{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Statistical Estimation" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "Warning message:\n", "“package ‘dplyr’ was built under R version 3.4.1”" ] } ], "source": [ "suppressPackageStartupMessages(library(tidyverse))\n", "suppressPackageStartupMessages(library(pwr))" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": true }, "outputs": [], "source": [ "options(repr.plot.width=4, repr.plot.height=3)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Working with probability distributions" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 5 samples from standard normal distribution" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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  1. 0.834184312954893
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  2. -1.42097831766073
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  3. 0.613256845929185
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  4. 1.41229807396209
    5. \n", "\t
  5. -1.21942815702242
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  6. 0.323522322297916
    7. \n", "
\n" ], "text/latex": [ "\\begin{enumerate*}\n", "\\item 0.834184312954893\n", "\\item -1.42097831766073\n", "\\item 0.613256845929185\n", "\\item 1.41229807396209\n", "\\item -1.21942815702242\n", "\\item 0.323522322297916\n", "\\end{enumerate*}\n" ], "text/markdown": [ "1. 0.834184312954893\n", "2. -1.42097831766073\n", "3. 0.613256845929185\n", "4. 1.41229807396209\n", "5. -1.21942815702242\n", "6. 0.323522322297916\n", "\n", "\n" ], "text/plain": [ "[1] 0.8341843 -1.4209783 0.6132568 1.4122981 -1.2194282 0.3235223" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "n <- 6\n", "(x <- rnorm(n))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 5 samples from normal distribution" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
    \n", "\t
  1. 6.10092036878306
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  2. 10.4553327181587
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  3. 7.60688063076039
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  4. 12.7554584484242
    5. \n", "\t
  5. 4.49824865096167
    6. \n", "\t
  6. 14.7050261258722
    7. \n", "
\n" ], "text/latex": [ "\\begin{enumerate*}\n", "\\item 6.10092036878306\n", "\\item 10.4553327181587\n", "\\item 7.60688063076039\n", "\\item 12.7554584484242\n", "\\item 4.49824865096167\n", "\\item 14.7050261258722\n", "\\end{enumerate*}\n" ], "text/markdown": [ "1. 6.10092036878306\n", "2. 10.4553327181587\n", "3. 7.60688063076039\n", "4. 12.7554584484242\n", "5. 4.49824865096167\n", "6. 14.7050261258722\n", "\n", "\n" ], "text/plain": [ "[1] 6.100920 10.455333 7.606881 12.755458 4.498249 14.705026" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "n <- 6\n", "mu <- 10\n", "sigma <- 5\n", "(x <- rnorm(n, mu, sigma))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### PDF" ] }, { "cell_type": "code", "execution_count": 57, "metadata": {}, "outputs": [ { "data": { "image/png": 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Csfgco4YDvPiyV/GtEOdGUprfkd8dck/z74OclOGGtMAAfcmE3RY1ZSBV6ROAd6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\n+Avvx+ozIrR8BNzX3efd930OYO8SwAG/y4K1DhHAAb8X5LIKuku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gAAEIQAACEIAABCAAAQhAAAIQgAAEIAABCEAAAhCAAAQgAAEIQAAC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\ngAAE6hH4/1VRDeky5lesAAAAAElFTkSuQmCC", "text/plain": [ "Plot with title “Standard normal”" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x <- seq(-3,3,length.out = 100)\n", "plot(x, dnorm(x), main=\"Standard normal\", type='l')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### CDF" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "data": { "image/png": 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5Zc/zbMH1R8iNZFViKaCA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\nUEABBRRQQAEFFFBAAQUUUECBqQT+HywK/myjJPCwAAAAAElFTkSuQmCC", "text/plain": [ "plot without title" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x <- seq(-3,3,length.out = 100)\n", "plot(x, pnorm(x), type=\"l\", ylab='CDF')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Percentiles" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
    \n", "\t
  1. 0
    2. \n", "\t
  2. 0.385
    3. \n", "\t
  3. 1.645
    4. \n", "\t
  4. 2.326
    5. \n", "
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    \n", "\t
  1. 0.5
    2. \n", "\t
  2. 0.65
    3. \n", "\t
  3. 0.95
    4. \n", "\t
  4. 0.99
    5. \n", "
