Yx1UZ0Ioku+w9b9/CfiStj5ok09Xy5aAOvTl1Dn799lvCC4ZXiH7lowi7fuMt3K275xl0M+sdX/jJHEat19uaMHyqm1kee34qN0QOdlfexZVcp/f5W1USXLJ7+chnnPjmhynanMxsXUXEFsmZAEvDzbNwwegZPfbnMlempeMtu/vTePH60TJhlHsYQFq/fyT8+8W5uXFGyy1ZtUj0zPbuE32Zb2CMlGa3DyAdzzPTipLTO48X9jrbXX/7CZP743jzX+18f0Rt6Y9rBKeaxTiXS66RyWQ7yypTVFYPffqpVr1dyTwItpeiKMvLcV8WIubJlV6njuZ3x6Nes9dAC9/ro2NWr4XAkbgi7t2Yqbq7H7VGxiPaXH6hw/Xx5clXX0uh93vLqLP49cWXccOBBcs1/g100xy9ZJfyxuOXVWfzvu1WBhoidXby10osfGSgrEpHg7ZM79pZROPxDbopYjQnc+TZH9hLufcffoi7xmL5qCw+8H1rFyc+pe71eibjFluzcx7SVm6vG+ZfKJrzoI1zy/HeO+0y2mLgPEXHwc3ix+0xh8orNvBsjWGKVMbuomEmTlld2rgi7EwP8Z1L8OQXpCpKW6EJIQZBVwh+vxf2vz5c4PvyOOWO8Yf/6fGmlh8dLV7Fi9z6tTG9aHhFVFwD3b7byG1ukrPwA5z4xocLHOUwq1xeNnAvgdaGW9dv2csEzk7jjjajImoaY5q+F63ZE3P/Umgu9ViyTlm3ip09+W2nbxKUbeWhcagYT7fh84QZueXWW7W9u4iNFszVGo+fnz0yMa06xWzksGWRCqz+rhD8ekV3hGRFeOHYP2ZzibUxfFb93UBaxQo+Tu9nOfWUpE0E3FYlT69iPzRJC4ji72HlWrR9J9FohXhsx8/fsfx20dUc3Buxu0S6roggvTO+ITd6qtn5/iMSuZKL50aUZKXyv7VxxLx05mae+9Bbn5u63UxMlNhy3PijijZUB3P6G9/k21ZWsEv7ox8QuHn+Y8fMOTjapEpIAOPuJCZz/tLcFJJyE//sERNGz7ddFmujl5NLBXXHCTCfSfvY7wzfQQvjgd0kSnnXb9ga2oE1877WDTF7uvwW9d395lRndBkPh8A9Z58H7RrHHVVhmEWkMjASOI6SvVwO3Ap2sJI2BrcaYbjZ5VwI7gHKgzBhTlHix3XHba+5epIkJdPEiB//8zCxMvwTb4+VcljoskpKIb3o6rkv0Mb2EIAhiPdlkhDwwBvqO+CzloQLueWcO//uuaiXhFFhu7/7ySs/L2zPdLZQUSaaEQ6gOuI3H/xgwzhhzgYjkA/WMMReFfxSRR4BYTYpTjTH+16RzSwI33umhiecNFDmIlO4HL5Fl/6LxIkJPfF61x1SdCMKd98GPnOPWu8XJOSBR0vFc2ok+2JucADrfO47mDfIpqBNakMiuzLEq13Fz1/FVjIVRlMrEFX4RKQAGAFcCGGNKgdKI3wW4EBiYnCKml+e+dl6BKBo/9t4grQmXjpxMq4LgFmNxi1OPadWmXazZvJv3ZqfHdXDm6i00bxD/ekSPeTgNDsfyHHITKVOxJ3ztNu4spaCu8+p3x9w33vG3RJYlzUXc2PjbAyXAiyIyU0RGikj9iN9PAtYbY5Y45DfAxyIyXUSGOR1ERIaJyDQRmVZSkvqaOwj7arRrZSxue22Wq8We12zx1grMpMloi9fvTHkYgEjOe2qi5+Nv37ufvn+zN0dc/e9p7E3jOql+SHcv1A3zIsMuW+XNhJj12Ywb4c8DegBPG2O6A7uA4RG/XwK8EiN/f2NMD+BM4DciMsAukTHmOWNMkTGmqEWLFu5KH72PBGw9qe4mvj3zByav2Bx3sOzut5PjZ5/pJGr2iHZdjEVkZRkv6um/kmTWSlaFPWl58i2siWK3FsMLET75iVKW5DUrqiNubPzFQLExZrL1fQyW8ItIHvAzwHGlZWPMWuv/BhF5G+gNfJ1IobOF3785mzWbgzURBBU/JBURJpOJ38lK1aGF7IWr/51+n3FPJKH+u2PM9/ET5RhxW/zGmHXAGhEJe/CcBsy3Pg8CFhpjbEPbiUh9EWkY/gwMBpLWhK1uL23Qog/+JpEpB0nXspRBuVumgq27kzgnJcDLv2HHXv5v3EJfHkLZjluvnpuA0ZZHz3LgKmv7xUSZeUTkUGCkMWYo0Ap42wrYlQe8bIwZF0TBk8mcGH73Snbz6YINaTmumxADqcZpUZCHXK7I5oflAU2IAxj+5pyKaJ9KZVwJvzFmFlDF/94Yc6XNtrXAUOvzcqBrYkV0T1At/rOfqBrlUMkO7B6RvRGB8FK9jmsm0/uv9tFFq0vPutRnCJJcIKtm7iqKHyIXfq82qpZWqsc1mhDgvJZsI6uEP132WSU7KDugLUQ3vDJlTfxESkaTXcKvuq8kwJrNe3i8ms9CVhQ3ZJXw60COEo9kBUJTlOpEVgm/ujIqiqLEJ6uEX1EURYmPCr+iKEqOocKvKIqSY6jwK4qi5Bgq/IqiKDmGCr+iKEqOocKvKIqSY6jwK4qi5Bgq/IqiKDmGCr+iKEqOocKvKIqSY6jwK4qi5Bgq/IqiKDmGCr+iKEqOocKvKIqSY6jwK4qi5Bgq/IqiKDmGK+EXkcYiMkZEForIAhHpKyKvicgs62+liMxyyDtERBaJyFIRGR5s8RVFURSv5LlM9xgwzhhzgYjkA/WMMReFfxSRR4Bt0ZlEpCbwJHA6UAxMFZH3jDHzEy+6oiiK4oe4wi8iBcAA4EoAY0wpUBrxuwAXAgNtsvcGlhpjlltpXwXOBVT4FUVR0oQbU097oAR4UURmishIEakf8ftJwHpjzBKbvIcBayK+F1vbqiAiw0RkmohMKykpcVl8RVEUxStuhD8P6AE8bYzpDuwCIm31lwCvOOQVm23GLqEx5jljTJExpqhFixYuiqUoiqL4wY3wFwPFxpjJ1vcxhCoCRCQP+BnwWoy8bSO+twHW+iuqoiiKEgRxhd8Ysw5YIyKdrE2ncdBGPwhYaIwpdsg+FThKRNpZg8IXA+8lWGZFURQlAdx69dwEjLbEezlwlbX9YqLMPCJyKDDSGDPUGFMmIjcC44GawChjzLxgiq4oiqL4wZXwG2NmAUU226+02bYWGBrx/SPgI/9FVBRFUYJEZ+4qiqLkGCr8iqIoOYYKv6IoSo6hwq8oipJjqPAriqLkGCr8iqIoOYYKv6IoSo6hwq8oipJjqPAriqLkGCr8iqIoOYYKv6IoSo6hwq8oipJjqPAriqLkGCr8iqIoOYYKv6IoSo6hwq8oipJjqPAriqLkGCr8iqIoOYYKv6IoSo6hwq8oipJjqPAriqLkGCr8iqIoOYYr4ReRxiIyRkQWisgCEelrbb9JRBaJyDwRecgh70oRmSMis0RkWpCFVxRFUbyT5zLdY8A4Y8wFIpIP1BORU4FzgS7GmH0i0jJG/lONMRsTLayiKIqSOHGFX0QKgAHAlQDGmFKgVESuB0YYY/ZZ2zcksZyKoihKQLgx9bQHSoAXRWSmiIwUkfpAR+AkEZksIl+JSC+H/Ab4WESmi8iwgMqtKIqi+MSN8OcBPYCnjTHdgV3AcGt7E6APcAfwuoiITf7+xpgewJnAb0RkgN1BRGSYiEwTkWklJSU+TkVRFEVxgxvhLwaKjTGTre9jCFUExcBbJsQU4ADQPDqzMWat9X8D8DbQ2+4gxpjnjDFFxpiiFi1aeD8TRVEUxRVxhd8Ysw5YIyKdrE2nAfOBd4CBACLSEcgHKg3gikh9EWkY/gwMBuYGVnpFURTFM269em4CRlsePcuBqwiZfEaJyFygFLjCGGNE5FBgpDFmKNAKeNuyAOUBLxtjxgV9EoqiKIp7XAm/MWYWUGTz0y9t0q4FhlqflwNdEymgoiiKEiw6c1dRFCXHUOFXFEXJMVT4FUVRcgwVfkVRlBxDhV9RFCXHUOFXFEXJMVT4FUVRcgwVfkVRlBxDhV9RFCXHUOFXFEXJMVT4FUVRcgwVfsWRRy/qlu4iKIqSBFT4FUd+2v2wdBchUC7p3TbdRVCUjCCrhL9mDbsFwBQlxIVFKvyKAlkm/ONvtV3VUalGnHRUlUXcAuOYQwuStm9FqU5klfAf2bIBf/npcY6/P3Fp9xSWRvHDg+cdn7R9C9ojzBUWPDCE0zq3THcxMpasEn4AY4zjb/riK0puUDe/JrVrxZa3G089MkWlyTyyT/jTXQAlIUTr5pTRvkX9dBchrXRt2zjdRUgb2Sf8qvwZz5K/nsniv5xp+5sEoPzN6ucnvI/qSvvm7sU8HXXsT7q0Ttmx4mlBLvuCZJ3wH/Co/McdFn/A78QjkzfgmIvUqlmD/Dz7Ry+Id9HpCTDaH6xEvEr2+lM6pKgkySG+8Gee8n/xu1NScpwsFH7n32rbiM0/Low/SenxS1IzKBxrYDpXCOJdzLzXOXV4qdpijYdBcj2svHBoozq+8sV7lpx+73xIQ1/HC4J2HnpsiZB1wu/0MK8ccZZvP/9UdQl7HtEkNQeySHeDp4ONjTmIAfhUuG1+8/tTk34MP8QTc7+c0K5pIPvxY8rr0LKBp/S3DeoIwP3nHMvFvdrStU0j12UpOqKJ5+NVR1wJv4g0FpExIrJQRBaISF9r+00iskhE5onIQw55h1hplorI8CALb0fLgoOtg3s8WEwsAAAYoElEQVTOOrrSb3Zd/chb375FfQ5rXJfnLy+qlKZGipQ/WULcvIG9zfulq09IzgFd0qJh7YTyO7XMUtGFb9u0nu+8H918UoAlqYyXZzWeCCfDC85rxdS1bWPbnnos+h/ZDAhpwYjzu5BX0z6/3aVqVeCvd/GzHtVrlrvbK/oYMM4Y0xnoCiwQkVOBc4EuxphjgYejM4lITeBJ4EzgGOASETkmkJI7cHaX1hxuvZTRD3a8Z+6JS3rw7fCBnH5Mq0rba3oUknO6HuopfbJJpBF4709Ct+uMY1vFSekdu3IFodmZbsm3q7Dq5ddkwQNDEt73Vf0KOdJlizUoWS8KqKf60AVdAtlPYZS5xKmyaVinViDHA2har3o5FMQVfhEpAAYALwAYY0qNMVuB64ERxph91vYNNtl7A0uNMcuNMaXAq4Qqi6QhIpx29MGJG+/+pj+jriyyTXtY47quhMariWjE+faTkD646cSY+YJsYf1haOe4adyc+69ObMfKEWfxjwu70cWhy+wXu9fRyxUIwgNo3K0n0bswGDOGW+xa5Tec0oG6+TU97aegTl6VbbVr1eTjFM5gXzniLAYH1CgIwra+5K9n0ryBu55kt7aNXc32v3lgVX//R37e1XPZwuTn1eDOIZ2ZdNdA3/tIFDct/vZACfCiiMwUkZEiUh/oCJwkIpNF5CsR6WWT9zBgTcT3YmtbSjDG0LVtYwZ2bmV9j52+tc0gUsuGtT2bDurl51GzhtC4XuUWxXGHBSucsRg2IL5HRtsmoZ7RHWd0qvLbi1dWvp31a+cFP9iXoPIL8K8EB947H1JApzQO5oXxU4ld1vcI2+1uzT1b9+yP+XukaTRW8bz0KGMljeWY4ZZaNmadWLv1e+/P79nGVz6AYSe15/pTOtC6UV3f+0gUN8KfB/QAnjbGdAd2AcOt7U2APsAdwOtS9em1e1xs74OIDBORaSIyraSkxG35bXFqOdsduLBZfYYcewgf3HQiTWz8v49uXeBrUHjBA0OY8odBnvKkerD18Gb1mH7PIG6Icturn1+TU22mu/d3cGut77GlGgsvvR4RXLfu7Bh6/CG+8zqRynsoCMe0Loja5h4vJsxErrNbnO3/iV3UX53Yzn0ZMt5QGAxuhL8YKDbGTLa+jyFUERQDb5kQU4ADQLQyFAORIRHbAGvtDmKMec4YU2SMKWrRooWXc3BN9IMlAnk1a/DMZT1jtsZr1hBGX+NuILRXYcjemZ9X2VfdjR20Sb38Ki9ysmnWoHbVsRCHtP06NKdP++DMIraD7UHY+F02QZ/6RU/HcvihoE4e8+8fUmWMyA1+z/vN6/sx497THT1XYtHNw8zVVPj0x2vx+zWv/KTLoRXhGW4/vWPMtI3q5vseJPrzucf6y5gG4gq/MWYdsEZEwvaA04D5wDvAQAAR6QjkAxujsk8FjhKRdiKSD1wMvBdQ2T0TfT+9dFEjW7vv/qa/YwCol35lX0G8/Os+Mfd/WZ8jaNGwNh/dchL3/SSp498JkVej6iMTPZgWi0g77kW9Dq/ye4GHAbd0u6PaUTe/Js/8siez7js96ccSCR2vaf18jmwZuq5eNOs3LmPVnNCuqa0JxQ92t+ywxnWtuTKxS9/QZkwjaKI9