Created on Cognitive Class Labs

Lab-Week2-3.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Week 2\n",
    "## Question 1."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import the data set and take a look at the first 10 rows."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>meltpoint</th><th scope=col>burner</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>94.10</td><td>b    </td></tr>\n",
       "\t<tr><td>94.17</td><td>b    </td></tr>\n",
       "\t<tr><td>94.67</td><td>a    </td></tr>\n",
       "\t<tr><td>93.83</td><td>b    </td></tr>\n",
       "\t<tr><td>94.82</td><td>b    </td></tr>\n",
       "\t<tr><td>94.16</td><td>a    </td></tr>\n",
       "\t<tr><td>93.96</td><td>a    </td></tr>\n",
       "\t<tr><td>94.05</td><td>b    </td></tr>\n",
       "\t<tr><td>93.95</td><td>b    </td></tr>\n",
       "\t<tr><td>93.66</td><td>a    </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       " meltpoint & burner\\\\\n",
       "\\hline\n",
       "\t 94.10 & b    \\\\\n",
       "\t 94.17 & b    \\\\\n",
       "\t 94.67 & a    \\\\\n",
       "\t 93.83 & b    \\\\\n",
       "\t 94.82 & b    \\\\\n",
       "\t 94.16 & a    \\\\\n",
       "\t 93.96 & a    \\\\\n",
       "\t 94.05 & b    \\\\\n",
       "\t 93.95 & b    \\\\\n",
       "\t 93.66 & a    \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "meltpoint | burner | \n",
       "|---|---|---|---|---|---|---|---|---|---|\n",
       "| 94.10 | b     | \n",
       "| 94.17 | b     | \n",
       "| 94.67 | a     | \n",
       "| 93.83 | b     | \n",
       "| 94.82 | b     | \n",
       "| 94.16 | a     | \n",
       "| 93.96 | a     | \n",
       "| 94.05 | b     | \n",
       "| 93.95 | b     | \n",
       "| 93.66 | a     | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "   meltpoint burner\n",
       "1  94.10     b     \n",
       "2  94.17     b     \n",
       "3  94.67     a     \n",
       "4  93.83     b     \n",
       "5  94.82     b     \n",
       "6  94.16     a     \n",
       "7  93.96     a     \n",
       "8  94.05     b     \n",
       "9  93.95     b     \n",
       "10 93.66     a     "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "melt = read.table('melt.dat', header=TRUE)\n",
    "head(melt, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can visualise the data with various plots. We can also show multiple plots at once by setting the parameter 'mfrow' to a 2 by 2 grid."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "  The decimal point is 1 digit(s) to the left of the |\n",
      "\n",
      "  934 | 76\n",
      "  936 | 60\n",
      "  938 | 0136912256777\n",
      "  940 | 00023599001677\n",
      "  942 | 0444889913488\n",
      "  944 | 22691288\n",
      "  946 | 0770\n",
      "  948 | 281\n",
      "  950 | 7\n",
      "\n"
     ]
    },
    {
     "data": {
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CmKCKmgEFK+EFJBiUdInNmQiZDk4hES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\nuYhJAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title “Histogram of melt$meltpoint”"
      ]
     },
     "metadata": {},
     "output_type": "display_data",
     "source": "R display func"
    }
   ],
   "source": [
    "par(mfrow=c(2,2))\n",
    "boxplot(melt$meltpoint)\n",
    "hist(melt$meltpoint)\n",
