{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<figure>\n",
    "  <IMG SRC=\"https://raw.githubusercontent.com/mbakker7/exploratory_computing_with_python/master/tudelft_logo.png\" WIDTH=250 ALIGN=\"right\">\n",
    "</figure>\n",
    "\n",
    "# Exploratory Computing with Python\n",
    "*Developed by Mark Bakker*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Notebook 11: Distribution of the mean, hypothesis tests, and the central limit theorem\n",
    "In this notebook we first investigate the distribution of the mean of a dataset, we simulate several hypothesis tests, and finish with exploring the central limit theorem. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy.random as rnd\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider a dataset of 100 points. The data are drawn from a normal distribution with mean 4 and standard deviation 2. As we noticed before, the sample mean of the 100 data points almost always differs from 4. And every time we generate a new set of 100 points, the mean will be somewhat different. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean a: 3.5749106487481073\n",
      "mean a: 3.9357430095703756\n",
      "mean a: 3.9711565636741524\n",
      "mean a: 3.8555191640408335\n",
      "mean a: 3.873414227135426\n"
     ]
    }
   ],
   "source": [
    "for i in range(5):\n",
    "    a = 2 * rnd.standard_normal(100) + 4\n",
    "    print('mean a:', np.mean(a))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In fact, the mean of the dataset itself can be considered as a random variable with a distribution of its own. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sample standard deviation\n",
    "The sample standard deviation $s_n$ of a dataset of $n$ values is defined as\n",
    "\n",
    "$s_n = \\sqrt{ \\frac{1}{n-1} \\sum_{i=1}^n (x_i - \\overline{x}_n)^2 }$\n",
    "\n",
    "and can be computed with the `std` function of the `numpy` package. By default, the `std` function devides the sum by $n$ rather than by $n-1$. To divide by $n-1$, as we want for an unbiased estimate of the standard deviation, specify the keyword argument `ddof=1` in the `np.std` function."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 1. <a name=\"back1\"></a>Histogram of the means of datasets with 100 values\n",
    "Generate 1000 datasets each with 100 values drawn from a normal distribution with mean 4 and standard deviation 2; use a seed of 22. Compute the mean of each dataset and store them in an array of length 1000. Compute the mean of the means and the standard deviation of the means, and print them to the screen. Draw a boxplot of the means. In a separate figure, draw a histogram of the means. Make sure the vertical axis of the boxplot and the horizontal axis of the histogram extend from 3 to 5."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a href=\"#ex1answer\">Answers to Exercise 1</a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 2. <a name=\"back2\"></a>Histogram of the means of datasets with 1000 values\n",
    "Repeat exercise 1 but now generate 1000 datasets each with 1000 values (rather than 100 values) drawn from the same normal distribution with mean 4 and standard deviation 2, and again with a seed of 22. Make sure the vertical axis of the boxplot and the horizontal axis of the histogram extend from 3 to 5, so that the graphs can be compared to the graphs you created in the previous exercise. Is the spread of the mean much smaller now as compared to the datasets consisting of only 100 values?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a href=\"#ex2answer\">Answers to Exercise 2</a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sample standard deviation of the sample mean\n",
    "The histogram of the means looks like the bell-shaped curve of a Normal distribution, but you may recall that it is actually a Student's $t$-distribution, also simply called a $t$-distribution. A $t$-distribution arises when estimating the mean of a normally distributed variable in situations where the sample size is relatively small and the standard deviation is unknown (as it pretty much always is in practice) and needs to be estimated from the data. \n",
    "\n",
    "The sample mean of a dataset of $n$ values is commonly written as $\\overline{x}_n$, while the sample standard deviation is written as $s_n$ (as defined above). Here, we are computing the sample standard deviation of the sample means, which we write as $\\hat{s}_n$ for a dataset of size $n$. Theoretically, the value of the sample standard deviation of the sample mean $\\hat{s}_n$ is related to the sample standard deviation as (see [here](http://en.wikipedia.org/wiki/Standard_deviation#Standard_deviation_of_the_mean))\n",
    "\n",
    "$\\hat{s}_n = s_n / \\sqrt{n}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Percentiles of $t$-distribution\n",