AcH9Nb2sbyFN6sV/fiOf26PS5Drq9m7eBIwWke+BbsCDwCigvYjMJTRoe4UxxojIoSLyEYAxpgy4ERgPLABeN8bMC/ok3BKUi3PXto35WY+QjW/o8YdUcgurU8ve7OE0UzXMnwOYvHX/OalvcRx7aAH/vbq36/SRrowX9GzDWzf0q/R7vOt082lHVUx0E4RaNe3Vf8a97oU3aLfF0PiONy+PRMvgpxLMz6vB0TF6mJFlOjwB99VI7F7BS084nHO6Hhq3xZ/I6+umV9f/yGbUr+2+cokU7fYtQp+dXEedOMTn5LREcVVKY8wsywzTxRjzU2PMFsu755fGmOOMMT2MMZ9badcaY4ZG5P3IGNPRGNPBGPPXZJ2IOxJT/kivll7tQqabK/u5tx8mm3O72buRJtNqObBzS5o52H87tqrcmjmhXdMqA49e9eq3p3fklE4hU2Cf9k3peUQTbh10VJV0TW3GazI5OmtYuC/ulbrFYrxUFk4NGj/YecmAc8MsyJ5drH15fT7evKEfX91xCm9e38/T6m6HNk7foG6YrJu5G4vwg+UmPo8dL/+6D1/fEZqx2bJhHVaOOIve7ZoG4laYjUS/SK9d27dKGj+VUpsm9fjid6dw55DOiAi3Dqpst+3TvpltPj+2/DevP9gj+XtAfuZ2hK/UiPOrHiPsTdWmSeKC0TtqBm6sJzeRsY9HL+rmOCDc3GHint3YTIPalSucPwztzHs39vddrlg4vcZO16igTi2OaFafnkc0sdWAz28/2TZfJszzyUrhd3pge1gDrJf1sXeDs2NAx4MDzQ1q53F4s8S6vG7MMWd1aV1lVmurgvheFUG1aGOZxKKv7aCjnQcyvdaHY64LVQzHRoVcsLODtmte37ZbPeHOU7n+ZP8DkS9e2YuHIsQ3MozG4GNiewFFV0CRRLv2Rv92dgwxeOlXJ1Q0MiJpVPfgPhtYJop4s1zr59es2E+eS281u/t482lHVXH5jeTkji2Yds8gfnFCaBynotKK8WxFxqlp3qA2d53ZmUcvOuiua0zITblLm+SEU3by/vFb/YXNP9FEXs90RRPOSuEPE10LtyoItdL7dXDvj351/8L4x/FQpiv6FdpOvImkVUEdpt49qNJLPO6WAcy9/wwPR0oNbmOaO7nERm4tsiZSvX5t30oVipcKpE2Tep5DbET6cp/SqQUX+jC3rBxxFlfHcBt88tIejr9N/sNprrr/9aLcZq/sV1jx+Y4zOnHHGZ34SZf4rcknL+3Bn8891lGY3PCTLq1tXX6jCU+mdNPKbRkVLuHakzvQomHtlBnoTunkbsWunyfgw58pZLXwB4EbM8553f3NSQu34t20vOrUqlnRqvNKkIG7/O5qpgcvl/q18ypNpru6f3LHUcKtUq/UibPCUyROcyC8cOeQzpXGMyJ7PPVr5/GbU4+MO+ekdeO6tGhYm8v6FgKVK9XoMRm/HN26gAKrNzKwcyuW/vXMSoHz3IwXVGoV2/x+55DOPHtZz0SL6ou/u3ArbWi9q5GVc7j3mB+Qh1QipL8ESSRZkQqj+dvPvK0TG65Mwt1Yv4GhWjeqw/Ex5h/0btc0qeuO1vUw4OfFTRMqv/hDjgt+olXlY/lrUy78s/1iMm4Yd+tB7ya3JrqGdWrFNCe5YehxzguhRJuSwnh9jcbeclKlCiivZo1K+zi/RxvuHBI7pIjdFYm8Tdef0oEzjnX/XKTKpBIW9c8s+/6fzjm2Yj2PWwcdxcoRZ1WqsNM1PJiVwh/vRQr6Ynt14QrjtwUf5oR2TWlpjQU0qJ1Xxc/59Wv78sKVvTwLm5N3EBx8gX7esw1jb3EXZTLWqlBuyuY1hk11oPMhzg4GyRz8i77cvXzGKYoUUq+aWrOG+JoQFoR4+6nkvSxR+dKvenPzaUdVGqPLs1yOM8mrLPkzInKcn3U/jLdm/mD7W6IVUORD/OhF3QIJHz37vsHUrx1faM/rfpiriVuz/zjYc1hdO5rVz2fTrlLH328/vSO9AooZn2qin4NmDmG0gyDaaeDus47m3xNXhsrhIEyBLI7jcR+R6ZMhl4d5cKn8abfDKGxWn/Oemhg3bfsWDfht1Ozg/zu/C899vZy+Hap6m6VrcFeFP8k8cmFXVzbBeNi9OHYvxKS7BrJlV+XgW9Emrz+dfQzPf7PC9jiN4sw89OriF+l54hdB+OiWk1hWstMxzU2nVfXl79u+GZOWb0r4+NnCRzefRMdWlYOSRc7IzZU4NaOvOYGjPIxniAjdD29C/yObcbHNbPN4tCqoUxHePFPISlNPPJpYsypTMVlGRKoMuAU29mCj/K0b1Y27AtWV/dvx7fAEQ8I6NMO8Dlw5teait7cqqOPJGwvg5V+H3CDdEFRM+Ugeuzj+sp6pJMiVyfxWEm7z2fc+En9vjDH0P7I5LRt6H1cbfU2fmG631YmcbPHXr53HsgeHpmxJxSCZdNdAPpqzjj9/MD+jbIYXFrWhVUEdbhvUkQXrtnPW4xPSXSRP9twXr+rFqk27PduAP7r5JA5zmFjl1sSVOXcxeSTyrLq5JR/fNoB12/YmtI8wudDzyUnhB++Lq2QKrRvVrXAVc/swu3mMh5/ZmX42Nsgq+3LY2UMXHDRneXnJ3bRCU+H50LBOrbjrJdTJryrkqVjfN1U4hjO3WyktydWV13vesVXDKmasSGJ1sps3qM3Gnfu8HbCak7PCr1TmOo+zXYN68Z2iPrZpEkxQsCCpnZcc76LqEvLj8Kb1WL15t6+8XlvRl1vzDILG7lpPuPNUyoNYBcYHV/QrZMLS6KDGyScnbfzVBTctcC+8/OsTeP/G2Ms/xiNVr0esWbB+Sb1prHoIejzCWhkOqQHJN4fYuXsmywOmTq2anqJyBsnpx7RyPQ4VJFkp/NliowsvFALRftOhL0KEj7ALjTn6kAKOD2jd3GQ3UmvWkMDHYPLzasQMneCGG07pwP8c1lzwS9Vl69JfYdjFR2pZUIdOMcwpsXB7Ttee3J4jouJhxcr7+yGd4q5lrVQlK4U/TKZ3oeMVr25+TdsBwnAlIAJ/Pe94rjmxHSd3dF61LBySIIiJUGd3Cc38jH45IwkPdgYVaiHIu3hWl9YVk9788PshnTkx4LWHg5h/ETSf/NY+smQyaB4xb+GuM4/