minjejeon/week2.ipynb

## week2.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "later-bandwidth",
   "metadata": {},
   "source": [
    "# 경영통계분석 week2(ch4, 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "unlikely-parking",
   "metadata": {},
   "source": [
    "## 패키지 불러오기"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "registered-ceremony",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message:\n",
      "\"package 'tidyverse' was built under R version 3.6.3\"-- Attaching packages ------------------------------------------------------------------------------- tidyverse 1.3.0 --\n",
      "√ ggplot2 3.3.3     √ purrr   0.3.4\n",
      "√ tibble  2.1.3     √ dplyr   0.8.3\n",
      "√ tidyr   1.1.3     √ stringr 1.4.0\n",
      "√ readr   1.3.1     √ forcats 0.4.0\n",
      "Warning message:\n",
      "\"package 'ggplot2' was built under R version 3.6.3\"Warning message:\n",
      "\"package 'tidyr' was built under R version 3.6.3\"Warning message:\n",
      "\"package 'purrr' was built under R version 3.6.3\"-- Conflicts ---------------------------------------------------------------------------------- tidyverse_conflicts() --\n",
      "x dplyr::filter() masks stats::filter()\n",
      "x dplyr::lag()    masks stats::lag()\n",
      "Warning message:\n",
      "\"package 'gganimate' was built under R version 3.6.3\"Warning message:\n",
      "\"package 'magick' was built under R version 3.6.3\"Linking to ImageMagick 6.9.11.57\n",
      "Enabled features: cairo, freetype, fftw, ghostscript, heic, lcms, pango, raw, rsvg, webp\n",
      "Disabled features: fontconfig, x11\n",
      "Warning message:\n",
      "\"package 'gapminder' was built under R version 3.6.3\"Warning message:\n",
      "\"package 'transformr' was built under R version 3.6.3\""
     ]
    }
   ],
   "source": [
    "list.of.packages <- c(\"tidyverse\", \"gganimate\", \"magick\", \"gapminder\", \"transformr\")\n",
    "new.packages <- list.of.packages[!(list.of.packages %in% installed.packages()[,\"Package\"])]\n",
    "if (length(new.packages)) install.packages(new.packages)\n",
    "for (x in list.of.packages) library(x, character.only = T)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "danish-palace",
   "metadata": {},
   "source": [
    "## 이항분포의 시각화"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "major-cooling",
   "metadata": {},
   "outputs": [],
   "source": [
    "n <- 20\n",
    "p <- 0.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "supported-necessity",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<style>\n",
       ".list-inline {list-style: none; margin:0; padding: 0}\n",
       ".list-inline>li {display: inline-block}\n",
       ".list-inline>li:not(:last-child)::after {content: \"\\00b7\"; padding: 0 .5ex}\n",
       "</style>\n",
       "<ol class=list-inline><li>0</li><li>1</li><li>2</li><li>3</li><li>4</li><li>5</li><li>6</li><li>7</li><li>8</li><li>9</li><li>10</li><li>11</li><li>12</li><li>13</li><li>14</li><li>15</li><li>16</li><li>17</li><li>18</li><li>19</li><li>20</li></ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 3\n",
       "\\item 4\n",
       "\\item 5\n",
       "\\item 6\n",
       "\\item 7\n",
       "\\item 8\n",
       "\\item 9\n",
       "\\item 10\n",
       "\\item 11\n",
       "\\item 12\n",
       "\\item 13\n",
       "\\item 14\n",
       "\\item 15\n",
       "\\item 16\n",
       "\\item 17\n",
       "\\item 18\n",
       "\\item 19\n",
       "\\item 20\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0\n",
       "2. 1\n",
       "3. 2\n",
       "4. 3\n",