    "stem(melt$meltpoint)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also use the tilde operator to graph one variable BY another and use optional arguments to label our axis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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IgUIQECCAkQQEiAAEKSxc6GSBGSLC5/R4qQZBFSpAhJFiFFipBkEVKkCEkWOxsi\nRUiy2NkQKUICBBASIICQAAGEJIudDZEiJFlc/o4UIckipEgRkixCihQhySKkSBGSLHY2RIqQ\nZLGzIVKEBAggJEAAIQECCEkWOxsiRUiyuPwdKUKSRUiRIiRZhBQpQpJFSJEiJFnsbIgUIcli\nZ0OkCAkQQEiAAEICBBCSLHY2RIqQZHH5O1KEJIuQIkVIsggpUgFCenLiYX1GL9hpG0JIUCbV\nkOovyd38vIfJa2y1DNQbEjsbIpVqSKYpSd7tU3nl2o33DTR3WwbqDYmdDZFKPaTbzKX5u0+Z\nL1kG6g0JkUo9pAvNS4X7Df0tAwkJyqQe0iyztXD/zCrLQEKCMqmHdK15p3D/pMMtA/WGxM6G\nSKUbUmVNTZX5deH+4DGWgXpD4vJ3pFIN6diC6/J3V5oLLAMJCcqE2tnw9I3PWj5LSFCGLUKy\nCClShCSLnQ2RChXS+hUrLJ/VGxI7GyIVKqSFxvZV9IaESIUK6c5hwyyfJSQow2skQAAhyWJn\nQ6QISRaXvyMVJqSNm+yfJyQok25I6772b5duSFaMNBUnrraNIyQok2pIG/7JGDPmnXozsIc5\n4r19Prl+0hc7fNrs54+szCKkSKUa0lXmK7+92Jw2+MVk02TzX/t8ctOVl3WYZra7PkZg7GyI\nVKohjR6wM2kdahbn7r5bd4Jl4B/UhsTOhkilGtLBp+duprZ9Z9+Egy0D9YaESKUaUu3ZuZu5\nbYdP6WkZSEhQJtWQPnVy7mZe78L9k+stAwkJyqQa0qmDdt8f2j1fI7GzIVKphnSNeaP9brP5\njmWg3pC4/B2pVEPa9VHHzylefv3LloGEBGWyudeOkKAMIckipEilHtLG5Q8+1LxjP4PChTT/\nED81NZ5fYH6gE4efdENquX1M4d90qWtaax0XLqS3H/WzeLHnF2BnhE6phrSzsZDRUeOqTd+H\nbQP1/tUOkUo1pJvMGS9sWTW5duWmBXW91lkGEhKUSTWkUUfn+9gxfGKSLDFzLQMJCcqku9eu\nLZ5L+uZuhg+xDCQkKJNqSL1nFD7M7pW7mVJjGUhIUCbVkMYfuj53u2FAQ+72FNu/2EdIUCbV\nkO4wg29+4NahZmGStA6aYBlISFAm3feRZhYuf0/alSSrx95iGUdIUCblnQ3LZk2csbhlv8MI\nCcqw1w4QQEiAAEICBBASIICQAAGEBAggJEAAIQECCAkQQEiAAEICBBASICCbIa0wgDIrDniZ\nd31IyXPPKHXaiYvg4cTTQv8XdPXcga/yFEJSi5+06ieq54+QOhfVQugCUT1/hNS5qBZCF4jq\n+SOkzkW1ELpAVM8fIXUuqoXQBaJ6/gipc1EthC4Q1fNHSJ2LaiF0gaieP0LqXFQLoQtE9fwR\nUueiWghdIKrnj5A6F9VC6AJRPX+E1Lk5c0LPQLeonj9C6tzGjaFnoFtUzx8hAQIICRBASIAA\nQgIEEBIggJAAAYQECCAkQAAhAQIICRBASIAAQgIEEBIggJAAAYQECCAkdIVm0xR6CukiJHQF\nQgIEEBIggJDQ5seNg2v7nfjL0NPQKhfSS5MO6fX5x0NPJC2E1ImKE2Z997wB5geh56FUs5nQ\n76QrZ9f1WBJ6JikhpE68kb/ZcnxdTD8JR1CzMZflPjxbdfiW0FNJByF1qvX9d/52rfm/0NPQ\nqdkcvDn/scn8IvRU0kFInXj2y30L/yzvf4eeiE7N5qTCx9sLfzBFgJBKW1l3yHfuuf/B/zQL\nQ89Ep2bzH4WPy8wFgWeSEkIq7VzzaP7D9wnJDX8ioeBzpvBX/JMJyQ2vkVAww9yXu73HEJIb\nrtqh4E89ar561aQeZxOSm473ke4LPZOUEFInfvP5gw46+bFFhOSmsLPh4LoJj4WeSFoICRBA\nSIAAQgIEEBIggJAAAYQECCAkQAAhAQIICRBASIAAQgIEEBIggJAAAYQECCAkQAAhAQIICRBA\nSIAAQgIEEBIggJAAAYQECCAkQAAhAQIICRBASIAAQgIEEBIggJAAAYQECCAkQAAhAQIICRBA\nSIAAQsquj0xe9ZDz1oSeCfaLkLLrI1M9d+7caUPMQa+Engr2h5Cy6yPTL/+hZbq5MPRUsD+E\nlF3FkJLFZmrudpm5uvCrfsOSwj8a/uq0/hXLcx/fmH5Y7fG/ajvij2fWVw0895U9BgSZeIwI\nKbvaQ2oyC5KPhXTyYcfOOLM597H+M18/q0flkzN4rRAAAAHOSURBVPlP/biy/6zLplb3fmr3\ngFBzjw4hZVfuNdK8efNmDu85fWvysZDMxbvaPl7ZmiSLzKTcL16uOjU/8Pk+o3YPQEoIKbva\nrtoZM/Le/K/2CenwLUnh46d25j609qvP3V5sntiQ12he7xiAlBBSdrX91W7z8i+Y65KPhfTF\nwi+aTWPh44jq3M1Y0255xwCkhJCyq/010j9qa9/9WEhfLfyi2TQVPo7ukbsZbJY+2ub9jgFI\nCSFlV3tIyQjzmyT5lbkif39Hz+JVu8Jn9gpptHm649j2TyAlhJRdHSENyIf0x7Y/Y/5kOg1p\nrvlmx7GElDJCyq72kG41vTfnXivV9vtbkrz/uc5DWtWz6rH8Lzb/gpBSR0jZ1bZF6NwGU/GT\n/C+/YerP/+rAMw7qNKTkf3pWnPrdb0/qPYKQUkdI2dV2+bvnEWcV3m1Ndl19ZNWRV27v13lI\nSfOMT1YfMuKC3xBS6ggJEEBIgABCAgQQEiCAkAABhAQIICRAACEBAggJEEBIgABCAgQQEiCA\nkAABhAQIICRAACEBAggJEEBIgABCAgQQEiCAkAABhAQIICRAACEBAggJEEBIgABCAgQQEiCA\nkAABhAQIICRAwP8DBMm5/OTl/REAAAAASUVORK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data",
     "source": "R display func"
    }
   ],
   "source": [
    "boxplot(melt$meltpoint ~ melt$burner, xlab = 'Burner', ylab='Melt Point')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Examine some numerical summaries."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Summary:\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. \n",
       "  93.47   93.97   94.17   94.21   94.42   95.07 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Mean:\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "94.2056666666667"
      ],
      "text/latex": [
       "94.2056666666667"
      ],
      "text/markdown": [
       "94.2056666666667"
      ],
      "text/plain": [
       "[1] 94.20567"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Standard Deviation: \n"
     ]
    },
    {
     "data": {
      "text/html": [
       "0.338608151446459"
      ],
      "text/latex": [
       "0.338608151446459"
      ],
      "text/markdown": [