mKysCrhtuOOXIpK5l7dUEWl3Iaht/qkI2eOWZy3ry3NfLPYcxCBM+K0Fo3qA298TxEf7d4E7cNqij7xnGkfyyzxFc2KttTHt3o7q10tJ9jUVkgLMPbj6RNZv3pOzYgz0uxRhEj7Vv+2b0bteUxz5b4jnvEBehEPy+Wnb5ft4ztlt1eEZ6vEV6gubJS3twVhfnEBfVmawU/kzoKseiX4fm9OvQnN2lZQBc3tc5TLRdryCyxe8GEakwCSWKiCRtkDOaoKrt+Q+cUemZaNmwji8/7kyha5tGzC7eFjPNK8NCS32Ghd/tEn9T7x4Uc9JdrOcxFk7P6iEFdeL2eO47+xiObNmAU11Gz7TDy7N0/clH8t3yzbYzbbOFrBT+6kK9/LyEWsYZbskKjERNdvXyvT/mj13cjQ4JhC12w8e3DWDx+h2e87386z5sjhG+wo4x1/eLn4jKIR0a1w2ZYJJ1HabePciV+bFhnVpcm0KTy/FtGnlaujMILul9ON8sKUnZ8VT4qyHZMnidyZzbzV+o7VhcesLhbN9zMJyGk+95vB5r/dp5nr1Q/ITOOObQAkZfcwJFhcHPaoaqcYNyGa8RfhNFhb8acrBrnSNN/izhwfNS+3K7JZZ/vtM6Ahk6fKa4JKu9erKJyFZ+xeBujuh+jpxm2ggvOZgrz5Oiwh8oibgJOmHb7TcH/fgVRVG84srUIyKNgZHAcYQanFcDZwC/BsIjEn8wxnxkk3clsAMoB8qMMUWJFzvz+PS3J9OsfvBx1JvUq8WebeWVKoBca/HnGsmMx58o4bLl5x18+No2dR/bPl0moqv6FTJ1xWYuSkFE3uqAWxv/Y8A4Y8wFIpIP1CMk/P80xjzsIv+pxpjUB6RIIUe6dJfzymvX9uWrxSWVvB+k4n92K3+u2pGHDWhPiwa1+f2b36fkeF7muzx+cXfGzl3HkS0PDkqfdXxrWgyrzUXPfcexDkHr0v2ktiyo49qzKReIK/wiUgAMAK4EMMaUAqWZPis2W2jbtB6/7FPZz//nRW2Z+8N2bh+c2Pqr1YVce9Rq1azBhb3apkz4vdCsQe0qz6OIcEL7Znxw04m0bZp5wfWUqrhp8bcnZM55UUS6AtOBW6zfbhSRy4FpwO3GmC02+Q3wsYgY4FljzHMBlDsm4Rl+TpEfqzt1atXk/y7oku5iKEnm1WF92Lm3LOnHCaoRl8zQCUqwuFHGPKAH8LQxpjuwCxgOPA10ALoBPwKPOOTvb4zpAZwJ/EZEBtglEpFhIjJNRKaVlCQ2keGmgUdy3ckduLi32vOqM8lYFSuVDOjYnL7tmzH8zM6+8vdp34xBHsM9ZCrhnkDPan5PswU3Lf5ioNgYM9n6PgYYboxZH04gIs8DH9hlNsastf5vEJG3gd7A1zbpngOeAygqKkrIulu/dp7vl03JHEZd1YvVPlbFCpo5fxrsqwz18vMqQifkOscd1ogvfncKhTGC+ympI26L3xizDlgjIp2sTacB80UkMnrRecDc6LwiUl9EGoY/A4Pt0imKHQUuVsVKBQ3r1KJBmuK1p4JUBTNs17x+2itxJYTbp/kmYLTl0bMcuAp4XES6EbLhrwSuBRCRQ4GRxpihQCvgbetm5wEvG2PGBXoGiqIEgkpy7uBK+I0xs4Bo//vLHNKuBYZan5cDXe3SKYqiKOkhO91eFEVRFEey13CZIDPvPT0jV0ZSlKCpabk9B7FQj1I9UOF3oEkSwi8oSiZyYVEbVm3cxS2Djkp3UZQUocKvKDlO7byacZfvVLIL7dspiqLkGCr8iqIoOYaaepS088jPu7Jj7/74CRVFCQQVfiXtnN+zTbqLoCg5hZp6FEVRcgwVfkVRlBxDhV9RFCXHUBu/omQxY67ry/KSXekuhpJhqPArShZTVNiUosKm6S6GkmGoqUdRFCXHUOFXFEXJMVT4FUVRcgwVfkVRlBxDhV9RFCXHUOFXFEXJMVT4FUVRcgwVfkVRlBxDjDHpLkMVRKQEWOUze3NgY4DFqQ7oOWc/uXa+oOfslSOMMS3cJMxI4U8EEZlmjClKdzlSiZ5z9pNr5wt6zslETT2Koig5hgq/oihKjpGNwv9cuguQBvScs59cO1/Qc04aWWfjVxRFUWKTjS1+RVEUJQZZI/wiMkREFonIUhEZnu7yJIKItBWRL0RkgYjME5FbrO1NReQTEVli/W9ibRcRedw69+9FpEfEvq6w0i8RkSvSdU5uEJGaIjJTRD6wvrcTkclW2V8TkXxre23r+1Lr98KIfdxlbV8kImek50zcIyKNRWSMiCy07nffbL7PInKb9UzPFZFXRKRONt5nERklIhtEZG7EtsDuq4j0FJE5Vp7HRUQ8FdAYU+3/gJrAMqA9kA/MBo5Jd7kSOJ/WQA/rc0NgMXAM8BAw3No+HPg/6/NQYCwgQB9gsrW9KbDc+t/E+twk3ecX47x/C7wMfGB9fx242Pr8DHC99fkG4Bnr88XAa9bnY6x7XxtoZz0TNdN9XnHO+T/ANdbnfKBxtt5n4DBgBVA34v5emY33GRgA9ADmRmwL7L4CU4C+Vp6xwJmeypfuCxTQRe4LjI/4fhdwV7rLFeD5vQucDiwCWlvbWgOLrM/PApdEpF9k/X4J8GzE9krpMukPaAN8BgwEPrAe6I1AXvQ9BsYDfa3PeVY6ib7vkeky8Q8osIRQorZn5X22hH+NJWR51n0+I1vvM1AYJfyB3Ffrt4UR2yulc/OXLaae8AMVptjaVu2xurfdgclAK2PMjwDW/5ZWMqfzr07X5VHg98AB63szYKsxpsz6Hln2ivOyft9mpa9O5wuhHmoJ8KJl4hopIvXJ0vtsjPkBeBhYDfxI6L5NJ/vvc5ig7uth1ufo7a7JFuG3s29Ve3clEWkAvAncaozZHiupzTYTY3tGISI/ATYYY6ZHbrZJauL8Vi3ON4I8QuaAp40x3YFdhEwATlTr87Zs2ucSMs8cCtQHzrRJmm33OR5ezzPh888W4S8G2kZ8bwOsTVNZAkFEahES/dHGmLeszetFpLX1e2tgg7Xd6fyry3XpD5wjIiuBVwmZex4FGotInpUmsuwV52X93gjYTPU53zDFQLExZrL1fQyhiiBb7/MgYIUxpsQYsx94C+hH9t/nMEHd12Lrc/R212SL8E