29:null 2a:["$","title",null,{"children":"Hypothesis test for a mean in a normal population | Mattias Villani | Observable"}] 2f:T11d2,plt = Plot.plot({ color: { legend: false }, x: { label: "t-statistic", axis: true, domain: [-10,10] }, y: { axis: false, domain: [0,max_y] }, marks: [ Plot.ruleY([0]), Plot.line(pdfdata, {filter: d => d.dist == "student-t" && d.type == "pdf", x: "x", y: "pdf", stroke : "dist", strokeWidth: 2}), Plot.areaY(pdfdata, {filter: d => showpvalue && side == "left" && d.x <= tobs && d.dist == "student-t" && d.type == "pdf", x: "x", y: "pdf", fill: "orange", opacity: 0.5}), Plot.areaY(pdfdata, {filter: d => showpvalue && side == "right" && d.x >= tobs && d.dist == "student-t" && d.type == "pdf", x: "x", y: "pdf", fill: "orange", opacity: 0.5}), Plot.areaY(pdfdata, {filter: d => showpvalue && side == "both" && d.x <= -Math.abs(tobs) && d.dist == "student-t" && d.type == "pdf", x: "x", y: "pdf", fill: "orange", opacity: 0.5}), Plot.areaY(pdfdata, {filter: d => showpvalue && side == "both" && d.x >= Math.abs(tobs) && d.dist == "student-t" && d.type == "pdf", x: "x", y: "pdf", fill: "orange", opacity: 0.5}), Plot.dot([{x: tobs, y: 0}], {x: "x", y: "y", fill: conclusioncolor, symbol: "circle", r: 5}), Plot.arrow([{x1: tobs, y1: max_y/12, x2:tobs, y2: 0.01}], {x1: "x1", y1: "y1", x2: "x2", y2: "y2", stroke: conclusioncolor}), Plot.text([{x1: tobs, y1: bignudge*max_y/12, texter: "t(obs)"}], {x: "x1", y: "y1", text: "texter", fontSize: textfontsize}), Plot.ruleX([{x: tcrit, y: max_y/4}], {filter: d => showcrit && side == "both", x: "x", y: "y", fill: "green"}), Plot.ruleX([{x: -tcrit, y: max_y/4}], {filter: d => showcrit && side == "both", x: "x", y: "y", fill: "green"}), Plot.text([{x1: tcrit, y1: nudge*max_y/4, texter: "t(crit)"}], {filter: d => showcrit && side == "both",x: "x1", y: "y1", text: "texter", fontSize: textfontsize}), Plot.text([{x1: -tcrit, y1: nudge*max_y/4, texter: "-t(crit)"}], {filter: d => showcrit && side == "both",x: "x1", y: "y1", text: "texter", fontSize: textfontsize}), Plot.ruleX([{x: tcrit, y: max_y/4}], {filter: d => showcrit && side == "right", x: "x", y: "y", fill: "green"}), Plot.text([{x1: tcrit, y1: nudge*max_y/4, texter: "t(crit)"}], {filter: d => showcrit && side == "right",x: "x1", y: "y1", text: "texter", fontSize: textfontsize}), Plot.ruleX([{x: -tcrit, y: max_y/4}], {filter: d => showcrit && side == "left", x: "x", y: "y", fill: "green"}), Plot.text([{x1: -tcrit, y1: nudge*max_y/4, texter: "-t(crit)"}], {filter: d => showcrit && side == "left",x: "x1", y: "y1", text: "texter", fontSize: textfontsize}), Plot.arrow([{x1: tcrit*bignudge, y1: max_y/6, x2:10, y2: max_y/6}], {filter: d => showcrit && side == "right", x1: "x1", y1: "y1", x2: "x2", y2: "y2", stroke: "red"}), Plot.arrow([{x1: -tcrit*bignudge, y1: max_y/6, x2:-10, y2: max_y/6}], {filter: d => showcrit && side == "left", x1: "x1", y1: "y1", x2: "x2", y2: "y2", stroke: "red"}), Plot.arrow([{x1: tcrit*bignudge, y1: max_y/6, x2:10, y2: max_y/6}], {filter: d => showcrit && side == "both", x1: "x1", y1: "y1", x2: "x2", y2: "y2", stroke: "red"}), Plot.arrow([{x1: -tcrit*bignudge, y1: max_y/6, x2:-10, y2: max_y/6}], {filter: d => showcrit && side == "both", x1: "x1", y1: "y1", x2: "x2", y2: "y2", stroke: "red"}), Plot.text([{x1: (-10 - tcrit*bignudge)/2, y1: max_y/4, texter: "Reject H₀"}], {filter: d => showcrit && side == "left", x: "x1", y: "y1", text: "texter", fontSize: textfontsize}), Plot.text([{x1: (10 + tcrit*bignudge)/2, y1: max_y/4, texter: "Reject H₀"}], {filter: d => showcrit && side == "right", x: "x1", y: "y1", text: "texter", fontSize: textfontsize}), Plot.text([{x1: (-10 - tcrit*bignudge)/2, y1: max_y/4, texter: "Reject H₀"}], {filter: d => showcrit && side == "both", x: "x1", y: "y1", text: "texter", fontSize: textfontsize}), Plot.text([{x1: (10 + tcrit*bignudge)/2, y1: max_y/4, texter: "Reject H₀"}], {filter: d => showcrit && side == "both", x: "x1", y: "y1", text: "texter", fontSize: textfontsize}) ] })30:[] 2d:{"id":"0e0e67cef3873adb","slug":"hypothesis-test-mean","title":"Hypothesis test for a mean in a normal population","thumbnail":"db1f98d6ab4a6b05a3a4c4e05bf28794201c15989d583de0c83371f4f0c9179b","access_level":"public","create_time":"$D2023-10-22T12:21:17.142Z","update_time":"$D2023-12-02T22:45:42.023Z","view_time":null,"first_public_time":"$D2023-10-22T19:39:29.688Z","has_unpublished_edits":null,"fork":{"id":"4594bdcee9954fb1","slug":"student-t-distribution","title":"Student-t distribution (three-parameter)","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"stars":0,"starred":false,"owner":{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"education_2026"},"version":1119,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null}],"members":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":true,"trashed":false,"lapsed":null,"subscribed":null,"forks":3,"collection":{"id":"636adfa34cc7a030","slug":"sda3","title":"SDA3","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Hypothesis test for a mean in a normal population","cells":[{"id":38,"value":"# Hypothesis test for a mean in a normal population","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":523,"value":"

Model: ${tex`X_1,\\ldots,X_n \\overset{\\mathrm{iid}}{\\sim} N(\\mu,\\sigma)\\text{ with }\\sigma\\text{ unknown}`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":842,"value":"hypotext = \n{\n var hyposign;\n if (alternative == 'equal to'){\n hyposign = \"\\\\neq\"\n }else if (alternative == 'less than'){\n hyposign = \"<\"\n }else{\n hyposign = \">\"\n }\n return tex`\\textbf{Hypotheses }\\\\\n H_0: \\mu = \\mu_0 = ${mu0}\\\\\n H_A: \\mu ${hyposign} \\mu_0 = ${mu0}`\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":800,"value":"tobstext = \n{\n return tex`\\textbf{Test statistic }\\\\\n t_{obs} = \\frac{\\bar x -\\mu_0}{\\frac{s}{\\sqrt{n}}} = \\frac{${xbar.toFixed(3)} - ${mu0.toFixed(3)}}{\\frac{${s.toFixed(3)}}{\\sqrt{${n}}}} = ${tobs.toFixed(3)}`\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":817,"value":"tcrittext = \n{\n return tex`\\textbf{Critical value }\\\\\nt_{crit} = t_{${alphaside}, \\,${n-1}} = ${tcrit.toPrecision(4)}`\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1035,"value":"pvaluetext = {\n var rejecttext;\n var sign;\n if (pvaluestudentt < alpha){\n rejecttext = \" \\\\textbf{we reject} \";\n sign = \"<\";\n }else{\n rejecttext = \"we \\\\textbf{do not reject} \";\n sign = \"\\\\geq\";\n }\n return tex`\\textbf{p-value}\\\\\np\\text{-value}=${pvaluestudentt.toPrecision(4)}`\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":868,"value":"pconclusion = {\n var rejecttext;\n var sign;\n if (pvaluestudentt < alpha){\n rejecttext = \"We\\\\textcolor{red}{ \\\\textbf{reject}}\\ \";\n sign = \"<\";\n }else{\n rejecttext = \"We\\\\textcolor{steelblue}{ \\\\textbf{do not reject}} \";\n sign = \"\\\\geq\";\n }\n return tex`\\text{${rejecttext}} H_0 \\text{ at } ${(alpha*100).toFixed(1)}\\% \\text{ significance level.} `\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":118,"value":"viewof inputs = Inputs.form([\n Inputs.range([2, 1000], {value: 5, step: 1, label: tex`\\text{sample size }n`}), \n Inputs.range([-10, 10], {value: 1.7, step: 0.01, label: tex`\\text{sample mean }\\bar x`}),\n Inputs.range([Number.EPSILON, 10], {value: 0.5, step: 0.1, label: tex`\\text{sample stdev }s`}),\n Inputs.range([0.001, 1], {value: 0.05, step: 0.001, label: tex`\\text{significance level }\\alpha`}),\n Inputs.range([-10, 10], {value: 1, step: 0.1, label: tex`\\text{null value }\\mu_0`}),\n Inputs.select([\"equal to\",\"greater than\", \"less than\"], {label: tex`\\text{alternative}`})\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":450,"value":"viewof toggles = Inputs.form([\n Inputs.toggle({value: true, label: \"show critical regions\"}),\n Inputs.toggle({value: false, label: \"show p-values\"})\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":305,"value":"$2f","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHere is only code and details that you probably don't want to read.