{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Foundations for statistical inference - Confidence intervals" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### If you have access to data on an entire population, say the size of every house in Ames, Iowa, it's straight forward to answer questions like, \"How big is the typical house in Ames?\" and \"How much variation is there in sizes of houses?\". If you have access to only a sample of the population, as is often the case, the task becomes more complicated. What is your best guess for the typical size if you only know the sizes of several dozen houses? This sort of situation requires that you use your sample to make inference on what your population looks like." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### In the previous lab, \"Sampling Distributions\", we looked at the population data of houses from Ames, Iowa. Let's start by loading that data set." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import warnings\n", "warnings.filterwarnings(\"ignore\")\n", "\n", "import numpy as np\n", "import pandas as pd\n", "import io\n", "import requests\n", "\n", "df_url = 'https://raw.githubusercontent.com/akmand/datasets/master/openintro/ames.csv'\n", "url_content = requests.get(df_url, verify=False).content\n", "ames = pd.read_csv(io.StringIO(url_content.decode('utf-8')))" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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OrderPIDMS.SubClassMS.ZoningLot.FrontageLot.AreaStreetAlleyLot.ShapeLand.Contour...Pool.AreaPool.QCFenceMisc.FeatureMisc.ValMo.SoldYr.SoldSale.TypeSale.ConditionSalePrice
0152630110020RL141.031770PaveNaNIR1Lvl...0NaNNaNNaN052010WDNormal215000
1252635004020RH80.011622PaveNaNRegLvl...0NaNMnPrvNaN062010WDNormal105000
2352635101020RL81.014267PaveNaNIR1Lvl...0NaNNaNGar21250062010WDNormal172000
3452635303020RL93.011160PaveNaNRegLvl...0NaNNaNNaN042010WDNormal244000
4552710501060RL74.013830PaveNaNIR1Lvl...0NaNMnPrvNaN032010WDNormal189900
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5 rows × 82 columns

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" ], "text/plain": [ " Order PID MS.SubClass MS.Zoning Lot.Frontage Lot.Area Street \\\n", "0 1 526301100 20 RL 141.0 31770 Pave \n", "1 2 526350040 20 RH 80.0 11622 Pave \n", "2 3 526351010 20 RL 81.0 14267 Pave \n", "3 4 526353030 20 RL 93.0 11160 Pave \n", "4 5 527105010 60 RL 74.0 13830 Pave \n", "\n", " Alley Lot.Shape Land.Contour ... Pool.Area Pool.QC Fence Misc.Feature \\\n", "0 NaN IR1 Lvl ... 0 NaN NaN NaN \n", "1 NaN Reg Lvl ... 0 NaN MnPrv NaN \n", "2 NaN IR1 Lvl ... 0 NaN NaN Gar2 \n", "3 NaN Reg Lvl ... 0 NaN NaN NaN \n", "4 NaN IR1 Lvl ... 0 NaN MnPrv NaN \n", "\n", " Misc.Val Mo.Sold Yr.Sold Sale.Type Sale.Condition SalePrice \n", "0 0 5 2010 WD Normal 215000 \n", "1 0 6 2010 WD Normal 105000 \n", "2 12500 6 2010 WD Normal 172000 \n", "3 0 4 2010 WD Normal 244000 \n", "4 0 3 2010 WD Normal 189900 \n", "\n", "[5 rows x 82 columns]" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ames.head()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### In this lab we'll start with a simple random sample of size 60 from the population." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "population = ames['Gr.Liv.Area']\n", "samp = population.sample(60)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "count 2930.000000\n", "mean 1499.690444\n", "std 505.508887\n", "min 334.000000\n", "25% 1126.000000\n", "50% 1442.000000\n", "75% 1742.750000\n", "max 5642.000000\n", "Name: Gr.Liv.Area, dtype: float64" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "population.describe()" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "1494.9833333333333" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "samp.mean()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Based only on this single sample, the best estimate of the average living area of houses sold in Ames would be the sample mean, usually denoted as (here we're calling it sample_mean). That serves as a good point estimate but it would be useful to also communicate how uncertain we are of that estimate. This can be captured by using a confidence interval.\n", "\n", "### We can calculate a 95% confidence interval for a sample mean by adding and subtracting 1.96 standard errors to the point estimate" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1376.238157814993 1613.7285088516737\n" ] } ], "source": [ "import numpy as np\n", "\n", "se = np.std(samp)/np.sqrt(60)\n", "lower = samp.mean() - (1.96 * se)\n", "upper = samp.mean() + (1.96 * se)\n", "print(lower, upper)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The distribution of sample mean" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "sample_means50 = [population.sample(60).mean() for i in range(0, 3000)]\n", "\n", "\n", "import matplotlib.pyplot as plt\n", "plt.rcParams['figure.figsize'] = (10,5)\n", "plt.hist(sample_means50, edgecolor = 'black', linewidth = 1.2)\n", "plt.show();" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### In this case we have the luxury of knowing the true population mean since we have data on the entire population." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Does your confidence interval capture the true average size of houses in Ames?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Now we're going to recreate many samples to learn more about how sample means and confidence intervals vary from one sample to another." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "samp_mean = np.empty(50)\n", "samp_sd = np.empty(50)\n", "n = 60" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "for i in range(50):\n", " samp = population.sample(n) # obtain a sample of size n = 60 from the population\n", " samp_mean[i] = samp.mean() # save sample mean in ith element of samp_mean\n", " samp_sd[i] = np.std(samp) # save sample sd in ith element of samp_sd" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [], "source": [ "se_array = samp_sd/np.sqrt(n)\n", "lower_array = samp_mean - (1.96 * se_array)\n", "upper_array = samp_mean + (1.96 * se_array)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercise 1: What proportion of your confidence intervals include the true population mean? " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercise 2: Pick a confidence level of your choosing (again for size=60), provided it is not 95% and calculate 50 confidence intervals at the confidence level you chose. What proportion of your confinence intervals include the true population mean?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercise 3: Repeat exercise 2 for the same size of confidence interval but for sample size =100. What do you notice?" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "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": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 4 }