       "0.338608151446459"
      ],
      "text/plain": [
       "[1] 0.3386082"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Length:\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "60"
      ],
      "text/latex": [
       "60"
      ],
      "text/markdown": [
       "60"
      ],
      "text/plain": [
       "[1] 60"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "message(\"Summary:\")\n",
    "summary(melt$meltpoint)\n",
    "message(\"Mean:\")\n",
    "mean(melt$meltpoint)\n",
    "message(\"Standard Deviation: \")\n",
    "sd(melt$meltpoint)\n",
    "message(\"Length:\")\n",
    "length(melt$meltpoint)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 2.\n",
    "Now we will conduct a one sample, two sided T test to see if our data is different to the required mu.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tOne Sample t-test\n",
       "\n",
       "data:  melt$meltpoint\n",
       "t = -4.4456, df = 59, p-value = 3.93e-05\n",
       "alternative hypothesis: true mean is not equal to 94.4\n",
       "95 percent confidence interval:\n",
       " 94.11819 94.29314\n",
       "sample estimates:\n",
       "mean of x \n",
       " 94.20567 \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t.test(melt$meltpoint, mu = 94.4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will reject the null hypothesis at .05 significance because our P value is 3.93e-05.\n",
    "We would also reject at .01 significance because 3.93e-05 < .01"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also change the level of significance and confidence interval using the optional argument 'conf.level'. Even with conf.level of .99 our confidence interval does not include mu (94.4)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tOne Sample t-test\n",
       "\n",
       "data:  melt$meltpoint\n",
       "t = -4.4456, df = 59, p-value = 3.93e-05\n",
       "alternative hypothesis: true mean is not equal to 94.4\n",
       "99 percent confidence interval:\n",
       " 94.08931 94.32202\n",
       "sample estimates:\n",
       "mean of x \n",
       " 94.20567 \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t.test(melt$meltpoint, mu = 94.4, conf.level=.99)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "t.test(melt$meltpoint, mu = 94.4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 3.\n",
    "We can see the confidence intervals for .95 and .99 above or we can access them directly from the result of the t.test as below. We use the $ operator to access a component of the test result."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "95% interval:\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>94.1181949058223</li>\n",
       "\t<li>94.293138427511</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 94.1181949058223\n",
       "\\item 94.293138427511\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 94.1181949058223\n",
       "2. 94.293138427511\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 94.11819 94.29314\n",
       "attr(,\"conf.level\")\n",
       "[1] 0.95"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "99% interval:\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>94.0893102135013</li>\n",
       "\t<li>94.322023119832</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 94.0893102135013\n",
       "\\item 94.322023119832\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 94.0893102135013\n",
       "2. 94.322023119832\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 94.08931 94.32202\n",
       "attr(,\"conf.level\")\n",
       "[1] 0.99"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "message(\"95% interval:\")\n",
    "t.test(melt$meltpoint, mu = 94.4, conf.level=.95)$conf.int\n",
    "message(\"99% interval:\")\n",
