Statistics_Lab2_(Probability_Distributions_&_CIs)_Jason_(EU)_18_Feb_2022_Student_(1).ipynb

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        "<a href=\"https://colab.research.google.com/github/coder-omer/Statistics/blob/main/Statistics_Lab2_(Probability_Distributions_%26_CIs)_Jason_(EU)_18_Feb_2022_Student_(1).ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "tMh2iHQMQRYm"
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import pandas as pd\n",
        "from scipy import stats\n",
        "import seaborn as sns\n",
        "import matplotlib.pyplot as plt"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "dY7J52nWQRYu"
      },
      "source": [
        "## Binomial Distribution\n",
        "*Source: Wackerly, D., Mendenhall, W., & Scheaffer, R. L. (2014). Mathematical statistics with applications. Cengage Learning.*"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "iO1h_auWQRYy"
      },
      "source": [
        "**Exercise 3.40**\n",
        "\n",
        "The probability that a patient recovers from a stomach disease is 0.8. \n",
        "\n",
        "Suppose 20 people are known to have contracted this disease. \n",
        "\n",
        "What is the probability that"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "F2HgF46xQRY0"
      },
      "source": [
        "a. exactly 14 recover?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "XSZxWB-VQRY3"
      },
      "source": [
        "this is P(X=14)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "chhP_QFHQRY3"
      },
      "outputs": [],
      "source": [
        "n = 20  #the number of trials\n",
        "p = 0.8 #the probability of success\n",
        "x = 14  #r.v."
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "ola = 1- (stats.binom.pmf(k=17, n=20, p=0.8) + stats.binom.pmf(k=18, n=20, p=0.8) + stats.binom.pmf(k=19, n=20, p=0.8) + stats.binom.pmf(k=20, n=20, p=0.8))\n",
        "ola"
      ],
      "metadata": {
        "colab": {
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        },
        "id": "UJ1Aw1jyYkSn",
        "outputId": "9dd80fb5-42a5-4a56-a51b-16c35f43d7a4"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.5885511380434316"
            ]
          },
          "metadata": {},
          "execution_count": 8
        }
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
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          "base_uri": "https://localhost:8080/"
        },
        "id": "Nf_4hswZQRY4",
        "outputId": "7e0fe295-c7e0-4706-8ab6-3502b2727fd8"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.10909970097304983"
            ]
          },
          "metadata": {},
          "execution_count": 5
        }
      ],
      "source": [
        "stats.binom.pmf(k=14, n=20, p=0.8)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
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        },
        "id": "kJkJrA-gQRY6",
        "outputId": "d44197e1-b6e0-44ca-e42c-b782a7ab0742"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.10909970097304983"
            ]
          },
          "metadata": {},
          "execution_count": 9
        }
      ],
      "source": [
        "stats.binom.pmf(x, n, p)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "MDzASyrHQRY8",
        "outputId": "ee9a3f1b-0f74-4e5b-9ee8-4972b8359e8f"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "16.0"
            ]
          },
          "metadata": {},
          "execution_count": 10
        }
      ],
      "source": [
        "stats.binom.mean(n, p)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "OLe0iR5OQRY9"
      },
      "source": [
        "b. at least 10 recover?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "zovU-2p0QRY-"
      },
      "source": [
        "this is P(X≥10)=P(10)+P(11)+P(12)+...+P(20)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "L8S82INgQRY_"
      },
      "outputs": [],
      "source": [
        "x = 9"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "stats.binom.cdf(x, n, p)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "xAwCXdZcbKJF",
        "outputId": "2f9ecd3e-d861-4da8-8ec4-3542fb065d53"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.0005634136976601897"
            ]
          },
          "metadata": {},
          "execution_count": 13
        }
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "bia9gy1wQRZA",
        "outputId": "a0ed71c2-73be-42c6-cc9d-4d9a296804af"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.9994365863023398"
            ]
          },
          "metadata": {},
          "execution_count": 12