       "5. 4\n",
       "6. 5\n",
       "7. 6\n",
       "8. 7\n",
       "9. 8\n",
       "10. 9\n",
       "11. 10\n",
       "12. 11\n",
       "13. 12\n",
       "14. 13\n",
       "15. 14\n",
       "16. 15\n",
       "17. 16\n",
       "18. 17\n",
       "19. 18\n",
       "20. 19\n",
       "21. 20\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1]  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "successes <- c(0:n)\n",
    "successes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "naughty-massachusetts",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<style>\n",
       ".list-inline {list-style: none; margin:0; padding: 0}\n",
       ".list-inline>li {display: inline-block}\n",
       ".list-inline>li:not(:last-child)::after {content: \"\\00b7\"; padding: 0 .5ex}\n",
       "</style>\n",
       "<ol class=list-inline><li>9.5367431640625e-07</li><li>1.9073486328125e-05</li><li>0.000181198120117188</li><li>0.00108718872070313</li><li>0.00462055206298827</li><li>0.0147857666015625</li><li>0.0369644165039063</li><li>0.0739288330078125</li><li>0.120134353637695</li><li>0.160179138183594</li><li>0.176197052001953</li><li>0.160179138183594</li><li>0.120134353637695</li><li>0.0739288330078124</li><li>0.0369644165039063</li><li>0.0147857666015625</li><li>0.00462055206298827</li><li>0.00108718872070313</li><li>0.000181198120117188</li><li>1.9073486328125e-05</li><li>9.5367431640625e-07</li></ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 9.5367431640625e-07\n",
       "\\item 1.9073486328125e-05\n",
       "\\item 0.000181198120117188\n",
       "\\item 0.00108718872070313\n",
       "\\item 0.00462055206298827\n",
       "\\item 0.0147857666015625\n",
       "\\item 0.0369644165039063\n",
       "\\item 0.0739288330078125\n",
       "\\item 0.120134353637695\n",
       "\\item 0.160179138183594\n",
       "\\item 0.176197052001953\n",
       "\\item 0.160179138183594\n",
       "\\item 0.120134353637695\n",
       "\\item 0.0739288330078124\n",
       "\\item 0.0369644165039063\n",
       "\\item 0.0147857666015625\n",
       "\\item 0.00462055206298827\n",
       "\\item 0.00108718872070313\n",
       "\\item 0.000181198120117188\n",
       "\\item 1.9073486328125e-05\n",
       "\\item 9.5367431640625e-07\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 9.5367431640625e-07\n",
       "2. 1.9073486328125e-05\n",
       "3. 0.000181198120117188\n",
       "4. 0.00108718872070313\n",
       "5. 0.00462055206298827\n",
       "6. 0.0147857666015625\n",
       "7. 0.0369644165039063\n",
       "8. 0.0739288330078125\n",
       "9. 0.120134353637695\n",
       "10. 0.160179138183594\n",
       "11. 0.176197052001953\n",
       "12. 0.160179138183594\n",
       "13. 0.120134353637695\n",
       "14. 0.0739288330078124\n",
       "15. 0.0369644165039063\n",
       "16. 0.0147857666015625\n",
       "17. 0.00462055206298827\n",
       "18. 0.00108718872070313\n",
       "19. 0.000181198120117188\n",
       "20. 1.9073486328125e-05\n",
       "21. 9.5367431640625e-07\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] 9.536743e-07 1.907349e-05 1.811981e-04 1.087189e-03 4.620552e-03\n",
       " [6] 1.478577e-02 3.696442e-02 7.392883e-02 1.201344e-01 1.601791e-01\n",
       "[11] 1.761971e-01 1.601791e-01 1.201344e-01 7.392883e-02 3.696442e-02\n",
       "[16] 1.478577e-02 4.620552e-03 1.087189e-03 1.811981e-04 1.907349e-05\n",
       "[21] 9.536743e-07"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "prob <- dbinom(x=0:n, size = n, prob=p)\n",