    "t.test(melt$meltpoint, mu = 94.4, conf.level=.99)$conf.int"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Week 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 1.\n",
    "Lets read out light data set and check the structure."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>Brand1</th><th scope=col>Brand2</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td> 75</td><td>85 </td></tr>\n",
       "\t<tr><td> 82</td><td>78 </td></tr>\n",
       "\t<tr><td>101</td><td>63 </td></tr>\n",
       "\t<tr><td> 92</td><td>50 </td></tr>\n",
       "\t<tr><td> 79</td><td>69 </td></tr>\n",
       "\t<tr><td> 63</td><td>45 </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       " Brand1 & Brand2\\\\\n",
       "\\hline\n",
       "\t  75 & 85 \\\\\n",
       "\t  82 & 78 \\\\\n",
       "\t 101 & 63 \\\\\n",
       "\t  92 & 50 \\\\\n",
       "\t  79 & 69 \\\\\n",
       "\t  63 & 45 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "Brand1 | Brand2 | \n",
       "|---|---|---|---|---|---|\n",
       "|  75 | 85  | \n",
       "|  82 | 78  | \n",
       "| 101 | 63  | \n",
       "|  92 | 50  | \n",
       "|  79 | 69  | \n",
       "|  63 | 45  | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  Brand1 Brand2\n",
       "1  75    85    \n",
       "2  82    78    \n",
       "3 101    63    \n",
       "4  92    50    \n",
       "5  79    69    \n",
       "6  63    45    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "light =  read.table('light.dat.txt', header=T)\n",
    "head(light)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 1. \n",
    "We have 2 samples brand 1 and brand 2. First we will check the normality of data using QQ plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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spRdCENusRbXLNbVDsUZRNSjGe0ruW97mJl86nim84fq5D2uUFm59GF1O8mb3Fv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",
      "text/plain": [
       "Plot with title “Normal Q-Q Plot”"
      ]
     },
     "metadata": {},
     "output_type": "display_data",
     "source": "R display func"
    }
   ],
   "source": [
    "par(mfrow=c(2,2))\n",
    "qqnorm(light$Brand1)\n",
    "qqnorm(light$Brand2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Both samples appear approximately normal so two sample T tests can be appropriate but we must check the standard deviation of samples to determine which test to use. We can use R functions max() and min() to ensure we divide the larger value by the smaller value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "1.1362578657718"
      ],
      "text/latex": [
       "1.1362578657718"
      ],
      "text/markdown": [
       "1.1362578657718"
      ],
      "text/plain": [
       "[1] 1.136258"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sd1 = sd(light$Brand1)\n",
    "sd2 = sd(light$Brand2)\n",
    "max(sd1, sd2)/ min(sd1, sd2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Standard devation ratio is near 1, much less than our rule of thumb: 2 so we will use a two sample t-test (not modified), we indicate that it is not a modified test by passing the optional argument var.equal=TRUE.\n",
    "\n",
    "\n",
    "Null hypothesis: The means are equal between brand 1 and brand 2.\n",
    "\n",
    "\n",
    "Alternative hypothesis: The means are not equal.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tTwo Sample t-test\n",
       "\n",
       "data:  light$Brand1 and light$Brand2\n",
       "t = 2.2316, df = 20, p-value = 0.03724\n",
       "alternative hypothesis: true difference in means is not equal to 0\n",
       "95 percent confidence interval:\n",
       "  0.8427061 24.9754757\n",
       "sample estimates:\n",
       "mean of x mean of y \n",