    "You may recall that the 90% interval around the mean for a Normally distributed variable runs from $\\mu-1.64\\sigma$ to $\\mu+1.64\\sigma$. In other words, 5% of the data is expected to lie below $\\mu-1.64\\sigma$ and 5% of the data is expected to lie above $\\mu+1.64\\sigma$. What now if you forgot it is $1.64\\sigma$ to the left and right of the mean? Or what if you want to know the value for some other percentile. You may look that up in a table in a Statistics book (or on the web), or use the percent point function `ppf`, which is part of any statistical distribution function defined in the `scipy.stats` package. The `ppf` function is the inverse of the cumulative distribution function. For example, `ppf(0.05)` returns the value of the data such that the cumulative distribution function is equal to 0.05 at the returned value. To find the 5% and 95% values, type (recall that by default the `norm` distribution has mean zero and standard deviation 1; you can specify different values with the `loc` and `scale` keyword arguments, respectively)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5% limit: -1.6448536269514729\n",
      "95% limit: 1.6448536269514722\n",
      "check if it works for 5%: 0.049999999999999975\n",
      "check if it works for 95%: 0.95\n",
      "5% limit with mu=20, sig=10: 3.5514637304852705\n",
      "check: 0.049999999999999975\n"
     ]
    }
   ],
   "source": [
    "from scipy.stats import norm\n",
    "xvalue_05 = norm.ppf(0.05)\n",
    "xvalue_95 = norm.ppf(0.95)\n",
    "print('5% limit:', xvalue_05)\n",
    "print('95% limit:', xvalue_95)\n",
    "print('check if it works for 5%:', norm.cdf(xvalue_05))\n",
    "print('check if it works for 95%:', norm.cdf(xvalue_95))\n",
    "# Next, specify a mean and standard deviation\n",
    "xvalue_05_musig = norm.ppf(0.05, loc=20, scale=10) # mu = 20, sigma = 10\n",
    "print('5% limit with mu=20, sig=10:', xvalue_05_musig)\n",
    "print('check:', norm.cdf(xvalue_05_musig, loc=20, scale=10))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A similar function exists for the $t$ distribution. The $t$-distribution takes one additional argument: the number of degrees of freedom, which is equal to the number of data points minus 1. For example, consider a sample with 40 data points, a sample mean of 20, and a sample standard deviation of the mean of 2, then the 5 and 95 percentiles are"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5% limit:   16.6302497610052\n",
      "95% limit:  23.369750238994797\n",
      "check if it works for 5%: 0.050000000215266564\n",
      "check if it works for 95%: 0.9499999997847333\n"
     ]
    }
   ],
   "source": [
    "from scipy.stats import t\n",
    "xvalue_05 = t.ppf(0.05, 39, loc=20, scale=2)\n",
    "xvalue_95 = t.ppf(0.95, 39, loc=20, scale=2)\n",
    "print('5% limit:  ',xvalue_05)\n",
    "print('95% limit: ',xvalue_95)\n",
    "print('check if it works for 5%:', t.cdf(xvalue_05, 39, loc=20, scale=2))\n",
    "print('check if it works for 95%:', t.cdf(xvalue_95, 39, loc=20, scale=2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 3. <a name=\"back3\"></a>Count the number of means outside 95 percentile\n",
    "Go back to Exercise 1. Generate 1000 datasets each with 100 values drawn from a normal distribution with mean 4 and standard deviation 2. For each dataset, evaluate whether the sample mean is within the 95 percentile of the $t$-distribution around the true mean of 4 (the standard deviation of the sample mean is different every time, of course). Count how many times the sample mean is outside the 95 percentile around the true mean of the $t$ distribution. If the theory is correct, it should, of course, be the case for about 5% of the datasets. Try five different seeds and report the percentage of means in the dataset that is outside the 95 percentile around the true mean. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a href=\"#ex3answer\">Answers to Exercise 3</a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 4. <a name=\"back4\"></a>$t$ test on dataset of 20 values\n",
    "Generate 20 datapoints from a Normal distribution with mean 39 and standard deviation 4. Use a seed of 2. Compute and report the sample mean and sample standard deviation of the dataset and the sample standard deviation of the sample mean."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If you computed it correctly, the mean of the 20 data points generated above is 38.16. Somebody now claims that the 20 datapoints are taken from a distribution with a mean of 40. You are asked to decide wether the true underlying mean could indeed be 40. In statistical terms, you are asked to perform a Hypothesis test, testing the null hypothesis that the mean is 40 against the alternative hypothesis that the mean is not 40 at significance level 5%. Hence, you are asked to do a two-sided $t$-test. All you can do in Hypothesis testing it trying to reject the null hypothesis, so let's try that. Most statistics books give a cookbook recipe for performing a $t$-test. Here we will visualize the $t$-test. We reject the null hypothesis if the sample mean is outside the 95% interval around the mean of the corresponding $t$-distribution. If the mean is inside the 95% interval we can only conclude that there is not enough evidence to reject the null hypothesis. Draw the probability density function of a $t$-distribution with mean 40 and standard deviation equal to the sample standard deviation of the sample mean you computed above. Draw red vertical lines indicating the left and right limits of the 95% interval around the mean. Draw a heavy black vertical line at the position of the sample mean you computed above. Decide whether you can reject the null hypothesis that the mean is 40 and add that as a title to the figure."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a href=\"#ex4answer\">Answers to Exercise 4</a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 5. <a name=\"back5\"></a>Hypothesis tests on Wooden beam data\n",