\n" ], "text/latex": [ "\\begin{enumerate*}\n", "\\item 0.5\n", "\\item 0.65\n", "\\item 0.95\n", "\\item 0.99\n", "\\end{enumerate*}\n" ], "text/markdown": [ "1. 0.5\n", "2. 0.65\n", "3. 0.95\n", "4. 0.99\n", "\n", "\n" ], "text/plain": [ "[1] 0.50 0.65 0.95 0.99" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pnorm(c(0, 0.385320466407568, 1.64485362695147, 2.32634787404084))" ] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "plot without title" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x <- seq(-3,3,length.out = 100)\n", "plot(x, pnorm(x), type=\"l\", ylab='CDF')\n", "abline(h=0.65, col='red', lty=2)\n", "abline(v=0.385, col='blue', lty=2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exercise**\n", "\n", "Assume that IQ has a normal distribution with mean = 100 and standard deviation = 15." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exercise 1**\n", "\n", "If your IQ is 154, what percentile are you?" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exercise 2**\n", "\n", "What percentage of the population has IQ between 70 and 120?" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exercise 3**\n", "\n", "What IQ do you need to be in the top 10 percentile?" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "## One sample model" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "mu <- 0\n", "sigma <- 1\n", "n <- 6" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "x <- rnorm(n, mu, sigma)" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
    \n", "\t
  1. 0.964404467414068
    2. \n", "\t
  2. 0.233532508165446
    3. \n", "\t
  3. -1.41560321218151
    4. \n", "\t
  4. -1.16570049438986
    5. \n", "\t
  5. -1.03987265775232
    6. \n", "\t
  6. -2.37490060325525
    7. \n", "
\n" ], "text/latex": [ "\\begin{enumerate*}\n", "\\item 0.964404467414068\n", "\\item 0.233532508165446\n", "\\item -1.41560321218151\n", "\\item -1.16570049438986\n", "\\item -1.03987265775232\n", "\\item -2.37490060325525\n", "\\end{enumerate*}\n" ], "text/markdown": [ "1. 0.964404467414068\n", "2. 0.233532508165446\n", "3. -1.41560321218151\n", "4. -1.16570049438986\n", "5. -1.03987265775232\n", "6. -2.37490060325525\n", "\n", "\n" ], "text/plain": [ "[1] 0.9644045 0.2335325 -1.4156032 -1.1657005 -1.0398727 -2.3749006" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Point estimates" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "-0.799689998666571" ], "text/latex": [ "-0.799689998666571" ], "text/markdown": [ "-0.799689998666571" ], "text/plain": [ "[1] -0.79969" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "mean(x)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/html": [ "-1.10278657607109" ], "text/latex": [ "-1.10278657607109" ], "text/markdown": [ "-1.10278657607109" ], "text/plain": [ "[1] -1.102787" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "median(x)" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "1.20265264827186" ], "text/latex": [ "1.20265264827186" ], "text/markdown": [ "1.20265264827186" ], "text/plain": [ "[1] 1.202653" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sd(x)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Interval estimates" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": true }, "outputs": [], "source": [ "me <- qt(0.975, df=n-1) * sd(x)/sqrt(n)" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
    \n", "\t
  1. -2.06179655017835
    2. \n", "\t
  2. 0.46241655284521
    3. \n", "
\n" ], "text/latex": [ "\\begin{enumerate*}\n", "\\item -2.06179655017835\n", "\\item 0.46241655284521\n", "\\end{enumerate*}\n" ], "text/markdown": [ "1. -2.06179655017835\n", "2. 0.46241655284521\n", "\n", "\n" ], "text/plain": [ "[1] -2.0617966 0.4624166" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "xbar <- mean(x)\n", "c(xbar - me, xbar + me)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exercise 4**\n", "\n", "Simulate 10 samples from a normal distribution with mean = 100 and standard deviation = 15.\n", "\n", "- Find the sample mean, median, standard deviation, margin of error and 90% confidence interval of the mean" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exercise 5**\n", "\n", "Simulate 200 samples from a normal distribution with mean = 100 and standard deviation = 15.\n", "\n", "- What percentile is a person with IQ = 154 in this population?\n", "- What percentage of this population has IQ between 70 and 120?\n", "- What IQ do you need to be in the top 10 percentile in this population?" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exercise 6**\n", "\n", "Compare the t and normal distributions. Overlay plots of the PDF for the standard t-distribution with 1, 5 and 30 degrees of freedom on the standard normal distribution on the same plot. Use different colors and/or line styles." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "R", "language": "R", "name": "ir" }, "language_info": { "codemirror_mode": "r", "file_extension": ".r", "mimetype": "text/x-r-source", "name": "R", "pygments_lexer": "r", "version": "3.4.0" } }, "nbformat": 4, "nbformat_minor": 2 }