       " 80.27273  67.36364 \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t.test(light$Brand1, y= light$Brand2, var.equal=TRUE)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Conclusion: We will reject the null hypothesis at the .05 significance level bceause our P value is 0.03725, which is less than .05.\n",
    "\n",
    "The brands do not have equal lumens."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 2."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lets read our farm data and check the structure."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>yield</th><th scope=col>plot</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td> 65.3  </td><td>sprayed</td></tr>\n",
       "\t<tr><td> 78.1  </td><td>sprayed</td></tr>\n",
       "\t<tr><td> 93.0  </td><td>sprayed</td></tr>\n",
       "\t<tr><td> 80.7  </td><td>sprayed</td></tr>\n",
       "\t<tr><td> 89.0  </td><td>sprayed</td></tr>\n",
       "\t<tr><td> 79.9  </td><td>sprayed</td></tr>\n",
       "\t<tr><td> 90.6  </td><td>sprayed</td></tr>\n",
       "\t<tr><td>102.4  </td><td>sprayed</td></tr>\n",
       "\t<tr><td> 64.7  </td><td>sprayed</td></tr>\n",
       "\t<tr><td>106.1  </td><td>sprayed</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       " yield & plot\\\\\n",
       "\\hline\n",
       "\t  65.3   & sprayed\\\\\n",
       "\t  78.1   & sprayed\\\\\n",
       "\t  93.0   & sprayed\\\\\n",
       "\t  80.7   & sprayed\\\\\n",
       "\t  89.0   & sprayed\\\\\n",
       "\t  79.9   & sprayed\\\\\n",
       "\t  90.6   & sprayed\\\\\n",
       "\t 102.4   & sprayed\\\\\n",
       "\t  64.7   & sprayed\\\\\n",
       "\t 106.1   & sprayed\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "yield | plot | \n",
       "|---|---|---|---|---|---|---|---|---|---|\n",
       "|  65.3   | sprayed | \n",
       "|  78.1   | sprayed | \n",
       "|  93.0   | sprayed | \n",
       "|  80.7   | sprayed | \n",
       "|  89.0   | sprayed | \n",
       "|  79.9   | sprayed | \n",
       "|  90.6   | sprayed | \n",
       "| 102.4   | sprayed | \n",
       "|  64.7   | sprayed | \n",
       "| 106.1   | sprayed | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "   yield plot   \n",
       "1   65.3 sprayed\n",
       "2   78.1 sprayed\n",
       "3   93.0 sprayed\n",
       "4   80.7 sprayed\n",
       "5   89.0 sprayed\n",
       "6   79.9 sprayed\n",
       "7   90.6 sprayed\n",
       "8  102.4 sprayed\n",
       "9   64.7 sprayed\n",
       "10 106.1 sprayed"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "farm =  read.table('farm.dat', header=T)\n",
    "head(farm, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We know that we have data from 2 similar fields at each farm, indicating that a paired test might be appropriate. We also know that we are looking specifically for an increase so a one sided test is required.\n",
    "\n",
    "Lets get the difference between our paired values by subtracting the  yield of values that satisfy 'plot == \"unsprayed\"' from the yield of values that satisfy 'plot == \"sprayed\"'.\n",
    "\n",
    "Then we will look at our differences to make sure they are sane."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>-4.7</li>\n",
       "\t<li>3.69999999999999</li>\n",
       "\t<li>6.40000000000001</li>\n",
       "\t<li>1.5</li>\n",
       "\t<li>4.3</li>\n",
       "\t<li>4.80000000000001</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item -4.7\n",
       "\\item 3.69999999999999\n",
       "\\item 6.40000000000001\n",
       "\\item 1.5\n",
       "\\item 4.3\n",
       "\\item 4.80000000000001\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. -4.7\n",
       "2. 3.69999999999999\n",
       "3. 6.40000000000001\n",
       "4. 1.5\n",
       "5. 4.3\n",