        }
      ],
      "source": [
        "1 - stats.binom.cdf(x, n, p)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "nxxgA5ohQRZA",
        "outputId": "46bc442b-24fc-4e7f-9af1-776a9f502236"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.9994365863023398"
            ]
          },
          "metadata": {},
          "execution_count": 14
        }
      ],
      "source": [
        "1 - stats.binom.cdf(9, 20, 0.8)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "QfSp_NWGQRZB"
      },
      "source": [
        "c. at least 14 but not more than 18 recover?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FExrJJhsQRZC"
      },
      "source": [
        "this is P(14≤X≤18)=P(14)+P(15)+P(16)+P(17)+P(18)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "jeeYoYuRQRZC"
      },
      "outputs": [],
      "source": [
        "x1 = 14\n",
        "x2 = 18"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "XVBrlj0ZQRZD",
        "outputId": "fdd91435-c87d-4848-f0d6-2aa15dcb31ba"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.844132196156189"
            ]
          },
          "metadata": {},
          "execution_count": 16
        }
      ],
      "source": [
        "stats.binom.cdf(x2, n, p) - stats.binom.cdf(x1-1, n, p)"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "stats.binom.cdf(18, n, p) - stats.binom.cdf(13, n, p)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "dRe_0jZ5ceJr",
        "outputId": "956764d0-787e-4bc5-9276-5bb5e3aaa6e5"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.844132196156189"
            ]
          },
          "metadata": {},
          "execution_count": 17
        }
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
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        "id": "OAUvyHYCQRZD",
        "outputId": "2bbed560-0aca-48e6-ce93-ef48239d9533"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.8441321961561871"
            ]
          },
          "metadata": {},
          "execution_count": 18
        }
      ],
      "source": [
        "stats.binom.pmf(k=14, n=20, p=0.8) + stats.binom.pmf(k=15, n=20, p=0.8) + stats.binom.pmf(k=16, n=20, p=0.8) + stats.binom.pmf(k=17, n=20, p=0.8) +stats.binom.pmf(k=18, n=20, p=0.8)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "y6Nn-DP8QRZE"
      },
      "source": [
        "d. at most 16 recover?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-7z6gAGeQRZE"
      },
      "source": [
        "this is P(X≤16)=1−P(X>16)=P(17)+P(18)+P(19)+P(20)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "Xx4H9Oz-QRZF"
      },
      "outputs": [],
      "source": [
        "x = 16"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "71plyEsqQRZF",
        "outputId": "dc7a36b4-f29d-4f8c-b9af-3e21f6a596e3"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.5885511380434305"
            ]
          },
          "metadata": {},
          "execution_count": 20
        }
      ],
      "source": [
        "stats.binom.cdf(x, n, p)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Zfb6X57eQRZG",
        "outputId": "415c34cf-c32e-4c17-94af-57bc3dbae324"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.5885511380434316"
            ]
          },
          "metadata": {},
          "execution_count": 21
        }
      ],
      "source": [
        "1 - (stats.binom.pmf(17, 20, 0.8) + stats.binom.pmf(18, 20, 0.8) + stats.binom.pmf(19, 20, 0.8) + stats.binom.pmf(20, 20, 0.8))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "v5CbDFgaQRZG"
      },
      "source": [
        "e. Construct discrete probability distribution table and draw a graph"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "QSPT3oy6QRZH",
        "outputId": "3a32efee-01e7-4d9a-d620-80a60e29e566"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "0\t1.048576000000001e-14\n",
            "1\t8.388607999999975e-13\n",
            "2\t3.187671040000001e-11\n",
            "3\t7.650410495999974e-10\n",
            "4\t1.3005697843199945e-08\n",
            "5\t1.6647293239295892e-07\n",
            "6\t1.664729323929588e-06\n",
            "7\t1.3317834591436803e-05\n",
            "8\t8.656592484433902e-05\n",
            "9\t0.000461684932503141\n",
            "10\t0.002031413703013826\n",
            "11\t0.007386958920050259\n",
            "12\t0.022160876760150803\n",
            "13\t0.0545498504865252\n",
            "14\t0.10909970097304983\n",
            "15\t0.17455952155687976\n",
            "16\t0.21819940194610007\n",
            "17\t0.2053641430080944\n",
            "18\t0.13690942867206304\n",