8FjrK8A/IJDQS9l+Yy+cYaoX8BWGCM+UfET+8B4ZH9KwjZ/sPbL7e8A/oA26yu5HhgsIg0sVpbg61tGYUx5i5jTBtjTCGhe/e5MeYXwBfABVay6PMNX4cLrPTG2n6x5Q3SDjiK0CBYRmKMWQesEZFO1qbTgPlk6X0mZOLpIyL1rGc8fL5ZfZ8jCOS+Wr/tEJE+1nW8PGJf7kj3AEiAAylDCXm/LAPuTnd5EjyXEwl13b4HZll/QwnZNz8Dllj/m1rpBXjSOvc5QFHEvq4Gllp/V6X73Fyc+ykc9OppT+iFXgq8AdS2ttexvi+1fm8fkf9u6zoswqOnQ5rOtxswzbrX7xDy3sja+wzcDywE5gIvEfLMybr7DLxCaBxjP6EW+q+CvK9AkXUNlwFPEOUgEO9PZ+4qiqLkGNli6lEURVFcosKvKIqSY6jwK4qi5Bgq/IqiKDmGCr+iKEqOocKvKIqSY6jwK4qi5Bgq/IqiKDnG/wN2XeJrWXQ/FQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "(array([  16.,  106.,  590., 1831., 2989., 2644., 1352.,  377.,   86.,\n",
       "           9.]),\n",
       " array([65.24631713, 65.56293231, 65.8795475 , 66.19616268, 66.51277787,\n",
       "        66.82939305, 67.14600823, 67.46262342, 67.7792386 , 68.09585379,\n",
       "        68.41246897]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "lower, upper : (65.9929132815752, 67.58402738281573)\n",
      "coffee_full[(coffee_full['drinks_coffee'] == False)]['height'].mean() : 66.44340776214705\n"
     ]
    }
   ],
   "source": [
    "\n",
    "np.random.seed(42)\n",
    "\n",
    "coffee_full = pd.read_csv('coffee_dataset.csv')\n",
    "coffee_red = coffee_full.sample(200) #this is the only data you might actually get in the real world.\n",
    "#coffee_red.head()\n",
    "drinks_coffee = coffee_red[(coffee_red['drinks_coffee'] == True)]\n",
    "nb_drinks_coffee = len(drinks_coffee)\n",
    "pv(\"nb_drinks_coffee/200\")\n",
    "pv(\"1-nb_drinks_coffee/200\")\n",
    "pv(\"drinks_coffee['height'].mean()\")\n",
    "pv(\"coffee_red[(coffee_red['drinks_coffee'] == False)]['height'].mean()\")\n",
    "\n",
    "# Simulate 200 \"new\" individuals from your original sample of 200. What are the proportion of coffee drinkers in your bootstrap sample? How about individuals that don't drink coffee?3\n",
    "coffee_new = coffee_red.sample(200, replace=True)\n",
    "#coffee_new.head()\n",
    "drinks_coffee = coffee_new[(coffee_new['drinks_coffee'] == True)]\n",
    "nb_drinks_coffee = len(drinks_coffee)\n",
    "pv(\"nb_drinks_coffee/200\")\n",
    "pv(\"1-nb_drinks_coffee/200\")\n",
    "pv(\"drinks_coffee['height'].mean()\")\n",
    "pv(\"coffee_new[(coffee_new['drinks_coffee'] == False)]['height'].mean()\")\n",
    "boot_means = []\n",
    "for _ in range(10_000):\n",
    "    nb_sample = 200\n",
    "    boot_sample = coffee_red.sample(nb_sample, replace=True)\n",
    "    boot_means.append(boot_sample[(boot_sample['drinks_coffee'] == False)]['height'].mean())\n",
    "plt.plot(all_mean_height_no_drinker)\n",
    "plt.show()\n",
    "plt.hist(all_mean_height_no_drinker)\n",
    "plt.show()\n",
    "lower, upper = np.percentile(boot_means, 2.5), np.percentile(boot_means, 97.5)\n",
    "pv(\"lower, upper\")\n",
    "\n",
    "pv(\"coffee_full[(coffee_full['drinks_coffee'] == False)]['height'].mean()\")\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "coffee_red[(coffee_red['drinks_coffee'] == False)]['height'].mean() : 66.78492279927877\n",
      "coffee_full[(coffee_full['drinks_coffee'] == False)]['height'].mean() : 66.44340776214705\n"