    "Load the data set of experiments on wooden beams stored in the file `douglas_data.csv`. First, consider the first 20 measurements of the bending strength. Compute the sample mean and the standard deviation of the sample mean. The manufacturer claims that the mean bending strength is only 50 N/mm$^2$. Perform a $t$-test (significance level 5%) with null hypothesis that the mean is indeed 50 N/mm$^2$ and alternative hypothesis that the mean is not 50 N/mm$^2$ using the approach applied in Exercise 4."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Repeat the $t$-test above but now with all the measurements of the bending strength. Do you reach the same conclusion?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a href=\"#ex5answer\">Answers to Exercise 5</a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Central limit theorem\n",
    "So far we looked at the distribution of the sample mean of a dataset while we knew that the data was taken from a normal distribution (except for the wooden beam data, but that looked very much like a Normal distribution). Such a sample mean has a Student $t$-distribtion, which approaches the Normal distribution when the dataset is large. Actually, 100 datapoints is already enough to approach the Normal distribution fairly closely. You may check this by comparing, for example, the percent point function `ppf` of a Normal distribution with a $t$-distribution with 99 degrees of freedom, or by simply plotting the pdf of both distributions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "95 percentile Standard Normal:   1.6448536269514722\n",
      "95 percentile t-dist with n=99:  1.6603911559963895\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('95 percentile Standard Normal:  ',norm.ppf(0.95))\n",
    "print('95 percentile t-dist with n=99: ',t.ppf(0.95,99)) \n",
    "x = np.linspace(-4,4,100)\n",
    "y1 = norm.pdf(x)\n",
    "y2 = t.pdf(x,99)\n",
    "plt.plot(x,y1,'b',label='Normal')\n",
    "plt.plot(x,y2,'r',label='t-dist')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Central limit theorem now states that the distribution of the sample mean approaches the Normal distribution in the limit even if the dataset is drawn from an entirely different distribution! We are going to test this theorem by drawing numbers from a Gamma distribution. The Gamma distribution is a skewed distribution and takes a shape parameter $k$ and a scale parameter $\\theta$, and is defined for $x>0$. Details on the Gamma distribution can be found, for example [here](http://en.wikipedia.org/wiki/Gamma_distribution). Let's set the shape parameter equal to 2 and the scale parameter equal to 1 (which happens to be the default). When the scale parameter is equal to 1, the mean is equal to the shape parameter. The pdf of the Gamma distribution for these values is shown below. The mean is indicated with the red vertical line."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "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": [
    "from scipy.stats import gamma\n",
    "x = np.linspace(1e-6, 10, 100)\n",
    "y = gamma.pdf(x, 2, scale=1)\n",
    "plt.plot(x, y)\n",
    "plt.axvline(2, color='r');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Random numbers may be drawn from any distribution in the `scipy.stats` package with the `rvs` function. Here, we draw 1000 numbers and add the histogram to the previous figure"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "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": [
    "x = np.linspace(1e-6, 10, 100)\n",
    "y = gamma.pdf(x, 2)\n",
    "plt.plot(x, y)\n",
    "plt.axvline(2, color='r')\n",
    "data = gamma.rvs(2, size=1000)\n",
    "plt.hist(data, bins=20, density=True);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 6. <a name=\"back6\"></a>Explore Central Limit Theorem for Gamma Distribution\n",
    "Generate $N$ datasets of 20 numbers randomly drawn from a Gamma distribution with shape parameter equal to 2 and scale equal to 1. Draw a histogram of the means of the $N$ datasets using 20 bins. On the same graph, draw the pdf of the Normal distribution using the mean of means and sample standard deviation of the means; choose the limits of the $x$-axis between 0 and 4. Make 3 graphs, for $N=100,1000,10000$ and notice that the distribution starts to approach a Normal distribution. Add a title to each graph stating the number of datasets."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a href=\"#ex6answer\">Answers to Exercise 6</a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Answers to the exercises"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a name=\"ex1answer\">Answers to Exercise 1</a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The mean of the means is: 4.004742118538674\n",
      "The standard deviation of the means is: 0.1904854817672302\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(3, 5)"
      ]
     },
     "execution_count": 8,
     "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"
    },
    {
     "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": [
    "rnd.seed(22)\n",
    "mean_of_data = np.mean(2 * rnd.standard_normal((1000, 100)) + 4, 1)\n",
    "print('The mean of the means is:', np.mean(mean_of_data))\n",
    "print('The standard deviation of the means is:', np.std(mean_of_data, ddof=1))\n",
    "plt.figure()\n",