            "19\t0.05764607523034236\n",
            "20\t0.011529215046068481\n"
          ]
        }
      ],
      "source": [
        "dist = []\n",
        "\n",
        "for i in range(n+1):\n",
        "    dist.append(stats.binom.pmf(i,n,p))\n",
        "    print(str(i) + \"\\t\" + str(stats.binom.pmf(i,n,p)))"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "dist"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "c0Fhtmg-eVYz",
        "outputId": "5af448e8-551a-4d3b-907b-ece4b7bac56a"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[1.048576000000001e-14,\n",
              " 8.388607999999975e-13,\n",
              " 3.187671040000001e-11,\n",
              " 7.650410495999974e-10,\n",
              " 1.3005697843199945e-08,\n",
              " 1.6647293239295892e-07,\n",
              " 1.664729323929588e-06,\n",
              " 1.3317834591436803e-05,\n",
              " 8.656592484433902e-05,\n",
              " 0.000461684932503141,\n",
              " 0.002031413703013826,\n",
              " 0.007386958920050259,\n",
              " 0.022160876760150803,\n",
              " 0.0545498504865252,\n",
              " 0.10909970097304983,\n",
              " 0.17455952155687976,\n",
              " 0.21819940194610007,\n",
              " 0.2053641430080944,\n",
              " 0.13690942867206304,\n",
              " 0.05764607523034236,\n",
              " 0.011529215046068481]"
            ]
          },
          "metadata": {},
          "execution_count": 23
        }
      ]
    },
    {
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          "height": 265
        },
        "id": "QsjiaExrQRZH",
        "outputId": "561a3a53-fcb1-456f-cc4a-279bbbb0e887"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": "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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "%matplotlib inline\n",
        "plt.bar(list(range(n+1)), dist)\n",
        "plt.show;"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "cMoVbhccQRZI",
        "outputId": "52addca6-2c91-41b5-a2eb-54c90e939520"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]"
            ]
          },
          "metadata": {},
          "execution_count": 25
        }
      ],
      "source": [
        "list(range(n+1))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "MJO-wt2JQRZI"
      },
      "source": [
        "## Poisson Distribution\n",
        "*Source: Wackerly, D., Mendenhall, W., & Scheaffer, R. L. (2014). Mathematical statistics with applications. Cengage Learning.*"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "6auGruoLQRZJ"
      },
      "source": [
        "Suppose that a random system of police patrol is devised so that a patrol officer may visit a given beat location Y = 0, 1, 2, 3, . . . times per half-hour period, with each location being visited an average of once per time period. \n",
        "\n",
        "Assume that Y possesses, approximately, a Poisson probability distribution. \n",
        "\n",
        "Calculate the probability that the patrol officer will miss a given location during a half-hour period. \n",
        "\n",
        "What is the probability that it will be visited once? \n",
        "\n",
        "Twice? \n",
        "\n",
        "At least once?"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "H45lF4OpQRZJ"
      },
      "outputs": [],
      "source": [
        "#For this example the time period is a half-hour, and the mean number of visits per half-hour interval is λ = 1. Then\n",
        "\n",
        "mu = 1"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "CA1Es_nGQRZJ"
      },
      "source": [
        "P(Y = 0) = ?"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "KSNItHMjQRZK",
        "outputId": "43417c86-75db-4db9-9681-1321c9f9dcd0"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.36787944117144233"
            ]
          },
          "metadata": {},
          "execution_count": 33
        }
      ],
      "source": [
        "stats.poisson.pmf(k=0, mu=1)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "pawuV00IQRZK"
      },
      "source": [
        "P(Y = 1) = ?"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "ZMl_ysMWQRZK",
        "outputId": "254692cb-ea2f-4b83-e645-485cd09e0984"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.36787944117144233"
            ]
          },
          "metadata": {},
          "execution_count": 34
        }
      ],
      "source": [
        "stats.poisson.pmf(k=1, mu=1)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "eXj09cd3QRZL"