    "prob"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "responsible-filename",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A data.frame: 21 × 2</caption>\n",
       "<thead>\n",
       "\t<tr><th scope=col>successes</th><th scope=col>prob</th></tr>\n",
       "\t<tr><th scope=col>&lt;int&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><td> 0</td><td>9.536743e-07</td></tr>\n",
       "\t<tr><td> 1</td><td>1.907349e-05</td></tr>\n",
       "\t<tr><td> 2</td><td>1.811981e-04</td></tr>\n",
       "\t<tr><td> 3</td><td>1.087189e-03</td></tr>\n",
       "\t<tr><td> 4</td><td>4.620552e-03</td></tr>\n",
       "\t<tr><td> 5</td><td>1.478577e-02</td></tr>\n",
       "\t<tr><td> 6</td><td>3.696442e-02</td></tr>\n",
       "\t<tr><td> 7</td><td>7.392883e-02</td></tr>\n",
       "\t<tr><td> 8</td><td>1.201344e-01</td></tr>\n",
       "\t<tr><td> 9</td><td>1.601791e-01</td></tr>\n",
       "\t<tr><td>10</td><td>1.761971e-01</td></tr>\n",
       "\t<tr><td>11</td><td>1.601791e-01</td></tr>\n",
       "\t<tr><td>12</td><td>1.201344e-01</td></tr>\n",
       "\t<tr><td>13</td><td>7.392883e-02</td></tr>\n",
       "\t<tr><td>14</td><td>3.696442e-02</td></tr>\n",
       "\t<tr><td>15</td><td>1.478577e-02</td></tr>\n",
       "\t<tr><td>16</td><td>4.620552e-03</td></tr>\n",
       "\t<tr><td>17</td><td>1.087189e-03</td></tr>\n",
       "\t<tr><td>18</td><td>1.811981e-04</td></tr>\n",
       "\t<tr><td>19</td><td>1.907349e-05</td></tr>\n",
       "\t<tr><td>20</td><td>9.536743e-07</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 21 × 2\n",
       "\\begin{tabular}{ll}\n",
       " successes & prob\\\\\n",
       " <int> & <dbl>\\\\\n",
       "\\hline\n",
       "\t  0 & 9.536743e-07\\\\\n",
       "\t  1 & 1.907349e-05\\\\\n",
       "\t  2 & 1.811981e-04\\\\\n",
       "\t  3 & 1.087189e-03\\\\\n",
       "\t  4 & 4.620552e-03\\\\\n",
       "\t  5 & 1.478577e-02\\\\\n",
       "\t  6 & 3.696442e-02\\\\\n",
       "\t  7 & 7.392883e-02\\\\\n",
       "\t  8 & 1.201344e-01\\\\\n",
       "\t  9 & 1.601791e-01\\\\\n",
       "\t 10 & 1.761971e-01\\\\\n",
       "\t 11 & 1.601791e-01\\\\\n",
       "\t 12 & 1.201344e-01\\\\\n",
       "\t 13 & 7.392883e-02\\\\\n",
       "\t 14 & 3.696442e-02\\\\\n",
       "\t 15 & 1.478577e-02\\\\\n",
       "\t 16 & 4.620552e-03\\\\\n",
       "\t 17 & 1.087189e-03\\\\\n",
       "\t 18 & 1.811981e-04\\\\\n",
       "\t 19 & 1.907349e-05\\\\\n",
       "\t 20 & 9.536743e-07\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 21 × 2\n",
       "\n",
       "| successes &lt;int&gt; | prob &lt;dbl&gt; |\n",
       "|---|---|\n",
       "|  0 | 9.536743e-07 |\n",
       "|  1 | 1.907349e-05 |\n",
       "|  2 | 1.811981e-04 |\n",
       "|  3 | 1.087189e-03 |\n",
       "|  4 | 4.620552e-03 |\n",
       "|  5 | 1.478577e-02 |\n",
       "|  6 | 3.696442e-02 |\n",
       "|  7 | 7.392883e-02 |\n",
       "|  8 | 1.201344e-01 |\n",
       "|  9 | 1.601791e-01 |\n",
       "| 10 | 1.761971e-01 |\n",
       "| 11 | 1.601791e-01 |\n",
       "| 12 | 1.201344e-01 |\n",
       "| 13 | 7.392883e-02 |\n",
       "| 14 | 3.696442e-02 |\n",
       "| 15 | 1.478577e-02 |\n",
       "| 16 | 4.620552e-03 |\n",
       "| 17 | 1.087189e-03 |\n",
       "| 18 | 1.811981e-04 |\n",
       "| 19 | 1.907349e-05 |\n",
       "| 20 | 9.536743e-07 |\n",
       "\n"
      ],
      "text/plain": [
       "   successes prob        \n",
       "1   0        9.536743e-07\n",
       "2   1        1.907349e-05\n",
       "3   2        1.811981e-04\n",
       "4   3        1.087189e-03\n",
       "5   4        4.620552e-03\n",
       "6   5        1.478577e-02\n",
       "7   6        3.696442e-02\n",