    "plt.boxplot(mean_of_data)\n",
    "plt.ylim(3, 5)\n",
    "plt.figure()\n",
    "plt.hist(mean_of_data, density=True)\n",
    "plt.xlim(3,5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a href=\"#back1\">Back to Exercise 1</a>\n",
    "\n",
    "<a name=\"ex2answer\">Answers to Exercise 2</a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The mean of the means is: 4.001281312353626\n",
      "The standard deviation of the means is: 0.0654148250988205\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(3, 5)"
      ]
     },
     "execution_count": 9,
     "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"
    },
    {
     "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": [
    "rnd.seed(22)\n",
    "mean_of_data = np.mean(2 * rnd.standard_normal((1000, 1000)) + 4, 1)\n",
    "print('The mean of the means is:', np.mean(mean_of_data))\n",
    "print('The standard deviation of the means is:', np.std(mean_of_data, ddof=1))\n",
    "plt.figure()\n",
    "plt.boxplot(mean_of_data)\n",
    "plt.ylim(3,5)\n",
    "plt.figure()\n",
    "plt.hist(mean_of_data)\n",
    "plt.xlim(3, 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a href=\"#back2\">Back to Exercise 2</a>\n",
    "\n",
    "<a name=\"ex3answer\">Answers to Exercise 3</a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "percentage of datasets where sample mean is outside 95 percentile: 4.0\n",
      "percentage of datasets where sample mean is outside 95 percentile: 5.8\n",
      "percentage of datasets where sample mean is outside 95 percentile: 6.2\n",
      "percentage of datasets where sample mean is outside 95 percentile: 5.9\n",
      "percentage of datasets where sample mean is outside 95 percentile: 5.7\n"
     ]
    }
   ],
   "source": [
    "from scipy.stats import t\n",
    "for s in [22, 32, 42, 52, 62]:\n",
    "    rnd.seed(s)\n",
    "    data = 2.0 * rnd.standard_normal((1000, 100)) + 4.0\n",
    "    mean = np.mean(data, 1)\n",
    "    sighat = np.std(data, axis=1, ddof=1) / np.sqrt(100)\n",
    "    count = 0\n",
    "    for i in range(1000):\n",
    "        low = t.ppf(0.025, 99, loc=4, scale=sighat[i])\n",
    "        high = t.ppf(0.975, 99, loc=4, scale=sighat[i])\n",
    "        if (mean[i] < low) or (mean[i] > high): count += 1\n",
    "    print('percentage of datasets where sample mean is outside 95 percentile:', count * 100 / 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a href=\"#back3\">Back to Exercise 3</a>\n",
    "\n",
    "<a name=\"ex4answer\">Answers to Exercise 4</a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean of the data: 38.16274004693891\n",
      "std of the data: 4.460326920868606\n",
      "std of the mean: 0.9973594196934527\n"
     ]
    }
   ],
   "source": [
    "rnd.seed(2)\n",
    "data = 4 * rnd.standard_normal(20) + 39\n",
    "mu = np.mean(data)\n",
    "sig = np.std(data, ddof=1)\n",
    "sighat = np.std(data, ddof=1) / np.sqrt(20)\n",
    "print('mean of the data:', mu)\n",
    "print('std of the data:', sig)\n",
    "print('std of the mean:', sighat)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "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": [
    "x = np.linspace(37, 43, 100)\n",
    "y = t.pdf(x, 19, loc=40, scale=sighat)\n",
    "plt.plot(x, y)\n",
    "perc025 = t.ppf(0.025, 19, loc=40, scale=sighat)\n",
    "perc975 = t.ppf(0.975, 19, loc=40, scale=sighat)\n",
    "plt.axvline(perc025, color='r')\n",
    "plt.axvline(perc975, color='r')\n",
    "plt.axvline(mu, color='k', lw=5)\n",
    "plt.title('H0 cannot be rejected');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a href=\"#back4\">Back to Exercise 4</a>\n",
    "\n",
    "<a name=\"ex5answer\">Answers to Exercise 5</a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sample mean, standard deviation of sample mean:  69.37100000000001 3.5096083077295406\n"
     ]
    },
    {
     "data": {
      "image/png": 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tBrOoMIv05ATWV9jBgMYbFvQmbvQcuHTxrPE9KC/B7+P8mbm8tr8OVbvEoBl/FvQmbry2v47inFSKc1PHfdkXz86jpvEMhxvaxn3ZxljQm7jQFQjy5sH6cR+26bFylrPc1+x0CMYDFvQmLrxV3URLRzcXz/Im6Evz0piWNcHOe2M8YUFv4sJr++sQgQtmenPQkohw0axcXj9QRyBo4/RmfFnQm7iwvqKOhdMyyUpN8qyGlbPzOdXezY6aJs9qMPHJgt7EvNaObrZXN3GRR8M2PS5yf03Y/vRmvFnQm5i39XAjgaB6fq6Z3InJnDNpIpsO2Xn9zPiyoDcxb+PBBvw+YUmJ95crXl6aQ3llg5222IwrC3oT8zYdamDBtEwmJod1stYxtaI0l9bOALuPnfK6FBNHLOhNTGvvCrC9uokVpTlelwJwto6NB234xowfC3oT07ZXN9EZCLJ8emQEfUFGCqV5aWy0cXozjizoTUzbeLABEVgWIUEPsHx6DpsrGwja/vRmnFjQm5i2qbKeuZMzyExN9LqUs1bMyKH5TBf7Tpz2uhQTJyzoTczq7A6ypaoxYsbneyw/O05v+9Ob8WFBb2LWziPNtHcFIy7oC7NTmZY1gU2VNk5vxocFvYlZGw85a8zLIyzowRm+2XSowc5Pb8aFBb2JWZsONTCrYCK5E5O9LuV9VpTmUNfSyYHaVq9LMXHAgt7EpEBQKa9sjMi1eXAOnIJ3f3UYM5Ys6E1M2nv8FC0d3RGz/3xvJbmp5E1MZktlo9elmDhgQW9i0pYqJ0CXRsD5bfoiIpSVZFNeZUFvxp4FvYlJ5ZWNTMpIpjB7gtel9KtsejaHG9o4eard61JMjLOgNzFpS1UjZSU5iIjXpfSr59eGrdWbsRZW0IvI1SKyT0QqROTOPh5PFpEn3Mc3ish0d3qiiDwsIjtFZI+I3DW65Rvzfseaz3Ck6UzEDtv0mD81k+QEH+U2Tm/G2KBBLyJ+4D5gNTAPuFlE5vVq9nmgUVVnAfcCP3an3wgkq+q5wFLg9p4vAWPGSk9wlk2P7KBPSvCxqCiLLVV24JQZW+Gs0S8HKlT1oKp2Ao8Da3q1WQM87N5+CrhcnN/MCqSJSAIwAegE7ETcZkxtqWpkQqKfD0zJ8LqUQZWVZLPr6CnOdAa8LsXEsHCCfhpQHXK/xp3WZxtV7QaagVyc0G8FjgGHgf9fVd+3+iIit4lIuYiU19bWDvlFGBOqvKqB84qySPRH/iaosunZdAeV7dV2wXAzdsL5n9DX1qzex23312Y5EACmAqXA/xKRGe9rqHq/qpapall+fn4YJRnTt9aObvYcOx3xwzY9lhQ7ddrwjRlL4QR9DVAUcr8QONpfG3eYJhNoAD4J/ElVu1T1JLABKBtp0cb0Z3t1E4GgRvyG2B5ZqUnMKphoe96YMRVO0G8GZotIqYgkATcBa3u1WQvc4t6+AXhJnbM1HQZWiSMNOB/YOzqlG/N+5ZWNiBARFwIPV1lJNlurGu1CJGbMDBr07pj7HcDzwB7gSVXdJSJ3i8h1brMHgVwRqQC+CvTsgnkfMBF4G+cL45equmOUX4MxZ5VXNTBnUjoZKZFzoZHBLC3J5lR7N/tPtnhdiolRCeE0UtV1wLpe074bcrsdZ1fK3s9r6Wu6MWMhEFS2H27iuvOmel3KkJS55+Mpr2pgzuR0j6sxsSjyd0swJkzvnDjN6Y7uqBmf7zE9N5XctKSz5+cxZrRZ0JuYsfWwe6BUSWSesbI/IsKSkmy2HbZdLM3YsKA3MWNLVSN5E5MpyoncE5n1Z2lJNofqWqlv6fC6FBODLOhNzNha1ciS4qyIPpFZf3r2p99qa/VmDFjQm5hQ19JBZX1b1I3P91hYmEmCT84OPxkzmizoTUzoGd+Opv3nQ6Uk+pk/NcM2yJoxYUFvYsKWqkYS/cK50zK9LmXYlpRks6Omia5A0OtSTIyxoDcxYWtVI/OnZpKS6Pe6lGFbWpJNe1eQPcfsBK9mdFnQm6jXFQjyVk3T2Q2a0erdE5zZ8I0ZXRb0JurtPnqKju5g1G6I7TE1awJTMlMs6M2os6A3Ua8nGJeUZHlcycjZgVNmLFjQm6i35XAjUzNTmJIZfQdK9bakOJsjTWc43tzudSkmhljQm6i3raoxaner7K1n+Mn2pzejyYLeRLVjzWc42twe9Rtie8ybkkFygu/sBc6NGQ0W9Caq9YzPR8ulAweTlOBjUWGWrdGbUWVBb6LalqpGUhJ9fGBKhteljJolJdnsOtpMe1fA61JMjLCgN1Fta1UjiwqzSPTHzkd5aUk2XQFl55Fmr0sxMSJ2/neYuHOmM8Cuo6eifv/53pYUO7uJ2v70ZrRY0JuotaOmie6gxlzQ505MpjQvzYLejBoLehO1trgbLBfHyB43oZYUZ7O1qhFV9boUEwMs6E3U2lrVxIz8NHLSkrwuZdQtLcmmvrWTqvo2r0sxMcCC3kQlVWXr4UaWxuDaPLx74JQN35jRYEFvolJlfRsNrZ0xNz7fY3bBRNKTE84OTxkzEhb0Jir1rOnGatD7fMLiEmec3piRsqA3UWlLVSMZKQnMzJ/odSljZmlxNvtOnOZUe5fXpZgoZ0FvotJW90RmPp94XcqYWVqSjSpst9MWmxEKK+hF5GoR2SciFSJyZx+PJ4vIE+7jG0VkeshjC0XkDRHZJSI7RSRl9Mo38aj5TBfvnDwdsxtieywqysQntkHWjNygQS8ifuA+YDUwD7hZROb1avZ5oFFVZwH3Aj92n5sAPAp8QVXnA5cC9jvUjMi2w42oEjOnJu5PekoicyZnWNCbEQtnjX45UKGqB1W1E3gcWNOrzRrgYff2U8DlIiLAlcAOVX0LQFXrVdXO1GRGpLyyEb9POK8o+q8oNZiykmy2HW6kOxD0uhQTxcIJ+mlAdcj9Gndan21UtRtoBnKBcwAVkedFZKuIfKOvBYjIbSJSLiLltbW1Q30NJs6UVzUwf2oGackJXpcy5sqmZ9PaGWDv8dNel2KiWDhB39fWrt7HZffXJgFYCXzK/fejInL5+xqq3q+qZapalp+fH0ZJJl51dgfZXt1EWUmO16WMi2XTnde5ubLB40pMNAsn6GuAopD7hcDR/tq44/KZQIM7/RVVrVPVNmAdsGSkRZv45ZynPciyGLnQyGCmZk1gWtYEu+KUGZFwgn4zMFtESkUkCbgJWNurzVrgFvf2DcBL6pyN6XlgoYikul8AHwR2j07pJh71BN7SOAl6cIZvNlc22AnOzLANGvTumPsdOKG9B3hSVXeJyN0icp3b7EEgV0QqgK8Cd7rPbQT+BefLYjuwVVX/OPovw8SLzZUNTM9NpSA9fvbSLZuew8nTHVQ3nPG6FBOlwtqaparrcIZdQqd9N+R2O3BjP899FGcXS2NGRFUpr2pk1dwCr0sZVz3DVOVVDRTnpnpcjYlGdmSsiRoH61ppaO2Mm/H5HucUpJOeksBmG6c3w2RBb6JGubvnydI42eOmh88nLC3JPvv6jRkqC3oTNTZXNpKdmsjM/DSvSxl3y6bnsP9kC42tnV6XYqKQBb2JGluqGimbnoNz0HV8KbMLkZgRsKA3UaH2dAeH6lrjbny+x6KiLBL9QrkFvRkGC3oTFeJ1fL5HSqKfc6dl2hGyZlgs6E1U2HiogQmJfhYWZnpdimeWl+ayo6aJM512XkAzNBb0Jiq8ebCepSXZJPrj9yO7YkYOXQHnoujGDEX8/q8xUaOxtZO9x0+zojQ+h216lJVk4/cJbx6s97oUE2Us6E3E2+SOS58/M9fjSryVnpLIgqkZbDxo4/RmaCzoTcTbeLCB5ARfXI/P91gxI5ft1U20d9k4vQmfBb2JeBsP1bOkOJvkBL/XpXhuRWkOnYEg2+yC4WYILOhNRGs+08XuY6dYMSO+x+d7lE3PwSfYOL0ZEgt6E9HKKxtQhRWl8T0+3yNzQiLzpmaw8ZAFvQmfBb2JaBsPNZDk97G4OPYvBB6uFaW5bDvcREe3jdOb8FjQm4j25sF6zivKIiXRxud7rCjNoaM7yFvVzV6XYqKEBb2JWKfbu3j7SDPn2/j8eywvzUFsnN4MgQW9iVjlVY0E1dml0LwrKzWJuZNtnN6Ez4LeRKw3DtST5PexpDg+z1g5kAtm5FJe2Wj705uwWNCbiLV+fx1LS7KZkGTj872tnJ1LR3eQrXbaYhMGC3oTkepbOth97BQrZ+d5XUpEWl6aS4JPWF9R53UpJgpY0JuI9PoBZ/z5olkW9H2ZmJzA4uIsNljQmzBY0JuItKGijvSUBM6dZue36c9Fs/LYcaSZ5rYur0sxEc6C3kQcVeW1/XVcODMXvy/+rg8brpWz8lCFNw7aWr0ZmAW9iThV9W0caTrDShu2GdCioizSkvy8tt+C3gzMgt5EnJ4NjDY+P7BEv4/zZ+TaOL0ZVFhBLyJXi8g+EakQkTv7eDxZRJ5wH98oItN7PV4sIi0i8rXRKdvEsg0VdUzNTKE0L83rUiLeRbPyqKxvo7qhzetSTAQbNOhFxA/cB6wG5gE3i8i8Xs0+DzSq6izgXuDHvR6/F3hu5OWaWBcIKq8fqOeiWXmI2Pj8YHp2P339gK3Vm/6Fs0a/HKhQ1YOq2gk8Dqzp1WYN8LB7+yngcnH/l4rI9cBBYNfolGxi2a6jzTSf6bL958M0u2Ai+enJrK+w0yGY/oUT9NOA6pD7Ne60PtuoajfQDOSKSBrwD8D3B1qAiNwmIuUiUl5bWxtu7SYG9WxYvCDOrw8bLhFh5aw8NlTUEQyq1+WYCBVO0Pf1+7n3J6q/Nt8H7lXVloEWoKr3q2qZqpbl5+eHUZKJVS/vPcmCaRkUpKd4XUrUuHROPg2tnbxVY5cXNH0LJ+hrgKKQ+4XA0f7aiEgCkAk0ACuA/y0ilcCXgW+KyB0jrNnEqMbWTrYebmTVnAKvS4kql8zOxyfw8j77NWz6Fk7QbwZmi0ipiCQBNwFre7VZC9zi3r4BeEkdF6vqdFWdDvwEuEdV/22Uajcx5tX9tQQVLptrQT8U2WlJLC7O5uW9J70uxUSoQYPeHXO/A3ge2AM8qaq7RORuEbnObfYgzph8BfBV4H27YBozmJf3niQnLYmFhXbZwKFaNbeAnUeaOXmq3etSTARKCKeRqq4D1vWa9t2Q2+3AjYPM43vDqM/EiUBQeeWdWi6bU2CnPRiGy+YU8P89v4+/vFPLx8uKBn+CiSt2ZKyJCNurG2ls67Jhm2H6wJR0Jmek2PCN6ZMFvYkIL++txe8TLplte10Nh4hw2dx8XttfR2d30OtyTISxoDcR4aW9J1lanE1maqLXpUSty+YU0NLRTXlVg9elmAhjQW88d7y5nd3HTtmwzQhdNCuPJL/Phm/M+1jQG8+9vM8JplUW9COSlpzAihk5vGRBb3qxoDeee2HXcaZlTeCcSRO9LiXqXT63gAO1rRyoHfBgdBNnLOiNp061d7G+oo7VCybb2SpHwVULJgPwp7ePe1yJiSQW9MZTL+05SVdAWX3uFK9LiQlTMiewuDiL594+5nUpJoJY0BtPrdt5jEkZySwusqNhR8vqBZN5+8gpDtfbxUiMw4LeeKa1o5tX3qnl6vmT8dnRsKNm9QLn19GfdtlavXFY0BvP/GVfLR3dQRu2GWVFOaksmJbBczZOb1wW9MYz694+Rt7EJJZNz/G6lJizesEUth1u4ljzGa9LMRHAgt54or0rwMt7T3LFvMl2ErMxcLXtfWNCWNAbT7z6Ti1tnQE+fO5kr0uJSTPzJzJnUroN3xjAgt545Nkdx8hKTeT8GXZt2LGy+tzJbK5s4HiznaM+3lnQm3F3ur2LF3Yf59qFU0j020dwrKw5bxqq8IftR7wuxXjM/peZcffczuO0dwX5qyWFXpcS00rz0lhSnMXTW2tQVa/LMR6yoDfj7umtNZTmpdlBUuPgr5YU8s6JFnYdPeV1KcZDFvRmXFU3tLHxUAMfWzLNzm0zDj6ycCpJfh9Pb63xuhTjIQt6M66e2eaMF1+/eJrHlcSHzNREPjSvgLXbj9IVsCtPxSsLejNuVJXfba3hghm5FGanel1O3PjYkkLqWzt5ZV+t16UYj1jQm3Gz9XATlfVt/NUSW5sfT5eck09uWiv9Xm4AABEsSURBVBK/22bDN/HKgt6Mm6e21DAh0W/nthlniX4f1503lf/efZLG1k6vyzEesKA346L5TBe/33aEaxZOYWJygtflxJ1PLCuiMxDkyfJqr0sxHrCgN+Pi6S01nOkKcOuF070uJS7NnZzBitIcHnmzikDQ9qmPNxb0ZswFg8qv3qhkaUk2C6Zlel1O3Lr1wunUNJ6xi4fHobCCXkSuFpF9IlIhInf28XiyiDzhPr5RRKa7068QkS0istP9d9Xolm+iwav7a6msb+OzF5R4XUpcu2LeJKZkpvCrNyq9LsWMs0GDXkT8wH3AamAecLOIzOvV7PNAo6rOAu4FfuxOrwM+oqrnArcAj4xW4SZ6PPx6JfnpyWevfGS8keD38enzS3htfx0VJ1u8LseMo3DW6JcDFap6UFU7gceBNb3arAEedm8/BVwuIqKq21T1qDt9F5AiIsmjUbiJDpV1rfzlnVo+ubyYpAQbKfTaJ5YVkeT38cgblV6XYsZROP/zpgGhm+pr3Gl9tlHVbqAZ6H3+2Y8B21S1Y3ilmmj0yJtV+EX41Ipir0sxQN7EZK5dNIWnttRwur3L63LMOAkn6Ps6IUnvzfYDthGR+TjDObf3uQCR20SkXETKa2vt6L1Y0dDayWObDnPNwikUZKR4XY5xfe7CUlo7AzzyZpXXpZhxEk7Q1wBFIfcLgaP9tRGRBCATaHDvFwLPAJ9V1QN9LUBV71fVMlUty8/PH9orMBHrwfUHOdMV4I7LZnldiglxbmEml83J5xevHqS1o9vrcsw4CCfoNwOzRaRURJKAm4C1vdqsxdnYCnAD8JKqqohkAX8E7lLVDaNVtIl8TW2dPPx6FR8+dwqzJ6V7XY7p5e8un01jWxeP2lp9XBg06N0x9zuA54E9wJOquktE7haR69xmDwK5IlIBfBXo2QXzDmAW8B0R2e7+KRj1V2EizkMbKmnp6ObvVtnafCRaUpzNxbPzuP/Vg5zpDHhdjhljYR2LrqrrgHW9pn035HY7cGMfz/sB8IMR1miiTPOZLn654RBXz5/M3MkZXpdj+vHlD83mYz97g19vrOJ/XjzD63LMGLL93cyo+88NlZxu7+bvLre1+Ui2tCSHi2bl8h+vHKS9y9bqY5kFvRlVnYEgD7x2kCvmTWL+VDvdQaT7+8vPoa6lgwfXH/K6FDOGLOjNqKpuaONMV4A7V8/1uhQThuWlOVw1fxL3vVxBZ7ddgSpWWdCbUdPS0U3t6Q4+d9F0ZuZP9LocE6ZvXzOP7qByuKHN61LMGLGgN6NCVamsbyPR7+PvLp/tdTlmCIpyUrnt4hnUtXRwut32q49FFvRmVPxh+1Fa2rsoykklIyXR63LMEP3tZTNJSvBRWd9K0M5XH3Ms6M2INbV1cs+6PaQlJ5Cfbuesi0apSQkU56TS2tHNrzfaQVSxxoLejNi3f/82Da2dzMhL6/OkRyY65E5MJjM1iXvW7eVQXavX5ZhRZEFvRuQP24/w7I5jfPlDs0mza8FGNQFm5qeRlODjK09spztge+HECgt6M2zHms/wnd+/zZLiLL7wwZlel2NGQZLfxw+uX8D26iZ+9pc+z0FoopAFvRmWQFD52m/fojuo/MvHzyPBbx+lWPGRRVO5btFUfvrn/WyvbvK6HDMK7H+nGZYfPbeHDRX1fPfaeUzPS/O6HDPK/mnNAiZlpHD7I+WcONXudTlmhCzozZA9WV7NL147xGcvKOGm5XblqFiUmZrIA7eUcbq9m7/+Vbmd4TLKWdCbIdl0qIFvPbOTlbPy+O61va8Rb2LJB6Zk8NObFrPzSDNff+otVG3/+mhlQW/Ctu/4ab7w6BaKslO575NLbFw+DlwxbxLfuGouz+44xo+e22thH6VsfzgTlt1HT/HpBzeS6BceunUZmal29Gu8+MIHZ3CkqY2fv3oQBe5aPRcRO2IimljQm0HtOtrMpx7YyIREP4/99fm28TXOiAj/tGYBPhHuf/UggaDy7Ws+YGEfRSzozYBer6jjb369lbQkP4/ddj4luRby8UhE+P518/GJ8OD6Q7R2dHP3mgUkJdjwXTSwoDd9UlV+uaGSH67bQ2leGr+8dRlFOalel2U8JCL840fmkZbs576XD7D/ZAs/+/QSCtJTvC7NDMK+js37tHV287Xf7uDuZ3dz2ZwCnvnbCy3kDeCE/devmsu/fXIxu4+e4rp/3cCWqkavyzKDsKA37/F6RR1X/eRVnt5aw5cun839n1lKup122PRy7cKpPP03F5LgF278j9f54R932772EcyC3gDQ0NrJXb/byScf2EiCz8eTt1/AV684B5/PNriZvs2bmsFzf38xNy0v5hevHWL1T19l/f46r8syfbAx+jh3ur2LB147xIPrD9HW2c3tl8zgK1ecQ0qi3+vSTBRIT0nkno+ey7ULp3Dn0zv59IMbWTkrj69dNYfzirK8Ls+4LOjj1PHmdn6zsYpH3qyisa2L1Qsm89UrzmH2pHSvSzNR6MKZebzwlUv49cbD3PdyBdfft4FVcwu49cLprJyVZ78MPWZBH0c6ugOs31/H01treH7XCYKqrJpTwN9/aDYLC23ty4xMSqKfz68s5RPLinho/SF+9UYln31oE6V5ady8vIhrFk5lWtYEr8uMSxb0Ma6xtZM3Dtbz5z0neWH3cU63d5M5IZHPryzl0ytKKM61vWnM6JqYnMCXLp/N7R+cwXM7j/PwG5Xcs24v96zby+LiLFYvmMzKWfnMnZxua/rjxII+hgSDSmV9Kztqmtle3UR5VQO7jp5CFdJTErhy3mSuXTiFi2bl2YEuZswlJ/i5fvE0rl88jUN1razbeYw/7jjGPev2AnvJTUvi/Bm5nFeUxcLCTBZMy7SrlI2RsHpVRK4Gfgr4gQdU9Ue9Hk8GfgUsBeqBT6hqpfvYXcDngQDwJVV9ftSqj0Md3QFqT3dwrLmd6oY2ahrPcKiulYqTLVScbOFMl7OL24REPwsLM/nKh87holl5LCrMtJOQGc+U5qXxxctm8cXLZnGs+QwbKup5vaKOjYca+OPOY2fbFWZPYHbBRGbmT6Q4N5XC7AkUZqcyKT2FjAkJdtqFYRo06EXED9wHXAHUAJtFZK2q7g5p9nmgUVVnichNwI+BT4jIPOAmYD4wFfhvETlHVWN+h9tgUOkOKoGg0hkI0h0I0hVQOruDdAYCdHQH6egO0t4ZoL07QFtngLaOAG2d3Zxu7+ZUexen27tpauuioa2TprZOak930NjW9b5lTclMYVbBRG5eXszcyeksLMpkVv5EC3YTkaZkTuCGpYXcsLQQgLqWDnbUNLHryCn2n2xh/8kWXj9QT0f3e69Zm5zgoyAjmZzUJLLTkshOTSJzQiLpKQmkpySQlpxAWlICE5L8TEj0k5LoJyXRR3KCn6QEH0kJPhL9QpLfR4LfR4JPSPAJfp/E/BdIOGv0y4EKVT0IICKPA2uA0KBfA3zPvf0U8G/i9Nwa4HFV7QAOiUiFO783Rqf8d+09foo7frPt7P3Q06m+58Sq+u4/PW2c2z3TFdWQ+6ooEFR3ujstqM60QNCZHggqAVWC7r8jPZtrSqKP9JREsiYkkp2WRGleGsum51CQnsKkjGQmZ6ZQlJPKtKwJtiukiWp5E5NZNXcSq+ZOOjstGFTqWjqobjxDTWMbtac7OHGqndrTHTS0dVHf0sn+Ey2cau+ipaN7xP/ffAIJPh8+H/hF8Ing8wk+AZ84XwQiTjuh57bz5SDi/uHd++BcbL3nC0TO/vV+H/qXV87evvScfL49Btd5CCfopwHVIfdrgBX9tVHVbhFpBnLd6W/2eu603gsQkduA2wCKi4d3xaKUBD9zeu8aKH3efE/nD/Sm9LyhQs+bzdk32Cfvvtn+ng+ET/C79xN8PhL8zu1Ev7MmkeDznV2zSPILyYl+UhKctY605ARSk/ykJiUwMTnBxtBNXPP5hIKMFAoyUlhakj1gW1WltTNAW0c3rZ0BWju6OdMVoL0rQHtX8OyvaOdfpTsQpDugdAWDBALv/vIOuCtugaCeXbELBBXFWbFTd1roSl/QvaEhtUDvlcf3rni+3Kv+0NyaMkZ7JYUT9H19D/X+/uyvTTjPRVXvB+4HKCsrG9Z38/S8NO771JLhPNUYE8VEhInJzgpSNLjvU73vj31uhbPaWAMUhdwvBI7210ZEEoBMoCHM5xpjjBlD4QT9ZmC2iJSKSBLOxtW1vdqsBW5xb98AvKTOb5W1wE0ikiwipcBsYNPolG6MMSYcg/7Wccfc7wCex9m98iFV3SUidwPlqroWeBB4xN3Y2oDzZYDb7kmcDbfdwBfjYY8bY4yJJGENaqnqOmBdr2nfDbndDtzYz3N/CPxwBDUaY4wZAdu1wxhjYpwFvTHGxDgLemOMiXEW9MYYE+NER3rs8CgTkVqgagSzyAMi8XpmVtfQWF1DY3UNTSzWVaKq+X09EHFBP1IiUq6qZV7X0ZvVNTRW19BYXUMTb3XZ0I0xxsQ4C3pjjIlxsRj093tdQD+srqGxuobG6hqauKor5sbojTHGvFcsrtEbY4wJYUFvjDExLmqDXkRSRGSTiLwlIrtE5Pvu9FIR2Sgi+0XkCffUypFQ13+KyCER2e7+OW886wqpzy8i20TkWfe+p/01QF2e95eIVIrITnf55e60HBF50e2vF0Vk4MsfjV9d3xORIyH99WEP6soSkadEZK+I7BGRCyKkv/qqy9P+EpE5IcveLiKnROTLY9VfURv0QAewSlUXAecBV4vI+TgXJr9XVWcDjTgXLo+EugC+rqrnuX+2j3NdPf4e2BNy3+v+6tG7LoiM/rrMXX7Pvs13An92++vP7v1IqAuc97Gnv9b1+8yx81PgT6o6F1iE835GQn/1VRd42F+quq9n2cBSoA14hjHqr6gNenW0uHcT3T8KrMK5QDnAw8D1EVKX50SkELgGeMC9L3jcX33VFeHW4PQTeNRfkUhEMoBLcK5Ngap2qmoTHvfXAHVFksuBA6paxRj1V9QGPZz9ub8dOAm8CBwAmlS1223S58XIx7suVd3oPvRDEdkhIveKSPJ41wX8BPgGEHTv5xIB/dVHXT287i8FXhCRLeJcwB5gkqoeA3D/LYiQugDucPvrIQ+GSGYAtcAv3SG4B0QkDe/7q7+6wNv+CnUT8Jh7e0z6K6qDXlUD7k+fQmA58IG+mo1vVe+vS0QWAHcBc4FlQA7wD+NZk4hcC5xU1S2hk/toOq791U9d4HF/uS5S1SXAauCLInKJBzX0pa+6fgbMxBkuPAb8n3GuKQFYAvxMVRcDrXg3rBWqv7q87i8A3G1i1wG/HcvlRHXQ93B/iv0FOB/IEucC5eDxxchD6rpaVY+5wzodwC9xvpjG00XAdSJSCTyOM2TzE7zvr/fVJSKPRkB/oapH3X9P4oyfLgdOiMgUAPffk5FQl6qecFcwgsAvGP/+qgFqQn69PoUTsF73V591RUB/9VgNbFXVE+79MemvqA16EckXkSz39gTgQzgbWV7GuUA5OBcs/0ME1LU35M0TnHG3t8ezLlW9S1ULVXU6zk/Fl1T1U3jcX/3U9Wmv+0tE0kQkvec2cKVbw1qcfgJvPl991tXTX66PMv6fr+NAtYjMcSddjnOtaE/7q7+6vO6vEDfz7rANjFV/qWpU/gEWAtuAHThv0nfd6TOATUAFzs+h5Aip6yVgpzvtUWCih313KfBsJPTXAHV52l9uv7zl/tkFfMudnouzN8R+99+cCKnrEbe/duCExRQP3r/zgHK3ht8D2V731wB1RUJ/pQL1QGbItDHpLzsFgjHGxLioHboxxhgTHgt6Y4yJcRb0xhgT4yzojTEmxlnQG2NMjLOgN8aYGGdBb4wxMe7/AaFfoL5PcrdxAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from pandas import read_csv\n",
    "w = read_csv('douglas_data.csv', skiprows=[1], skipinitialspace=True)\n",
    "mu20 = np.mean(w.bstrength[:20])\n",
    "sig20 = np.std(w.bstrength[:20], ddof=1) / np.sqrt(20)\n",
    "print('sample mean, standard deviation of sample mean: ', mu20, sig20)\n",