      },
      "source": [
        "P(Y = 2) = ?"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "y3MjPwNfQRZM",
        "outputId": "93e04f08-cbba-4cc7-afad-fc199c410ab0"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.18393972058572114"
            ]
          },
          "metadata": {},
          "execution_count": 35
        }
      ],
      "source": [
        "stats.poisson.pmf(k=2, mu=1)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "alBLtkBoQRZM"
      },
      "source": [
        "P(Y ≥ 1) = ?"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "9IywqZ6cQRZM",
        "outputId": "06eb6c4c-6635-4a4c-cff5-43178966a172"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.6321205588285577"
            ]
          },
          "metadata": {},
          "execution_count": 36
        }
      ],
      "source": [
        "1 - stats.poisson.pmf(0, 1)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "nuRtouR0QRZN",
        "outputId": "c1206047-e020-474c-9917-f0f1fb4b1c95"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.6321205588285576"
            ]
          },
          "metadata": {},
          "execution_count": 37
        }
      ],
      "source": [
        "1 - stats.poisson.cdf(0, 1)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "gmMRPl0XQRZN"
      },
      "source": [
        "P(Y ≥ 3) = ?"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "DcK6sI0BQRZO",
        "outputId": "3bb3225c-243f-4eb0-fbbf-030fec529d15"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.08030139707139416"
            ]
          },
          "metadata": {},
          "execution_count": 38
        }
      ],
      "source": [
        "1 - stats.poisson.cdf(2, 1)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "w3cJdGnzQRZO"
      },
      "source": [
        "## Normal Distribution\n",
        "*Source: Wackerly, D., Mendenhall, W., & Scheaffer, R. L. (2014). Mathematical statistics with applications. Cengage Learning.*"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "3seETe2PQRZO"
      },
      "source": [
        "**Exercise 4.74**\n",
        "\n",
        "Scores on an examination are assumed to be normally distributed with mean 78 and variance 36."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "YN-dkF5_QRZP"
      },
      "source": [
        "a. What is the probability that a person taking the examination scores higher than 72?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4pY0awIvQRZP"
      },
      "source": [
        "this is P(X>72)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "S40iYEqAQRZP"
      },
      "outputs": [],
      "source": [
        "mu = 78\n",
        "sigma = np.sqrt(36)\n",
        "x = 72"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "bz72FXKdQRZQ",
        "outputId": "46aa222d-59cc-49e3-8fc3-767ef38098fe"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.8413447460685429"
            ]
          },
          "metadata": {},
          "execution_count": 40
        }
      ],
      "source": [
        "1 - stats.norm.cdf(x, loc=mu, scale=sigma)"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "1 - stats.norm.cdf(72, 78, 6)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "mcEH5TLXj4DU",
        "outputId": "43644023-5996-411e-ead9-57df98946e85"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.8413447460685429"
            ]
          },
          "metadata": {},
          "execution_count": 41
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Db-WMzORQRZQ"
      },
      "source": [
        "b. Suppose that students scoring in the top 10% of this distribution are to receive an A grade.\n",
        "What is the minimum score a student must achieve to earn an A grade?"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "yoJgk4hgQRZQ"
      },
      "outputs": [],
      "source": [
        "q = 1 - 0.1  #90%"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "-LzOJKhrQRZQ",
        "outputId": "c07a12aa-9023-4d62-d6a9-fef3680083b7"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "1.2815515655446004"
            ]
          },
          "metadata": {},
          "execution_count": 43
        }
      ],
      "source": [
        "stats.norm.ppf(q)  #corresponding z-score"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "xveblZicQRZR",
        "outputId": "d4a3b006-1cf1-43db-cdef-756ff442d0ae"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "85.6893093932676"
            ]
          },
          "metadata": {},
          "execution_count": 44
        }
      ],
      "source": [
