32:T51b,plt_pdf = Plot.plot({ caption:html` probability of type I error: α = Pr(reject H0 | H0 true)
probability of type II error: β = Pr(not rejecting H0 | H0 not true)
test power: 1-β = Pr(reject H0 | H0 not true) `, color: { legend: true }, x: { label: "xBar", axis: true }, y: { axis: false, domain: [0,1.02*maxpdf] }, marks: [ Plot.ruleY([0]), Plot.areaY(pdfdata, {filter: d => d.x <= zcrit_right && d.x >= zcrit_left && d.type == "HA", x: "x", y: "pdf", fill: "#C0C0C0", opacity: 0.3}), Plot.areaY(pdfdata, {filter: d => d.x > zcrit_right && d.type == "HA", x: "x", y: "pdf", fill: "orange", opacity: 0.5}), Plot.areaY(pdfdata, {filter: d => d.x < zcrit_left && d.type == "HA", x: "x", y: "pdf", fill: "orange", opacity: 0.5}), Plot.areaY(pdfdata, {filter: d => d.x >= zcrit_right && d.type == "H0", x: "x", y: "pdf", fill: "steelblue", opacity: 0.7}), Plot.areaY(pdfdata, {filter: d => d.x <= zcrit_left && d.type == "H0", x: "x", y: "pdf", fill: "steelblue", opacity: 0.7}), Plot.line(pdfdata, {x: "x", y: "pdf", stroke : "type", strokeWidth: 2}) ] })33:[] 30:{"title":"Hypothesis test for a mean - Type I and II errors","cells":[{"id":38,"value":"# Hypothesis test for a mean - Type I and II errors","pinned":false,"hidden":false,"mode":"md"},{"id":496,"value":"modelltext = md`### Model and hypotheses\nLet ${tex`X_1,X_2,\\ldots,X_n \\overset{iid}{\\sim}N(\\mu,\\sigma)`} with ${tex`\\sigma`} known.\n\nWe are testing:\n\n${tex`\nH_0:\\mu = \\mu_0 \\\\\nH_A:\\mu \\neq \\mu_0\n`}\n
\nwith test statistic
\n\n${tex`Z = \\frac{\\bar X -\\mu_0}{\\sigma/\\sqrt n}`}\n
\nthat follows a standard normal distribution under ${tex`H_0`}.\n\nThe alternative hypothesis ${tex`H_A`} has a mean of ${tex`\\mu_A`}.\n\n${tex`\nH_0: \\bar X \\sim N(\\mu_0,\\sigma/\\sqrt n) \\\\\nH_A: \\bar X \\sim N(\\mu_A,\\sigma/\\sqrt n)\n`}\n
\nThe figure below shows the distribution for the sample mean ${tex`\\bar X`} under ${tex`H_0`} and ${tex`H_A`} on the original scale (i.e. without standardization).\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":118,"value":"viewof inputs = Inputs.form([\n Inputs.range([-3, 3], {value: 0, step: 0.1, label: tex`\\mu_0`}),\n Inputs.range([-5, 5], {value: 2, step: 0.1, label: tex`\\mu_A`}),\n Inputs.range([0.01, 2], {value: 1, step: 0.01, label: tex`\\sigma`}),\n Inputs.range([1, 20], {value: 1, step: 1, label: \"n\"}),\n Inputs.range([0, 1], {value: 0.05, step: 0.001, label: tex`\\alpha`}),\n ])","pinned":false,"hidden":false,"mode":"ojs"},{"id":529,"value":"probserrors = tex`\n\\begin{aligned}\n&\\text{Type I error: }\\alpha = P(\\text{rejecting } H_0 \\vert H_0 \\text{ true}) =${alpha.toPrecision(3)} \\\\[0.2em]\n&\\text{Type II error: }\\beta = P(\\text{not rejecting } H_0 \\vert H_0 \\text{ not true}) =${beta.toPrecision(3)} \\\\[0.2em]\n&\\text{Power: }1-\\beta = P(\\text{reject } H_0 \\vert H_0 \\text{ not true}) =${(1-beta).toPrecision(3)} \n\\end{aligned}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":305,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nDetails and data things that you probably don't want to read.","pinned":false,"hidden":false,"mode":"md"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":699,"value":"mu0 = inputs[0]","pinned":true,"hidden":false,"mode":"ojs"},{"id":703,"value":"muA = inputs[1]","pinned":true,"hidden":false,"mode":"ojs"},{"id":707,"value":"sigma = inputs[2]","pinned":true,"hidden":false,"mode":"ojs"},{"id":705,"value":"n = inputs[3]","pinned":true,"hidden":false,"mode":"ojs"},{"id":736,"value":"alpha = inputs[4]","pinned":true,"hidden":false,"mode":"ojs"},{"id":17,"value":"pdfdata = {\n const x = d3.range(-10, 10, 0.01);\n const normpdfH0 = x.map(x => ({x: x, pdf: jstat.normal.pdf(x, mu0, sigma/Math.sqrt(n)), type: \"H0\"}));\n const normpdfHA = x.map(x => ({x: x, pdf: jstat.normal.pdf(x, muA, sigma/Math.sqrt(n)), type: \"HA\"}));\n return normpdfH0.concat(normpdfHA)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":723,"value":"zcrit_right = mu0 + jstat.normal.inv(1-alpha/2, 0, 1)*(sigma/Math.sqrt(n))","pinned":true,"hidden":false,"mode":"ojs"},{"id":806,"value":"zcrit_left = mu0 - jstat.normal.inv(1-alpha/2, 0, 1)*(sigma/Math.sqrt(n))","pinned":true,"hidden":false,"mode":"ojs"},{"id":19,"value":"beta = jstat.normal.cdf(zcrit_right, muA, sigma/Math.sqrt(n)) - jstat.normal.cdf(zcrit_left, muA, sigma/Math.sqrt(n));","pinned":true,"hidden":false,"mode":"ojs"},{"id":621,"value":"maxpdf = Math.max(jstat.normal.pdf(mu0, mu0, sigma/Math.sqrt(n)), jstat.normal.pdf(muA, muA, sigma/Math.sqrt(n)))","pinned":false,"hidden":false,"mode":"ojs"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}