     ]
    }
   ],
   "source": [
    "pv(\"coffee_red[(coffee_red['drinks_coffee'] == False)]['height'].mean()\")\n",
    "pv(\"coffee_full[(coffee_full['drinks_coffee'] == False)]['height'].mean()\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Confidence Interval - Difference In Means¶"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>user_id</th>\n",
       "      <th>age</th>\n",
       "      <th>drinks_coffee</th>\n",
       "      <th>height</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2402</th>\n",
       "      <td>2874</td>\n",
       "      <td>&lt;21</td>\n",
       "      <td>True</td>\n",
       "      <td>64.357154</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2864</th>\n",
       "      <td>3670</td>\n",
       "      <td>&gt;=21</td>\n",
       "      <td>True</td>\n",
       "      <td>66.859636</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2167</th>\n",
       "      <td>7441</td>\n",
       "      <td>&lt;21</td>\n",
       "      <td>False</td>\n",
       "      <td>66.659561</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      user_id   age  drinks_coffee     height\n",
       "2402     2874   <21           True  64.357154\n",
       "2864     3670  >=21           True  66.859636\n",
       "2167     7441   <21          False  66.659561"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "(array([  24.,  109.,  546., 1516., 2623., 2826., 1664.,  568.,  106.,\n",
       "          18.]),\n",
       " array([-0.43028277, -0.07929544,  0.2716919 ,  0.62267923,  0.97366656,\n",
       "         1.3246539 ,  1.67564123,  2.02662857,  2.3776159 ,  2.72860323,\n",
       "         3.07959057]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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DVXV/N/0csIejf6U/Tut2kH7HQre+nu8eHtvdjjz4OpLPg+UaAKdW1QHo/UUATplm3KuS7EpyV5JhhsQgl7z42ZiqehF4BviVoXQ3TR+d6S7P8cfdZv+Xk6yZYv64WG6XG/ndbtfA15O8YdTNAHS7H95E75tqv7FctzP0C2OyfpOsSPIAcBDYWVXTrtthfh6M7X8JmeQbwGummPXhObzMr1XV/iSvA76Z5KGqemxxOpzRrJe8GHDMMAzSx78AN1XVC0neS++byh8ueWfzMy7rdRD3A6+tqueTrAf+GVg3yoaSnAB8BXh/VT175OwpnjLSdTtLv2OzfqvqJeCsJCcCX0vyxqrqPzY0knU7tlsAVfXWqnrjFLdbgScPb3p29weneY393f3jwH/Q+5YwDLNe8qJ/TJJjgF9mNLsLBrk8x4+r6oXu4eeA3xlSb/MxyLofC1X17OFdA9X7fcyxSVaOqp8kx9L7MP1iVX11iiFjtW5n63fc1m/Xx//Q+yy6+IhZI/k8GNsAmMV2YGM3vRG49cgBSU5Kcnw3vRI4jyW67PQUBrnkRf97+BPgm9UdARqyWXs9Yj/v2+jtbx1X24EruzNWzgWeOby7cNwkec3h/bxJzqH37/HHI+olwOeBPVX1qWmGjc26HaTfcVm/SVZ13/xJ8ovAW4HvHTFsNJ8Hwzwavlg3evvG7gAe7e5P7uoTwD90028BHqJ3VstDwFVD7nE9vTMTHgM+3NU+Brytm34V8E/AJHAP8LoRrs/Zev1LYHe3Lu8EfmOEvd4EHAD+j963pquA9wLv7eaH3n869Fj35z4xxr1e07de7wLeMsJef4/eLocHgQe62/oxXreD9DsW6xf4LeA/u14fBj7S1Uf+eeAvgSWpUct1F5AkaYEMAElqlAEgSY0yACSpUQaAJDXKAJCkRhkAktQoA0CSGvX/b1oBPwAgwPMAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "lower, upper : (0.10258900080921124, 0.10258900080921124)\n",
      "full_data[(full_data['drinks_coffee'] == True)]['height'].mean() - full_data[(full_data['drinks_coffee'] == False)]['height'].mean()  : 1.9568024933369657\n"
     ]
    }
   ],
   "source": [
    "np.random.seed(42)\n",
    "\n",
    "full_data = pd.read_csv('coffee_dataset.csv')\n",