        "#stats.norm.ppf(q, mu, sigma)\n",
        "stats.norm.ppf(q, loc=78, scale=6)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "UUBPB3aiQRZR"
      },
      "source": [
        "*c*. What must be the cutoff point for passing the examination if the examiner wants only the\n",
        "top 28.1% of all scores to be passing?"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "XznwVpkIQRZS",
        "outputId": "c7c7edc7-9c87-49c2-d273-a041b6a0cc6c"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.719"
            ]
          },
          "metadata": {},
          "execution_count": 46
        }
      ],
      "source": [
        "q = 1 - 0.281\n",
        "q"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "w6RNGDYDQRZS",
        "outputId": "427a660b-bd6d-4557-fa6b-ef92760e7535"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "81.47924035456623"
            ]
          },
          "metadata": {},
          "execution_count": 47
        }
      ],
      "source": [
        "stats.norm.ppf(q, mu, sigma)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "gS92jnMIQRZS"
      },
      "source": [
        "d. Approximately what proportion of students have scores 5 or more points above the score\n",
        "that cuts off the lowest 25%?"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "PBK3KvhFQRZS"
      },
      "outputs": [],
      "source": [
        "q1 =  #HOMEWORK\n",
        "x1 = "
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "4a2ZxaaZQRZT"
      },
      "outputs": [],
      "source": [
        "#0.43689605516226615"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "pe4v97rKQRZT"
      },
      "source": [
        "*OPTIONAL*\n",
        "\n",
        "f. If it is known that a student’s score exceeds 72, what is the probability that his or her score\n",
        "exceeds 84? "
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "sdJ9p1wXQRZT"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "R-tFDw8dQRZU"
      },
      "source": [
        "## t Distribution\n",
        "*Lane, D. (2006). Online statistics education: a multimedia course of study (http://onlinestatbook.com/). Rice University.*"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "WaHp8nenQRZU"
      },
      "source": [
        "For a t distribution with 15 degrees of freedom, 90% of the distribution is \n",
        "\n",
        "---\n",
        "\n",
        "within how many standard deviations of the mean?"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "xEyrsht-QRZU"
      },
      "outputs": [],
      "source": [
        "dof = 15\n",
        "q = 0.05"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "HP2qfu9KQRZU",
        "outputId": "4e0e3297-5806-4431-d6b5-ce6793880a7a"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "-1.7530503556925552"
            ]
          },
          "execution_count": 49,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "stats.t.ppf(q, dof)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "DzVM7Xa1QRZV"
      },
      "source": [
        "## Confidence Intervals Using the Normal Distribution"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4JB7HvGwQRZV"
      },
      "source": [
        "If we’re working with larger samples (n≥30), we can assume that the sampling distribution of the sample mean is normally distributed (thanks to the Central Limit Theorem) and can instead use the norm.interval() function from the scipy.stats library."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "iyM1ejWnQRZV"
      },
      "outputs": [],
      "source": [
        "sns.get_dataset_names()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_lG9c7BNQRZW"
      },
      "source": [
        "1. Load \"tips\" dataset from Seaborn library"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "pKzBrutVQRZW"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "UQ-cMmFRQRZW"
      },
      "source": [
        "2. Draw a set of vertical bar plots (based on \"total_bill\") grouped by \"day\" variable:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "eBbb7n8RQRZW"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "IpB0Qo4-QRZW"
      },
      "source": [
        "3. Subset a dataframe just including Sunday tips"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "l1lXxGx6QRZX"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "CKYDFqVOQRZX"
      },
      "source": [
        "4. Calculate mean and std"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "nowsj3rPQRZX"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "ahejOLtDQRZX"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cn7rgHqWQRZX"
