diff options
| -rw-r--r-- | sem6/prob/stat4/Opgaver.ipynb | 93 |
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": { |