    "sample_data = full_data.sample(200)\n",
    "sample_data.head(3)\n",
    "# 1. For 10,000 iterations, bootstrap sample your sample data, compute the difference in the average heights for coffee and non-coffee drinkers. Build a 99% confidence interval using your sampling distribution. Use your interval to start answering the first quiz question below.\n",
    "\n",
    "boot_diff = []\n",
    "for _ in range(10_000):\n",
    "    nb_sample = 200\n",
    "    boot_sample = sample_data.sample(nb_sample, replace=True)\n",
    "    coffee_mean = boot_sample[(boot_sample['drinks_coffee'] == True)]['height'].mean()\n",
    "    no_coffee_mean = boot_sample[(boot_sample['drinks_coffee'] == False)]['height'].mean()\n",
    "    boot_diff.append(coffee_mean - no_coffee_mean)\n",
    "plt.hist(boot_diff)\n",
    "plt.show()\n",
    "lower, upper = np.percentile(boot_diff, 0.5), np.percentile(boot_diff, 99.5)\n",
    "pv(\"lower, upper\")\n",
    "\n",
    "pv(\"full_data[(full_data['drinks_coffee'] == True)]['height'].mean() - full_data[(full_data['drinks_coffee'] == False)]['height'].mean() \")\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([   5.,   35.,  265., 1116., 2646., 3061., 2009.,  702.,  150.,\n",
       "          11.]),\n",
       " array([2.78783612, 3.06299912, 3.33816213, 3.61332513, 3.88848813,\n",
       "        4.16365114, 4.43881414, 4.71397715, 4.98914015, 5.26430316,\n",
       "        5.53946616]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "lower, upper : (3.350274589725852, 5.109059900189764)\n",
      "full_data[(full_data['age'] == '>=21')]['height'].mean() - full_data[(full_data['age'] == '<21')]['height'].mean()  : 3.88229124992111\n"
     ]
    }
   ],
   "source": [
    "\n",
    "# 2. For 10,000 iterations, bootstrap sample your sample data, compute the difference in the average heights for those older than 21 and those younger than 21. Build a 99% confidence interval using your sampling distribution. Use your interval to finish answering the first quiz question below. \n",
    "boot_diff = []\n",
    "for _ in range(10_000):\n",
    "    nb_sample = 200\n",
    "    boot_sample = sample_data.sample(nb_sample, replace=True)\n",
    "    older_mean = boot_sample[(boot_sample['age'] == \">=21\")]['height'].mean()\n",
    "    younger_mean = boot_sample[(boot_sample['age'] == \"<21\")]['height'].mean()\n",
    "    boot_diff.append(older_mean - younger_mean)\n",
    "plt.hist(boot_diff)\n",
    "plt.show()\n",
    "lower, upper = np.percentile(boot_diff, 0.5), np.percentile(boot_diff, 99.5)\n",
    "pv(\"lower, upper\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For 10,000 iterations bootstrap your sample data, compute the difference in the average height for coffee drinkers and the average height for non-coffee drinkers for individuals under 21 years old. Using your sampling distribution, build a 95% confidence interval. Use your interval to start answering question 2 below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([   4.,   54.,  391., 1414., 2679., 3021., 1750.,  553.,  120.,\n",
       "          14.]),\n",