      },
      "source": [
        "5. Calculate standart error of the mean"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "DQdmX3PrQRZY"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "JuS3R21CQRZY"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "n1345nmuQRZY"
      },
      "source": [
        "6. Calculate the margin of error (95%)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "z1l0Qiz6QRZY"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-cyjKRTYQRZY"
      },
      "source": [
        "7. Calculate the upper confidence limit"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "XI0NqxS0QRZZ"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "QSTKb4xpQRZZ"
      },
      "source": [
        "8. Calculate the lower confidence limit"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "krGO1YOrQRZZ"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "NAh3CsdwQRZZ"
      },
      "source": [
        "9. Calculate 95% Confidence Interval around the mean by using scipy.stats"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "lhTBSEIiQRZa"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "JNf1cVFPQRZa"
      },
      "source": [
        "### Exercise 8.7 (Large Samples)\n",
        "*Source: Wackerly, D., Mendenhall, W., & Scheaffer, R. L. (2014). Mathematical statistics with applications. Cengage Learning.*"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "7wK0o8hkQRZa"
      },
      "source": [
        "- The shopping times of n = 64 randomly selected customers at a local supermarket were recorded. \n",
        "\n",
        "- The average and variance of the 64 shopping times were 33 $minutes$ and 256 $minutes^2$, respectively. \n",
        "\n",
        "- Estimate μ, the true average shopping time per customer, with a confidence coefficient of 1 − α = .90."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_WPXayhBQRZa"
      },
      "source": [
        "![image.png](attachment:image.png)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "hGnvKhFxQRZb"
      },
      "outputs": [],
      "source": [
        "n = \n",
        "xbar = \n",
        "var = \n",
        "ci = \n",
        "s = \n",
        "\n",
        "sem = "
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "nPrvM4s2QRZb"
      },
      "outputs": [],
      "source": [
        "stats.norm."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "skdTIlwBQRZb"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "jHa4svd7QRZb"
      },
      "source": [
        "### Exercise 8.11 (Small Samples)\n",
        "*Source: Wackerly, D., Mendenhall, W., & Scheaffer, R. L. (2014). Mathematical statistics with applications. Cengage Learning.*"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ABGUpZcRQRZc"
      },
      "source": [
        "A manufacturer of gunpowder has developed a new powder, which was tested in eight shells. The resulting muzzle velocities, in feet per second, were as follows:\n",
        "\n",
        "3005 2925 2935 2965\n",
        "\n",
        "2995 3005 2937 2905\n",
        "\n",
        "Find a 95% confidence interval for the true average velocity μ for shells of this type. Assume that muzzle velocities are approximately normally distributed."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "L-GL7XL4QRZc"
      },
      "source": [
        "![image.png](attachment:image.png)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "rXtgGL2LQRZc"
      },
      "outputs": [],
      "source": [
        "x = [3005, 2925, 2935, 2965, 2995, 3005, 2937, 2905]"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "_8j-XoGiQRZc"
      },
      "outputs": [],
      "source": [
        "# Sample size\n",
        "n = \n",
        "\n",
        "# Sample mean\n",
        "xbar = \n",
        "\n",
        "# Sample standard deviation\n",
        "s = \n",
        "\n",
        "# Confidence level\n",
        "cl = "
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "znW6SoxHQRZd"
      },
      "outputs": [],
      "source": [
        "stats.t."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "6W3XIKH6QRZd"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "b6QKcggTQRZd"
      },
      "outputs": [],
      "source": [
        ""
      ]
    }
  ],
  "metadata": {
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      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.7.11"
    },
    "colab": {
      "name": "Statistics_Lab2_(Probability Distributions & CIs)_Jason_(EU)_18_Feb_2022-Student (1).ipynb",
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