       "8   7        7.392883e-02\n",
       "9   8        1.201344e-01\n",
       "10  9        1.601791e-01\n",
       "11 10        1.761971e-01\n",
       "12 11        1.601791e-01\n",
       "13 12        1.201344e-01\n",
       "14 13        7.392883e-02\n",
       "15 14        3.696442e-02\n",
       "16 15        1.478577e-02\n",
       "17 16        4.620552e-03\n",
       "18 17        1.087189e-03\n",
       "19 18        1.811981e-04\n",
       "20 19        1.907349e-05\n",
       "21 20        9.536743e-07"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "df <- data.frame(successes, prob)\n",
    "df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "banned-sentence",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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KQMEpGaIREpg0SkZkhEyiARqRkSkTJI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\nnDCIxDAnDCIxzAmDSAxzwiASw5ww/w/RfPekQYKXGwAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ggplot(df, aes(x=successes, y=prob)) + geom_col()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "three-forestry",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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F0MIUkNIXUxhCQ1hNTFEJLUEFIXQ0hS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\nFqtgERKLVbAIicUqWITEYhWs/wf9M1yAv3og6gAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "binomial <- function(size, p, x=size, graph=T, type = 'line', ylim = NULL) {\n",
    "  successes <- c(0:x)\n",
    "  prob <- dbinom(x=0:x, size = size, prob=p)\n",
    "  df <- data.frame(successes = successes, prob=prob)\n",
    "  if (graph) {\n",
    "    return(ggplot(df, aes(x=factor(successes), y=prob)) + geom_col() + coord_cartesian(ylim = ylim))\n",
    "  } else {\n",
    "    return(df)\n",
    "  }\n",
    "}\n",
    "\n",
    "binomial(20, 0.5, x=20, ylim=c(0,0.25))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "improved-design",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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IyQhMQEjJCExASMkITEBIyQhMQEjJCE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\nQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "binomial(10, 0.5, x=20, ylim=c(0,0.25))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "structured-constitution",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ppNdmqTfWqXfavmS7K6gwikv6f3fzri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\nDEiMJez/ARiawRgtL9D7AAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "df_for_linegraph <- binomial(20, 0.5, graph=F)\n",
    "df_for_linegraph$freq <- df_for_linegraph$prob * 200\n",
    "ggplot(df_for_linegraph, aes(x=successes, y=prob)) + geom_line()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "designed-notice",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<style>\n",
       ".list-inline {list-style: none; margin:0; padding: 0}\n",
       ".list-inline>li {display: inline-block}\n",
       ".list-inline>li:not(:last-child)::after {content: \"\\00b7\"; padding: 0 .5ex}\n",
       "</style>\n",