       " array([-3.44885318, -3.13776917, -2.82668516, -2.51560114, -2.20451713,\n",
       "        -1.89343312, -1.58234911, -1.27126509, -0.96018108, -0.64909707,\n",
       "        -0.33801306]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "lower, upper : (-2.6136701999388614, -1.075162498976531)\n"
     ]
    }
   ],
   "source": [
    "boot_diff = []\n",
    "for _ in range(10_000):\n",
    "    nb_sample = 200\n",
    "    boot_sample = sample_data.sample(nb_sample, replace=True)\n",
    "    drink_mean = boot_sample.query(\"age == '<21' and drinks_coffee == True\")['height'].mean()\n",
    "    no_drink_mean = boot_sample.query(\"age == '<21' and drinks_coffee == False\")['height'].mean()\n",
    "    boot_diff.append(drink_mean - no_drink_mean)\n",
    "plt.hist(boot_diff)\n",
    "plt.show()\n",
    "lower, upper = np.percentile(boot_diff, 2.5), np.percentile(boot_diff, 97.5)\n",
    "pv(\"lower, upper\")\n",
    "pv(\"upper-lower\")\n",
    "\n",
    "#pv(\"full_data[(full_data['age'] == '>=21')]['height'].mean() - full_data[(full_data['age'] == '<21')]['height'].mean() \")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "4. For 10,000 iterations bootstrap your sample data, compute the difference in the average height for coffee drinkers and the average height for non-coffee drinkers for individuals over 21 years old. Using your sampling distribution, build a 95% confidence interval. Use your interval to finish answering the second quiz question below. As well as the following questions. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([1.000e+01, 1.270e+02, 9.900e+02, 3.108e+03, 3.656e+03, 1.697e+03,\n",
       "        3.560e+02, 4.700e+01, 7.000e+00, 2.000e+00]),\n",
       " array([-5.80729099, -5.16742569, -4.52756038, -3.88769507, -3.24782976,\n",
       "        -2.60796445, -1.96809914, -1.32823383, -0.68836852, -0.04850321,\n",
       "         0.5913621 ]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "lower, upper : (-4.367928496025803, -1.8065555122633452)\n"
     ]
    }
   ],
   "source": [
    "boot_diff = []\n",
    "for _ in range(10_000):\n",
    "    nb_sample = 200\n",
    "    boot_sample = sample_data.sample(nb_sample, replace=True)\n",
    "    drink_mean = boot_sample.query(\"age == '>=21' and drinks_coffee == True\")['height'].mean()\n",
    "    no_drink_mean = boot_sample.query(\"age == '>=21' and drinks_coffee == False\")['height'].mean()\n",
    "    boot_diff.append(drink_mean - no_drink_mean)\n",
    "plt.hist(boot_diff)\n",
    "plt.show()\n",
    "lower, upper = np.percentile(boot_diff, 2.5), np.percentile(boot_diff, 97.5)\n",
    "pv(\"lower, upper\")\n",
    "pv(\"upper-lower\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Traditional approaches"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import statsmodels.stats.api as sms\n",
    "# sms.CompareMeans() => T-test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(\"\")\n",
    "print(\"\")\n",
    "print(\"\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(\"\")\n",
    "print(\"\")\n",
    "print(\"\")"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}