mendel-inference.ipynb

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   "source": [
    "# Testing hypotheses with simulation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from datascience import *\n",
    "import numpy as np\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plots\n",
    "plots.style.use('fivethirtyeight')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Mendel and Pea Flowers ##"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Determine the observed statistic\n",
    "\n",
    "We'll use the observed number of purple pea plants as our statistic."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Mendel had 929 plants, of which 709 had purple flowers\n",
    "observed_purples = 709\n",
    "observed_purples"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Use `sample_proportions` function to create a simulated sample\n",
    "\n",
    "The `datascience` library has a function named `sample_proportions` that takes in a size of a sample, and a probability distribution as an array. It will return an array that shows the proportion of the sample that falls into each category. In this example, the first element in the array represents proportion of purple flowers we expect and the second element represents the proportion of white flowers we would expect."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "predicted_proportions = make_array(.75, .25)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sample_proportions(929, predicted_proportions)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now write a function to simulate growing 929 flowers in a greenhouse given that each flower has a 75% of being purple and a 25% chance of being white. This function will return the number of purple flowers in the greenhouse."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def number_purple_flowers():\n",
    "    return sample_proportions(929, predicted_proportions).item(0) * 929"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "number_purple_flowers()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can run the cell above several times to verify that this is a random process."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simulate growing many greenhouses of flowers\n",
    "\n",
    "Now all we need to do is simulate the greenhouse experiement many, many times. A loop is a programming concept that allows for the same set of operations to be run several times in sequence. The loop below will simulate growing 929 flowers in a greenhouse as many times as the value stored to `trials`. Each time it computes the statisticit is appended to an array named `statistics`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "statistics = make_array()\n",
    "trials = 10000\n",
    "\n",
    "for i in np.arange(trials):\n",
    "    one_sample = number_purple_flowers()\n",
    "    statistics = np.append(one_sample, statistics)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "statistics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Visualize the distribution\n",
    "\n",
    "Let's put our statistics array into a Table so we can create a histogram."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "Table().with_column('Number of purple flowers in sample of 929', statistics).hist()\n",
    "plots.scatter(observed_purples, -0.001, color = 'red', s = 60, zorder = 2, marker=\"^\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Compute p-value\n",
    "\n",
    "We can use some array functions to quickly compute how many of our simulation statistics were more extreme than our observed statistic."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "np.count_nonzero(statistics >= observed_purples)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And then, dividing this by the number of shufflings that took place, we can compute the p-value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "np.count_nonzero(statistics >= observed_purples) / trials"
   ]
  }
 ],
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