       "<ol class=list-inline><li>7</li><li>11</li><li>11</li><li>11</li><li>12</li><li>11</li><li>5</li><li>8</li><li>11</li><li>10</li><li>11</li><li>10</li><li>9</li><li>13</li><li>9</li><li>12</li><li>9</li><li>9</li><li>8</li><li>8</li><li>9</li><li>9</li><li>8</li><li>6</li><li>8</li><li>12</li><li>10</li><li>13</li><li>12</li><li>6</li><li>10</li><li>9</li><li>9</li><li>10</li><li>8</li><li>12</li><li>8</li><li>9</li><li>15</li><li>12</li><li>10</li><li>11</li><li>9</li><li>11</li><li>9</li><li>10</li><li>11</li><li>10</li><li>8</li><li>12</li><li>7</li><li>9</li><li>11</li><li>10</li><li>8</li><li>10</li><li>10</li><li>12</li><li>8</li><li>12</li><li>12</li><li>6</li><li>9</li><li>5</li><li>8</li><li>11</li><li>9</li><li>10</li><li>6</li><li>10</li><li>7</li><li>13</li><li>5</li><li>12</li><li>7</li><li>10</li><li>9</li><li>7</li><li>9</li><li>11</li><li>13</li><li>10</li><li>8</li><li>10</li><li>8</li><li>13</li><li>9</li><li>9</li><li>8</li><li>13</li><li>8</li><li>13</li><li>8</li><li>8</li><li>7</li><li>10</li><li>9</li><li>6</li><li>9</li><li>11</li><li>6</li><li>10</li><li>9</li><li>8</li><li>8</li><li>9</li><li>8</li><li>7</li><li>10</li><li>6</li><li>11</li><li>7</li><li>14</li><li>7</li><li>8</li><li>13</li><li>14</li><li>9</li><li>7</li><li>12</li><li>11</li><li>13</li><li>16</li><li>14</li><li>10</li><li>9</li><li>8</li><li>10</li><li>10</li><li>9</li><li>14</li><li>11</li><li>7</li><li>10</li><li>13</li><li>10</li><li>13</li><li>11</li><li>11</li><li>13</li><li>10</li><li>15</li><li>9</li><li>10</li><li>9</li><li>11</li><li>11</li><li>10</li><li>15</li><li>10</li><li>10</li><li>8</li><li>7</li><li>12</li><li>8</li><li>15</li><li>11</li><li>17</li><li>9</li><li>10</li><li>10</li><li>10</li><li>10</li><li>8</li><li>7</li><li>11</li><li>10</li><li>7</li><li>12</li><li>9</li><li>12</li><li>10</li><li>11</li><li>10</li><li>9</li><li>12</li><li>10</li><li>10</li><li>7</li><li>9</li><li>10</li><li>9</li><li>11</li><li>7</li><li>14</li><li>6</li><li>12</li><li>11</li><li>9</li><li>11</li><li>11</li><li>15</li><li>7</li><li>13</li><li>12</li><li>12</li><li>12</li><li>11</li><li>15</li><li>11</li></ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 7\n",
       "\\item 11\n",
       "\\item 11\n",
       "\\item 11\n",
       "\\item 12\n",
       "\\item 11\n",
       "\\item 5\n",
       "\\item 8\n",
       "\\item 11\n",
       "\\item 10\n",
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       "\\item 9\n",
       "\\item 12\n",
       "\\item 9\n",
       "\\item 9\n",
       "\\item 8\n",
       "\\item 8\n",
       "\\item 9\n",
       "\\item 9\n",
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       "\\item 12\n",
       "\\item 10\n",
       "\\item 13\n",
       "\\item 12\n",
       "\\item 6\n",
       "\\item 10\n",
       "\\item 9\n",
       "\\item 9\n",
       "\\item 10\n",
       "\\item 8\n",
       "\\item 12\n",
       "\\item 8\n",
       "\\item 9\n",
       "\\item 15\n",
       "\\item 12\n",
       "\\item 10\n",
       "\\item 11\n",
       "\\item 9\n",
       "\\item 11\n",
       "\\item 9\n",
       "\\item 10\n",
       "\\item 11\n",
       "\\item 10\n",
       "\\item 8\n",
       "\\item 12\n",
       "\\item 7\n",
       "\\item 9\n",
       "\\item 11\n",
       "\\item 10\n",
       "\\item 8\n",
       "\\item 10\n",
       "\\item 10\n",
       "\\item 12\n",
       "\\item 8\n",
       "\\item 12\n",
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       "\\item 9\n",
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       "\\item 10\n",
       "\\item 6\n",
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       "\\item 7\n",
       "\\item 13\n",
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       "\\item 9\n",
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       "\\item 8\n",
       "\\item 13\n",
       "\\item 8\n",
       "\\item 8\n",
       "\\item 7\n",
       "\\item 10\n",
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       "\\item 9\n",
       "\\item 11\n",
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       "\\item 10\n",