    "x = np.linspace(30,70,100)\n",
    "y = t.pdf(x, 19, loc=50, scale=sig20)\n",
    "plt.plot(x,y)\n",
    "perc025 = t.ppf(0.025, 19, loc=50, scale=sig20)\n",
    "perc975 = t.ppf(0.975, 19, loc=50, scale=sig20)\n",
    "plt.axvline(perc025, color='r')\n",
    "plt.axvline(perc975, color='r')\n",
    "plt.axvline(mu20, color='k', lw=4)\n",
    "plt.title('H0 is rejected: mean is not 50 N/mm2');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sample mean, standard deviation of sample mean:  48.65050561797753 0.9035436317023355\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": [
    "from pandas import read_csv\n",
    "w = read_csv('douglas_data.csv', skiprows=[1], skipinitialspace=True)\n",
    "N = len(w.bstrength)\n",
    "mu = np.mean(w.bstrength)\n",
    "sig = np.std(w.bstrength, ddof=1) / np.sqrt(N)\n",
    "print('sample mean, standard deviation of sample mean: ', mu, sig)\n",
    "x = np.linspace(30, 70, 100)\n",
    "y = t.pdf(x, N - 1, loc=50, scale=sig)\n",
    "plt.plot(x, y)\n",
    "perc025 = t.ppf(0.025, N - 1, loc=50, scale=sig)\n",
    "perc975 = t.ppf(0.975, N - 1, loc=50, scale=sig)\n",
    "plt.axvline(perc025, color='r')\n",
    "plt.axvline(perc975, color='r')\n",
    "plt.axvline(mu, color='k', lw=4)\n",
    "plt.title('Not enough evidence to reject H0: mean may very well be 50 N/mm2');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a href=\"#back5\">Back to Exercise 5</a>\n",
    "\n",
    "<a name=\"ex6answer\">Answers to Exercise 6</a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "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"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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2z2bgQgAzm0RQ6BpTkZibsmMDAKtT4QzR5iZMgO7dVegSN20WurvXAbcDi4E1BEezlJnZPWZ2RWS2rwI3m9nbwHzgs+7efFhGpMOm7KigAWNNTgoOuWRlwbRpKnSJm6j2Krn7IoKdnU3fu7vJ89XArNhGEznRlB0b2DAgl/e7dAs7yqkpLIT588E9+S/7KyknxQ4TkEw3aecGluU2Pw0i+eTfubDF96/d2Jn/2L+fc//pAV6596YEp5J0p1P/JWX0e/8AeQdqUuMKi61ozN54+zyRWFKhS8qY3LhDNJWu4dLMupx86qzT8Z27IrGkQpeU0ViCqbyFfrRzF8oHjtAWusSFCl1SxpSdFVT3HsTeHn3DjtIhZUPGMGWnttAl9lTokjKm7NiQ0lvnjVYPGcuQQ3tgx46wo0iaUaFLSuhWe4Qxe7ayenAKniHazPE/SjoeXWJMhS4pYWLNJrK8IT220Aer0CU+VOiSEqYevyl06m+hH+jWi819h6jQJeZU6JISJu+oYG+33mztkx5X6SwbMlaFLjGnQpeUMHVHBauGjE2b0+XLhoyB8nI4cCDsKJJGVOiS9LLraymoqUyL8fNGx4eO3n473CCSVlTokvTG79pC1/q6tBg/b1SmHaMSByp0SXqNZ1WuGjou5CSxs7PXABgyRPcYlZhSoUvSm7KjgkNdulPZf1jYUWLHLLiDkQpdYkiFLklv6o4KVg8ejZ94m9rUVlQEq1fD+++HnUTSRJr9hkjaqa9n0s6NaTV+ftyMGVBfDytXhp1E0oQKXZLb+vX0rD3CqiHpM35+XONNo5ctCzeHpA0VuiS3yBhzOh2yeNzIkTBggMbRJWZU6JLcli/naFY25QNHhJ0k9rRjVGJMhS7JbcUK1gzOpy4rTW9/O2NGMIZ+9GjYSSQNqNAlebnD8uVpccncVhUVQW0tlJWFnUTSgApdkldlJezbx6qhaV7ooGEXiYmoCt3M5pjZOjMrN7M7W5nnajNbbWZlZvb72MaUjNS4QzSFbwrdpjFjoE8fFbrERJsDk2aWBfwcuBioApaa2QJ3X91knvHAN4BZ7r7XzAbHK7BkkGXLoHNn1g4eHXaS+OnUKdhK16GLEgPRbKGfAZS7+wZ3PwY8CsxrNs/NwM/dfS+Au++MbUzJSCUlcNppHO3cJewk8VVUFFx1sbY27CSS4qIp9FxgS5PXVZH3mpoATDCz18xsiZnNiVVAyVDuQaEXF4edJP6KioKjXNauDTuJpLhoCr2lOwp4s9edgfHAbOBa4H4z63fCF5ndYmYlZlZSU1PT3qySSTZuhL17M6fQQcMu0mHRFHoV0PSsjjyguoV5/ubute6+EVhHUPAf4u73uXuxuxfn5KTHrcQkTkpKgp+ZUOgTJkDPnip06bBoCn0pMN7MRptZF+AaYEGzef4KXABgZoMIhmA2xDKoZJilS6FLF5g6Newk8ZeVFWylL10adhJJcW0WurvXAbcDi4E1wGPuXmZm95jZFZHZFgO7zWw18ALwdXffHa/QkgFKSuD004NSzwQzZ0JpKRw7FnYSSWFRnU/t7ouARc3eu7vJcwe+EnmIdExDQzD8cN11YSdJnJkzgx2jq1Z9MKYu0k5peoEMSSX5dy780Osxu6t4/uBBvl6ZzePNpqWtM84Ifi5dqkKXU6ZT/yXpnLZ9PQArh56wXz19jR4NAwdqHF06RIUuSWfa9nKOdO7C+kEjw46SOGbBET0qdOkAFbokndO2r6ds8BjqO2WFHSWxZs4Mrrp4+HDYSSRFqdAlqXRqqGfqjgreGZZBwy2NZs4M7jG6YkXYSSRFqdAlqYzdXUWP2qO8k0nj541mzgx+athFTpEKXZLKtO3lAJlZ6MOGQV4evPVW2EkkRanQJalM37aOg126s3HA8LCjhGPmTG2hyylToUtSKaxeR+mwAhoybYdoo5kzobw8uDCZSDvpxCJJGt2PHWHizo384qxPhh0lIZqfUAUwq7KBR4Drbv05r44upPJ7lyU+mKQsbaFL0jhtRzmdvYEVuRPDjhKalUPHATAtcnKVSHuo0CVpFFYHN3goHTYh5CThOdCtFxUDcimsXhd2FElBKnRJGoXV69jYfxh7e/QNO0qoVgyfGPxx8+b3kRE5ORW6JAd3irauZcXwzB1uabQsdxKDDu9n5L7tYUeRFKNCl6Qw/GANg9/by4rhBWFHCd2yyD6EGVvXhJxEUo0KXZJC4dZgzFhb6LB+0EgOdOmhQpd2U6FLUiisXsuRzl1Ym5MfdpTQuXWidHgBRZGdxCLRUqFLUiisXsc7Q8dRl6VTIwCW506koGYTHDwYdhRJISp0CV2Xulqm7qjQcEsTy3InkeUNuq6LtIsKXUI3aecGutbXaodoE6XDC2jA4PXXw44iKUSFLqFrHCtWoX/gYNeevDtopApd2kWFLqErrlpNVZ8cdvQeFHaUpLI8dxIsWQINDWFHkRShQpdwuXPGljLeHDE17CRJZ3nuRNi3D9bqaBeJjgpdwrV+PTmH97E0b0rYSZLOstxJwZM33gg3iKSMqArdzOaY2TozKzezO08y31Vm5mZWHLuIktZefhmAt7SFfoKN/YfDwIHw2mthR5EU0Wahm1kW8HNgLjAZuNbMJrcwX2/gi8CbsQ4paezll6np0Y8NA3LDTpJ8zOCcc+CVV8JOIikimi30M4Byd9/g7seAR4F5Lcz3HeAHwJEY5pN09/LLvDViSlBecqLZs4M7GFVVhZ1EUkA0hZ4LbGnyuiry3nFmVgiMcPcnT/ZFZnaLmZWYWUlNTU27w0qa2bQJNm3ScMvJzJ4d/HzppVBjSGqIptBb2nQ6fqFmM+sE/D/gq219kbvf5+7F7l6ck5MTfUpJT5GhBO0QPYlp06B/f3jxxbCTSAqIptCrgBFNXucB1U1e9wamAi+aWSVwFrBAO0alTa+8An37sjZnVNhJklenTnDeeSp0iUo0hb4UGG9mo82sC3ANsKBxorvvd/dB7p7v7vnAEuAKdy+JS2JJHy+/DOecQ0OnrLCTJDeNo0uU2ix0d68DbgcWA2uAx9y9zMzuMbMr4h1Q0tTOncEJM+edF3aS5KdxdIlSVMehu/sid5/g7mPd/d8j793t7gtamHe2ts6lTY2H4qnQ26ZxdImSzhSVcLz0EnTvDkVFYSdJfhpHlyip0CUczz4L558PXbqEnSQ1aBxdoqBCl8TbvDkYP7/44rCTpA6No0sUVOiSeM8+G/y85JJwc6QSjaNLFFToknjPPAPDhsEUnVAUtU6dgiGqZ58F97bnl4ykQpfEqq+H554Lts51/Zb2+ehHg8slrFsXdhJJUip0SawVK2DPHg23nIq5c4OfTz0Vbg5JWip0Saxnngl+XnRRuDlS0ahRMGmSCl1apUKXxHrmGSgshMGDw06SmubODY50ee+9sJNIEuocdgDJIAcPBnex//KXw06SMvLvXPih17O2DuCRY8f4/I0/4PlxZwBQ+b3LwogmSUhb6JI4L70EtbUaP++ApXlTOJzdlfM3Lgs7iiQhFbokzuLFwen+s2aFnSRlHeuczesjpzF7wzIdvignUKFLYrjDE0/AhRdCt25hp0lpL44pZtS+7YzeW932zJJRVOiSGKWlwTHUH/942ElS3otjZgAwe4MuaiofpkKXxPjLX4KzHS+/POwkKa+q31AqBuQFwy4iTegoF4mrxqM0nnrgtxzIncynfvhWyInSw9/HzuTG5