       "6. 4.80000000000001\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] -4.7  3.7  6.4  1.5  4.3  4.8"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "difference =farm$yield[farm$plot == \"sprayed\"] - farm$yield[farm$plot == \"unsprayed\"]\n",
    "head(difference)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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rIYcHGgc6SOkJBFLuQzkhQh1VNzG/zb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\nhAQI/H9ieBP0HMvA9AAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title “Normal Q-Q Plot”"
      ]
     },
     "metadata": {},
     "output_type": "display_data",
     "source": "R display func"
    }
   ],
   "source": [
    "qqnorm(difference)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Differences look approximately normal so we can conduct tests on this data. \n",
    "\n",
    "We use the ~ operator to say that we want the yield column explained by the plot column.\n",
    "\n",
    "We must pass 'paired=TRUE' to ensure a paired test and 'alternative=\"greater\"' to ensure a one sided test."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tPaired t-test\n",
       "\n",
       "data:  farm$yield by farm$plot\n",
       "t = 2.2226, df = 13, p-value = 0.0223\n",
       "alternative hypothesis: true difference in means is greater than 0\n",
       "95 percent confidence interval:\n",
       " 0.5631646       Inf\n",
       "sample estimates:\n",
       "mean of the differences \n",
       "               2.771429 \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t.test(farm$yield ~ farm$plot, paired=TRUE, alternative=\"greater\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also do this by separating our data into 2 vectors: sprayed and unsprayed then performing the same test."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tPaired t-test\n",
       "\n",
       "data:  sprayed and unsprayed\n",
       "t = 2.2226, df = 13, p-value = 0.0223\n",
       "alternative hypothesis: true difference in means is greater than 0\n",
       "95 percent confidence interval:\n",
       " 0.5631646       Inf\n",
       "sample estimates:\n",
       "mean of the differences \n",
       "               2.771429 \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sprayed = farm[farm$plot== \"sprayed\",\"yield\"]\n",
    "unsprayed = farm[farm$plot== \"unsprayed\",\"yield\"]\n",
    "t.test(sprayed, unsprayed, paired=TRUE, alternative=\"greater\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We find the same result with the 2 different ways to do the same test in R."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Conclusion: We reject the null hypothesis with a significance level of .05. P-value = 0.0223\n",
    "\n",
    "The yield of the sprayed plots were greater than the yield of similar unsprayed plots."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 3.\n",
    "Lets get our dataset and look at the structure."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>fertile</th><th scope=col>nonfertile</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td> 5.9</td><td>10.6</td></tr>\n",
       "\t<tr><td> 8.8</td><td> 5.4</td></tr>\n",
       "\t<tr><td> 6.5</td><td> 6.1</td></tr>\n",
       "\t<tr><td>13.8</td><td> 8.2</td></tr>\n",
       "\t<tr><td>20.2</td><td>11.5</td></tr>\n",
       "\t<tr><td>16.1</td><td> 9.1</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       " fertile & nonfertile\\\\\n",
       "\\hline\n",
       "\t  5.9 & 10.6\\\\\n",
       "\t  8.8 &  5.4\\\\\n",
       "\t  6.5 &  6.1\\\\\n",
       "\t 13.8 &  8.2\\\\\n",
       "\t 20.2 & 11.5\\\\\n",
       "\t 16.1 &  9.1\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "fertile | nonfertile | \n",
       "|---|---|---|---|---|---|\n",
       "|  5.9 | 10.6 | \n",
       "|  8.8 |  5.4 | \n",
       "|  6.5 |  6.1 | \n",
       "| 13.8 |  8.2 | \n",