       "\\item 9\n",
       "\\item 8\n",
       "\\item 8\n",
       "\\item 9\n",
       "\\item 8\n",
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       "\\item 11\n",
       "\\item 7\n",
       "\\item 14\n",
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       "\\item 13\n",
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       "\\item 12\n",
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       "\\item 16\n",
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       "\\item 10\n",
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       "\\item 14\n",
       "\\item 11\n",
       "\\item 7\n",
       "\\item 10\n",
       "\\item 13\n",
       "\\item 10\n",
       "\\item 13\n",
       "\\item 11\n",
       "\\item 11\n",
       "\\item 13\n",
       "\\item 10\n",
       "\\item 15\n",
       "\\item 9\n",
       "\\item 10\n",
       "\\item 9\n",
       "\\item 11\n",
       "\\item 11\n",
       "\\item 10\n",
       "\\item 15\n",
       "\\item 10\n",
       "\\item 10\n",
       "\\item 8\n",
       "\\item 7\n",
       "\\item 12\n",
       "\\item 8\n",
       "\\item 15\n",
       "\\item 11\n",
       "\\item 17\n",
       "\\item 9\n",
       "\\item 10\n",
       "\\item 10\n",
       "\\item 10\n",
       "\\item 10\n",
       "\\item 8\n",
       "\\item 7\n",
       "\\item 11\n",
       "\\item 10\n",
       "\\item 7\n",
       "\\item 12\n",
       "\\item 9\n",
       "\\item 12\n",
       "\\item 10\n",
       "\\item 11\n",
       "\\item 10\n",
       "\\item 9\n",
       "\\item 12\n",
       "\\item 10\n",
       "\\item 10\n",
       "\\item 7\n",
       "\\item 9\n",
       "\\item 10\n",
       "\\item 9\n",
       "\\item 11\n",
       "\\item 7\n",
       "\\item 14\n",
       "\\item 6\n",
       "\\item 12\n",
       "\\item 11\n",
       "\\item 9\n",
       "\\item 11\n",
       "\\item 11\n",
       "\\item 15\n",
       "\\item 7\n",
       "\\item 13\n",
       "\\item 12\n",
       "\\item 12\n",
       "\\item 12\n",
       "\\item 11\n",
       "\\item 15\n",
       "\\item 11\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 7\n",
       "2. 11\n",
       "3. 11\n",
       "4. 11\n",
       "5. 12\n",
       "6. 11\n",
       "7. 5\n",
       "8. 8\n",
       "9. 11\n",
       "10. 10\n",
       "11. 11\n",
       "12. 10\n",
       "13. 9\n",
       "14. 13\n",
       "15. 9\n",
       "16. 12\n",
       "17. 9\n",
       "18. 9\n",
       "19. 8\n",
       "20. 8\n",
       "21. 9\n",
       "22. 9\n",
       "23. 8\n",
       "24. 6\n",
       "25. 8\n",
       "26. 12\n",
       "27. 10\n",
       "28. 13\n",
       "29. 12\n",
       "30. 6\n",
       "31. 10\n",
       "32. 9\n",
       "33. 9\n",
       "34. 10\n",
       "35. 8\n",
       "36. 12\n",
       "37. 8\n",
       "38. 9\n",
       "39. 15\n",
       "40. 12\n",
       "41. 10\n",
       "42. 11\n",
       "43. 9\n",
       "44. 11\n",
       "45. 9\n",
       "46. 10\n",
       "47. 11\n",
       "48. 10\n",
       "49. 8\n",
       "50. 12\n",
       "51. 7\n",
       "52. 9\n",
       "53. 11\n",
       "54. 10\n",
       "55. 8\n",
       "56. 10\n",
       "57. 10\n",
       "58. 12\n",
       "59. 8\n",
       "60. 12\n",
       "61. 12\n",
       "62. 6\n",
       "63. 9\n",
       "64. 5\n",
       "65. 8\n",
       "66. 11\n",
       "67. 9\n",
       "68. 10\n",
       "69. 6\n",
       "70. 10\n",
       "71. 7\n",
       "72. 13\n",
       "73. 5\n",
       "74. 12\n",
       "75. 7\n",
       "76. 10\n",
       "77. 9\n",
       "78. 7\n",
       "79. 9\n",
       "80. 11\n",
