Changes to stat assignments

1 files changed, 84 insertions, 9 deletions

diff --git a/sem6/prob/stat4/Opgaver.ipynb b/sem6/prob/stat4/Opgaver.ipynb
index 563260b..e4b0984 100644
--- a/sem6/prob/stat4/Opgaver.ipynb
+++ b/sem6/prob/stat4/Opgaver.ipynb
@@ -9,7 +9,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 34,
+   "execution_count": 3,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -27,7 +27,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 9,
+   "execution_count": 4,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -46,7 +46,7 @@
     "What conclusion can be made if $\\alpha = 0.10$ level of significance.\n",
     "\n",
     "We let our $H_0: \\mu = 8.20$.\n",
-    "We can conclude the probability of a *type 1* error, thus a false positive.\n",
+    "We can conclude the probability of a *type 1* error.\n",
     "\n",
     "We can calculate the stats \n",
     "$$\n",
@@ -63,15 +63,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 37,
+   "execution_count": 25,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "3.320391543176363\n",
-      "0.0008989127881156023 0.1 False\n"
+      "8.179000000000002\n",
+      "5.143928459843998\n",
+      "2.6905202554772245e-07 0.1 False\n"
      ]
     },
     {
@@ -89,6 +90,7 @@
    ],
    "source": [
     "x_hat = np.mean(samples)\n",
+    "print(x_hat)\n",
     "\n",
     "def assign(alpha):\n",
     "    v = np.sqrt(n) / sigma * np.abs(x_hat - mu)\n",
@@ -105,7 +107,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 39,
+   "execution_count": 6,
    "metadata": {},
    "outputs": [
     {
@@ -153,7 +155,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 42,
+   "execution_count": 7,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -175,7 +177,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 49,
+   "execution_count": 8,
    "metadata": {},
    "outputs": [
     {
@@ -204,6 +206,79 @@
     "n = ((z_alpha2 + z_beta)**2 * sigma**2) / ((mu + sigma_reject - mu)**2)\n",
     "display(Markdown(f\"Thus $n$ must be ${np.ceil(n)}$\"))"
    ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 33,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "p_value: 0.0 ts: 13.472193585307627\n",
+      "H_0 is rejected\n"
+     ]
+    }
+   ],
+   "source": [
+    "## Part C\n",
+    "n = 6\n",
+    "\n",
+    "# We start by calculating a new test statistic\n",
+    "ts = np.sqrt(n) / sigma * (8.31 - mu)\n",
+    "p_value = 2 * (1 - norm.cdf(np.abs(ts)))\n",
+    "print(\"p_value:\", p_value, \"ts:\", ts)\n",
+    "if p_value < alpha:\n",
+    "    print(\"H_0 is rejected\")\n",
+    "else:\n",
+    "    print(\"H_0 is accepted\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Part D\n",
+    "\n",
+    "The probability of accepting $H_0$ when the true mean is $\\mu$ is:\n",
+    "\n",
+    "$$\n",
+    "\\beta(\\mu) = \\Phi\\left(\\frac {\\mu_0 - \\mu} {\\sigma / \\sqrt{n}} + z_{\\alpha / 2}\\right) - \\Phi\\left(\\frac {\\mu_0 - \\mu} {\\sigma / \\sqrt{n}} - z_{\\alpha / 2}\\right)\n",
+    "$$\n",
+    "\n",
+    "To find the probability of rejecting $H_0$ we subtract this from 1.\n",
+    "\n",
+    "$$\n",
+    "P(reject \\: H_0) = 1 - \\beta(\\mu)\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 35,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "beta: 1.8420324769867475e-37\n",
+      "There is a 1.0 change that H_0 is rejected\n"
+     ]
+    }
+   ],
+   "source": [
+    "real_mu = 8.32\n",
+    "blob = (mu - real_mu) / (sigma / np.sqrt(n))\n",
+    "za2 = - norm.ppf(alpha / 2)\n",
+    "\n",
+    "beta = norm.cdf(blob + za2) - norm.cdf(blob - za2)\n",
+    "print(\"beta:\", beta)\n",
+    "\n",
+    "prob = 1 - beta\n",
+    "print(f\"There is a {prob} change that H_0 is rejected\")"
+   ]
   }
  ],
  "metadata": {