       "| 20.2 | 11.5 | \n",
       "| 16.1 |  9.1 | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  fertile nonfertile\n",
       "1  5.9    10.6      \n",
       "2  8.8     5.4      \n",
       "3  6.5     6.1      \n",
       "4 13.8     8.2      \n",
       "5 20.2    11.5      \n",
       "6 16.1     9.1      "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fertile  =  read.table('fertile.dat', header=T)\n",
    "head(fertile)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are 2 samples but no indication that these are paired.\n",
    "\n",
    "Lets check the normality."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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M7omJAQJJ6GtKLzhpLxLIeJhATNcGYD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\nAEICBBASIICQAAGEBAggJEDA/wPROQHIeQ095gAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title “Normal Q-Q Plot”"
      ]
     },
     "metadata": {},
     "output_type": "display_data",
     "source": "R display func"
    },
    {
     "data": {
      "image/png": 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9p/GfIkSAUhZamPuw3/2F3Mb3l72JMgFYSUrd44vFW/IeVFt4Q9B1JCSFlrz2Oz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\nSUVORK5CYII=",
      "text/plain": [
       "Plot with title “Normal Q-Q Plot”"
      ]
     },
     "metadata": {},
     "output_type": "display_data",
     "source": "R display func"
    }
   ],
   "source": [
    "qqnorm(fertile$fertile)\n",
    "qqnorm(fertile$nonfertile)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Both data sets appear approximately normal. Now we will check standard deviation ratio."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "2.37758217433345"
      ],
      "text/latex": [
       "2.37758217433345"
      ],
      "text/markdown": [
       "2.37758217433345"
      ],
      "text/plain": [
       "[1] 2.377582"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "7"
      ],
      "text/latex": [
       "7"
      ],
      "text/markdown": [
       "7"
      ],
      "text/plain": [
       "[1] 7"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "max(sd(fertile$fertile), sd(fertile$nonfertile))/min(sd(fertile$fertile), sd(fertile$nonfertile))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ratio of sd1/sd2 is more than 2. Do we have a large data set?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "7"
      ],
      "text/latex": [
       "7"
      ],
      "text/markdown": [
       "7"
      ],
      "text/plain": [
       "[1] 7"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "7"
      ],
      "text/latex": [
       "7"
      ],
      "text/markdown": [
       "7"
      ],
      "text/plain": [
       "[1] 7"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "length(fertile$fertile)\n",
    "length(fertile$nonfertile)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our deviation ratio is over 2 and our dataset is small so we will specify that var.equal=FALSE so conduct a modified (Welch) 2 sample T Test."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tWelch Two Sample t-test\n",
       "\n",
       "data:  fertile$fertile and fertile$nonfertile\n",
       "t = 1.3525, df = 8.0584, p-value = 0.2129\n",
       "alternative hypothesis: true difference in means is not equal to 0\n",
       "95 percent confidence interval:\n",
       " -2.138671  8.224386\n",
       "sample estimates:\n",
       "mean of x mean of y \n",
       "11.271429  8.228571 \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t.test(fertile$fertile, fertile$nonfertile, var.equal=FALSE)"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "Conclusion: Our P value is 0.2129 so we do not reject the null hypothesis "
   ]
  }
 ],
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