U/Q+6gOX5QPaAtd4m7k3m1Mqqlk8fizw46SNp4qmEXX+jouWv9m2FEkiajQJe4++u4bADwz4ayQk6SP0uETqO49iEvXvRp2FEkiKnSJu0vWL6Fs8Biq+g4JO0racOvEUwWzOG/jcjhwIOw4kiRU6BJXg97by4yta1g8QcMtsbaw4By61tcFh4OKoEKXOLt4/Zt0wlXocbAit4BtvQbC44+HHUWShApd4mrOu6+zqd9Q1g0aFXaUtOPWiacLPgJPPx1cJ0cyngpd4qe6mnMqS/nbpPN1M4s4WTjxHDh6FJ58MuwokgRU6BI/v/sdWd7An6f+Q9hJ0tay3EkwfLiGXQSIstDNbI6ZrTOzcjO7s4XpXzGz1Wb2jpn93cz07+tM5w4PPURJ7iQqB+SGnSZtuXWCq66CRYuCO0FJRmuz0M0sC/g5MBeYDFxrZpObzbYCKHb3acAfgR/EOqikmGXLYPVq/jT1wrCTpL/PfS4YdnnkkbCTSMii2UI/Ayh39w3ufgx4FJjXdAZ3f8HdD0deLgHyYhtTUs6DD0K3bsEYr8TX9OlQXAy/+pUuqZvhoin0XGBLk9dVkfdacxPQ4k0PzewWMysxs5KamproU0pqOXoU5s+HK6/kQLdeYafJDDffDCtXwlu6Vk4mi6bQWzo8ocXNADO7DigG/rOl6e5+n7sXu3txTk5O9CkltSxcGIzn3nhj2EkyxzXXQI8ecP/9YSeREEVT6FXAiCav84ATrqxvZhcBdwFXuPvR2MSTlPTggzBsGFx8cdhJMkefPkGpz5+vY9IzWDSFvhQYb2ajzawLcA2woOkMZlYI/JKgzHfGPqakjPLy4Jjoz38esrLCTpNZbr4Z3nsPHn007CQSkjavh+7udWZ2O7AYyAIecPcyM7sHKHH3BQRDLL2Axy04gWSzu18Rx9ySrH78Y8jOhttvDztJxmi85jzuPD1oFEe+9Z9cWTH8+PTK710WUjJJtKhucOHui4BFzd67u8nzi2KcS1LRnj3wm9/AZz4DQ4eGnSbzmPH76XO457lfUlxVRknelLATSYLpTFGJnXvvhcOH4ctfDjtJxnps2sXs7t6H297QmaOZSIUusXH0KPzsZ3DJJXDaaWGnyVhHsrvx65lX8g8bSpi8Y0PYcSTBVOgSG/Pnw/bt8NWvhp0k4/2u8FIOdOnBbUu0lZ5pVOjScfX18F//FWyZ61DF0B3o1ovfFl3GpWtfZfSerWHHkQRSoUvHPfwwlJXBN7+py+QmiQeK53Gscza3Lvlj2FEkgVTo0jGHDwdFfuaZ8MlPhp1GInb37Mf80z/KP5Y9D+vXhx1HEiSqwxZFWpJ/50Juf/1RvlZdzVUX3EHJNxa1/SFJmF+cdTVXrXyO7DvuCC7HoH89pT1tocspG/TeXm598088PeFsHfOchGp69efHsz4NTz2lOxplCBW6nLI7XptP17pjfP/8z4YdRVrx0IzLYdIk+NKX4MiRsONInKnQ5dS8/jqfLn2aR6bPZaPuSJS06rI6B+cHbNgAP/xh2HEkzlTo0n6HDsH111PdJ4f/Ou+GsNNIWy68ED7xCfj3f9cO0jSnnaLSfl/9KmzcyFeu/Q8Ode0RdhppQ/6dCxk6+HKe8mfYeu5cPnHdf3K0c5cPzaMLeKUHbaFL+zz5JNx3H3z96ywdMTXsNBKl7X0G8bXLvsTUHRX86wu/DjuOxIkKXaK3aRPcdBNMmwb33BN2Gmmnv487k1/NvJIbly9k7tpXw44jcaBCl+js2QNz535wv9CuXcNOJKfgB+ffSOmwCXz/qZ8ybtfmsONIjKnQpW1HjsC8eVBRAX/7G0yeHHYiOUW1WdncPu9fOJrdhUf+8E1G7T3hbpKSwlTocnK1tXDddfDqq8E1W84/P+xE0kFVfYfwmU/9X7Lr63jk0bsYfkB3jUwXOspFWnfgAFx9NSxeDD/6EXzqU2Enkhh5Nyef66++h/mP3sXv59/FLIetfQef9DM6Eib5aQtdWrZlC5xzDvz973D//boLURoqGzqOGz/5bwx4/wBPPPQlZlWWhh1JOkiFLidavDi4euKmTcF1QG66KexEEicrcidyxQ0/oqZnPx5+7G5uefNP4B52LDlFGnKRD+zZE2yJP/ww5QPyuO2q7/Luc0fhuYVhJ5M4qhyQy8ev/yE/WPQT/vXF33BhxVK+ddH/Yu3g0WFHk3ZSoQu8914wrPLd7wal/s1vcun7MzjWOTvsZJIgh7t05/Z5/8Krb0/nn19+mIUP3sHDRZfxs49cw54efcOOJ1HSkEsm2749KPH8/OBqfJMmQUkJfOc7KvNMZMaj0+dwwc2/5JHpc7lh+ULe+MVn+f6inzBx58aw00kUzEMaLysuLvaSkpJQlp3Rtm6Fp58OTg564QVoaAhOGLrrLm27RmsAAAalSURBVJg16/hs+XdqmCXTjd29hc+VLOAfy56nR+1RVg4ZyzPjz+KZCWezbtCoFm+YoSNh4s/Mlrl7cYvToil0M5sD/ATIAu539+81m94VeBiYAewGPuXulSf7ThV6AuzdC6tWBY+SEnjppeDkIIBx4+Daa4PHpEknfFSFLo36HDnEJ995lkvXvcaM6rUA1PTsx/LhE1meO5GywWOpGJjH9t4D2fj9y0NOm/46VOhmlgW8C1wMVAFLgWvdfXWTeW4Dprn7rWZ2DfBxdz/pQcsq9Gbcg0dDA9TXQ11d8LO2NngcOxacsXnkCLz/fjDufeBA8Ni7F3btCh7btsHmzcERKnv2fPD9AwbAuecGJwZdcAGcfvpJb0mmQpeW5Bzaw4Xlb3FGVRmF1WsZvXfb8WmHs7uype8QdvQayPbeA6np2Z993Xqzr3svDnTtxeEu3Xgvuzt//tpF0K1b8OjaFbKzg0fnzpCV9cGjk0aEW9LRQj8b+La7fzTy+hsA7v4fTeZZHJnnDTPrDGwHcvwkX37Khf6Xv8ANSXIN7tb+5zV9v/nzlh4NDR0/VKxTJxg4EIYMgVGjYORIGD0apk4NHnl5xwtcZS2xMuDwfibs2sSYPVsZu7uKvP07GHJoN0MO7mHQ4X1kN9R3bAFmwX/bZh9+NE5rulHS2vPm35cMvvzlU77AXUcL/Spgjrt/IfL6euBMd7+9yTyrIvNURV5XRObZ1ey7bgFuibwsANad0v8iGATsanOuxEvWXJC82ZSrfZSrfdIx1yh3z2lpQjSHLbb0J635X4Fo5sHd7wPui2KZJw9kVtLaX6gwJWsuSN5sytU+ytU+mZYrmkGqKmBEk9d5QPNLtB2fJzLk0hfYg4iIJEw0hb4UGG9mo82sC3ANsKDZPAuAGyPPrwKeP9n4uYiIxF6bQy7uXmdmtwOLCQ5bfMDdy8zsHqDE3RcAvwZ+a2blBFvm18QzNDEYtomTZM0FyZtNudpHudono3KFdmKRiIjElg70FBFJEyp0EZE0kdSFbmZzzGydmZWb2Z0tTO9qZn+ITH/TzPKTJNdnzazGzEojjy8kKNcDZrYzcl5AS9PNzH4ayf2OmRUlSa7ZZra/yfq6OwGZRpjZC2a2xszKzOyOFuZJ+PqKMlfC11dkud3M7C0zezuS7d9amCfhv5NR5grrdzLLzFaY2ZMtTIv9unL3pHwQ7ICtAMYAXYC3gcnN5rkNuDfy/BrgD0mS67PA/w9hnZ0HFAGrWpl+KfAUwXkDZwFvJkmu2cCTCV5Xw4CiyPPeBJe3aP7/Y8LXV5S5Er6+Iss1oFfkeTbwJnBWs3nC+J2MJldYv5NfAX7f0v9f8VhXybyFfgZQ7u4b3P0Y8Cgwr9k884CHIs//CFxoFvdze6PJFQp3f5mTH/8/D3jYA0uAfmY2LAlyJZy7b3P35ZHnB4E1QG6z2RK+vqLMFYrIejgUeZkdeTQ/qiLhv5NR5ko4M8sDLgPub2WWmK+rZC70XGBLk9dVnPgf9vF53L0O2A8MTIJcAJ+I/DP9j2Y2ooXpYYg2exjOjvyT+Skzm5LIBUf+qVtIsGXXVKjr6yS5IKT1FRlCKAV2As+6e6vrLIG/k9HkgsT/Tv4Y+GegoZXpMV9XyVzoMbvkQIxFs8wngHx3nwY8xwd/hcMWxvqKxnKC61OcDvwM+GuiFmxmvYA/AV9y9wPNJ7fwkYSsrzZyhba+3L3e3acTnDF+hplNbTZLKOssilwJ/Z00s48BO9192clma+G9Dq2rZC70ZL3kQJu53H23ux+NvPwVwXXik0E06zTh3P1A4z+Z3X0RkG1mg+K9XDPLJijNR9z9zy3MEsr6aitXWOurWYZ9wIvAnGaTQr0MSGu5QvidnAVcYWaVBMOy/2Bmv2s2T8zXVTIXerJecqDNXM3GWa8gGAdNBguAGyJHb5wF7Hf3bW19KN7MbGjj2KGZnUHw3+XuOC/TCM5wXuPuP2pltoSvr2hyhbG+IsvKMbN+kefdgYuAtc1mS/jvZDS5Ev076e7fcPc8d88n6Ijn3f26ZrPFfF0l7U2iPTkvORBtri+a2RVAXSTXZ+OdC8DM5hMcATHIzKqAbxHsIMLd7wUWERy5UQ4cBj6XJLmuAv7JzOqA94FrEvCHeRZwPbAyMvYK8K/AyCa5wlhf0eQKY31BcATOQxbc9KYT8Ji7Pxn272SUuUL5nWwu3utKp/6LiKSJZB5yERGRdlChi4ikCRW6iEiaUKGLiKQJFbqISJpQoYuIpAkVuohImvgfOiSr/rrtvSUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import norm, gamma\n",
    "for N in [100, 1000, 10000]:\n",
    "    data = gamma.rvs(2, size=(N, 20))\n",
    "    mean_of_data = np.mean(data, 1)\n",
    "    mu = np.mean(mean_of_data)\n",
    "    sig = np.std(mean_of_data, ddof=1)\n",
    "    plt.figure()\n",
    "    plt.hist(mean_of_data, bins=20, density=True)\n",
    "    x = np.linspace(0, 4, 100)\n",
    "    y = norm.pdf(x, loc=mu, scale=sig)\n",
    "    plt.plot(x, y, 'r')\n",
    "    plt.title('N=' + str(N))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a href=\"#back6\">Back to Exercise 6</a>"
   ]
  }
 ],
 "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": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}