       "81. 13\n",
       "82. 10\n",
       "83. 8\n",
       "84. 10\n",
       "85. 8\n",
       "86. 13\n",
       "87. 9\n",
       "88. 9\n",
       "89. 8\n",
       "90. 13\n",
       "91. 8\n",
       "92. 13\n",
       "93. 8\n",
       "94. 8\n",
       "95. 7\n",
       "96. 10\n",
       "97. 9\n",
       "98. 6\n",
       "99. 9\n",
       "100. 11\n",
       "101. 6\n",
       "102. 10\n",
       "103. 9\n",
       "104. 8\n",
       "105. 8\n",
       "106. 9\n",
       "107. 8\n",
       "108. 7\n",
       "109. 10\n",
       "110. 6\n",
       "111. 11\n",
       "112. 7\n",
       "113. 14\n",
       "114. 7\n",
       "115. 8\n",
       "116. 13\n",
       "117. 14\n",
       "118. 9\n",
       "119. 7\n",
       "120. 12\n",
       "121. 11\n",
       "122. 13\n",
       "123. 16\n",
       "124. 14\n",
       "125. 10\n",
       "126. 9\n",
       "127. 8\n",
       "128. 10\n",
       "129. 10\n",
       "130. 9\n",
       "131. 14\n",
       "132. 11\n",
       "133. 7\n",
       "134. 10\n",
       "135. 13\n",
       "136. 10\n",
       "137. 13\n",
       "138. 11\n",
       "139. 11\n",
       "140. 13\n",
       "141. 10\n",
       "142. 15\n",
       "143. 9\n",
       "144. 10\n",
       "145. 9\n",
       "146. 11\n",
       "147. 11\n",
       "148. 10\n",
       "149. 15\n",
       "150. 10\n",
       "151. 10\n",
       "152. 8\n",
       "153. 7\n",
       "154. 12\n",
       "155. 8\n",
       "156. 15\n",
       "157. 11\n",
       "158. 17\n",
       "159. 9\n",
       "160. 10\n",
       "161. 10\n",
       "162. 10\n",
       "163. 10\n",
       "164. 8\n",
       "165. 7\n",
       "166. 11\n",
       "167. 10\n",
       "168. 7\n",
       "169. 12\n",
       "170. 9\n",
       "171. 12\n",
       "172. 10\n",
       "173. 11\n",
       "174. 10\n",
       "175. 9\n",
       "176. 12\n",
       "177. 10\n",
       "178. 10\n",
       "179. 7\n",
       "180. 9\n",
       "181. 10\n",
       "182. 9\n",
       "183. 11\n",
       "184. 7\n",
       "185. 14\n",
       "186. 6\n",
       "187. 12\n",
       "188. 11\n",
       "189. 9\n",
       "190. 11\n",
       "191. 11\n",
       "192. 15\n",
       "193. 7\n",
       "194. 13\n",
       "195. 12\n",
       "196. 12\n",
       "197. 12\n",
       "198. 11\n",
       "199. 15\n",
       "200. 11\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  [1]  7 11 11 11 12 11  5  8 11 10 11 10  9 13  9 12  9  9  8  8  9  9  8  6  8\n",
       " [26] 12 10 13 12  6 10  9  9 10  8 12  8  9 15 12 10 11  9 11  9 10 11 10  8 12\n",
       " [51]  7  9 11 10  8 10 10 12  8 12 12  6  9  5  8 11  9 10  6 10  7 13  5 12  7\n",
       " [76] 10  9  7  9 11 13 10  8 10  8 13  9  9  8 13  8 13  8  8  7 10  9  6  9 11\n",
       "[101]  6 10  9  8  8  9  8  7 10  6 11  7 14  7  8 13 14  9  7 12 11 13 16 14 10\n",
       "[126]  9  8 10 10  9 14 11  7 10 13 10 13 11 11 13 10 15  9 10  9 11 11 10 15 10\n",
       "[151] 10  8  7 12  8 15 11 17  9 10 10 10 10  8  7 11 10  7 12  9 12 10 11 10  9\n",
       "[176] 12 10 10  7  9 10  9 11  7 14  6 12 11  9 11 11 15  7 13 12 12 12 11 15 11"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "set.seed(1234)\n",
    "trial_200 <- rbinom(n=200, size=20, prob=0.5)\n",
    "trial_200"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "static-nomination",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "oriented-realtor",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "                                                                              \r"
     ]
    },
    {
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