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":972,"value":"#### Settings","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":974,"value":"textfontsize = 12","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":671,"value":"showcrit = toggles[0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":668,"value":"showpvalue = toggles[1]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":573,"value":"n = inputs[0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":558,"value":"xbar = inputs[1]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":559,"value":"s = inputs[2]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":562,"value":"alpha = inputs[3]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":583,"value":"alphaside = {\n if (alternative == 'equal to'){\n return alpha/2\n }else{\n return alpha\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":564,"value":"mu0 = inputs[4]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":566,"value":"alternative = inputs[5]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":570,"value":"tobs = (xbar-mu0)/(s/Math.sqrt(n))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":580,"value":"tcrit = jstat.studentt.inv(1-alphaside, n-1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":705,"value":"zcrit = jstat.normal.inv(1-alphaside, 0, 1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":630,"value":"pvaluestudentt = {\n if (alternative == 'equal to'){\n return 2*jstat.studentt.cdf(-Math.abs(tobs), n-1)\n }else if(alternative == 'less than'){\n return jstat.studentt.cdf(tobs, n-1)\n }else{\n return 1-jstat.studentt.cdf(tobs, n-1)\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":17,"value":"pdfdata = {\n const x = d3.range(-10, 10, 0.01);\n var data = x.map(x => ({x: x, pdf: jstat.normal.pdf(x, 0, 1), dist: \"normal\", type: \"pdf\"}));\n const studentpdf = x.map(x => ({x: x, pdf: jstat.studentt.pdf(x, n-1), dist: \"student-t\", type: \"pdf\"}));\n data = data.concat(studentpdf)\n data = data.concat(tcritdata)\n data = data.concat(tobsdata)\n return data\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":682,"value":"max_y = 1.05*jstat.normal.pdf(0, 0, 1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":687,"value":"twosided = alternative == 'equal to'","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":892,"value":"side = {\n if (alternative == 'less than'){\n return \"left\"\n }else if(alternative == 'greater than'){\n return \"right\"\n }else{\n return \"both\"\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":690,"value":"tcritdata = {\n if (twosided){\n return [{x: tcrit, y: max_y/4, dist: \"student-t\", type: \"crit\"}, {x: -tcrit, y: max_y/4, dist: \"student-t\", type: \"crit\"}]\n }else if (alternative == 'less than'){\n return [{x: -Math.abs(tcrit), y: max_y/4, dist: \"student-t\", type: \"crit\"}]\n }else\n {\n return [{x: Math.abs(tcrit), y: max_y/4, dist: \"student-t\", type: \"crit\"}]\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":730,"value":"tobsdata = {\n if (twosided){\n return [{x: tobs, y: 0, dist: \"student-t\", type: \"obs\"}, {x: -tobs, y: 0, dist: \"student-t\", type: \"obs\"}]\n }else if (alternative == 'less than'){\n return [{x: -Math.abs(tobs), y: 0, dist: \"student-t\", type: \"obs\"}]\n }else\n {\n return [{x: Math.abs(tobs), y: 0, dist: \"student-t\", type: \"obs\"}]\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1015,"value":"conclusioncolor = {\n if(pvaluestudentt < alpha){\n return \"red\"\n }else{\n return \"steelblue\"\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":930,"value":"nudge = 1.1","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":984,"value":"bignudge = 1.3","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W30","stdlib":"1","titling":"auto"}} 2e:"$2d:body"