31:T456,textresults = { if (tobs < tcrit_left){ const textresults = tex` \begin{aligned} &\text{teststatistika } t_{obs} = ${tobs.toFixed(3)} \\[0.2em] &\text{kritiskt värde } t_{crit} = ${tcrit_left.toFixed(3)} \\[0.2em] &\text{signifikansnivå } \alpha = P(T \leq t_{crit} \mid H_0 \text{ sann} ) =${(alpha).toPrecision(2)}\\[0.2em] &\text{p-värde: } = P( T \leq t_{obs} \mid H_0 \text{ sann}) =${(pvalue).toPrecision(3)} \\[0.2em] &H_0\text{ förkastas på signifikansnivå } ${(100*inputs[1]).toFixed(0)}\%. \end{aligned} ` return textresults }else{ const textresults = tex` \begin{aligned} &\text{teststatistika } t_{obs} = ${tobs.toFixed(3)} \\[0.2em] &\text{kritiskt värde } t_{crit} = ${tcrit_left.toFixed(3)} \\[0.2em] &\text{signifikansnivå } \alpha = P(T \leq t_{crit} \mid H_0 \text{ sann} )=${(alpha).toPrecision(2)} \\[0.2em] &\text{p-värde: } = P( T \leq t_{obs} \mid H_0 \text{ sann}) =${(pvalue).toPrecision(3)} \\[0.2em] &H_0\text{ kan inte förkastas på signifikansnivå } ${(100*inputs[1]).toFixed(0)}\%. \end{aligned} ` return textresults } }32:T42c,plt_pdf = Plot.plot({ caption:html` student-${tex`t`} fördelning med ${n-1} frihetsgrader under nollhypotesen och förkastelseregion (area).
observerat värde på statistikan ${tex`t_{obs}`} (linje) och ${tex`p`}-värde (area) `, color: { legend: false }, x: { label: "teststatistiska, T", axis: true }, y: { axis: false, domain: [0,1.02*maxpdf] }, marks: [ Plot.ruleY([0]), Plot.ruleX([tobs], {stroke: "orange", strokeWidth: 2}), Plot.line(pdfdata, {x: "x", y: "pdf", stroke : "type", strokeWidth: 2}), //Plot.areaY(pdfdata, {filter: d => d.x >= tcrit_right && d.type == "H0", x: "x", y: "pdf", fill: "steelblue", opacity: 0.2}), Plot.areaY(pdfdata, {filter: d => d.x <= tobs && d.type == "H0", x: "x", y: "pdf", fill: "orange", opacity: 0.2}), Plot.areaY(pdfdata, {filter: d => d.x <= tcrit_left && d.type == "H0", x: "x", y: "pdf", fill: "steelblue", opacity: 0.3}) ] })33:[] 2f:{"title":"Hypotestest för ett väntevärde","cells":[{"id":38,"value":"# Hypotestest för ett väntevärde","pinned":false,"hidden":false,"mode":"md"},{"id":926,"value":"### Data","pinned":false,"hidden":false,"mode":"md"},{"id":841,"value":"Ett stickprov med ${tex`n=${n}`} observationer:","pinned":false,"hidden":false,"mode":"md"},{"id":838,"value":"x = [15.77, 20.5, 8.26, 14.37, 21.09]","pinned":false,"hidden":false,"mode":"ojs"},{"id":496,"value":"modelltext = md`### Modell\nLåt ${tex`X_1,X_2,\\ldots,X_${n} \\overset{iid}{\\sim}N(\\mu,\\sigma)`} med skattningar av ${tex`\\mu`} och ${tex`\\sigma`} givna av\n\n${tex`\\bar x=`} ${xbar}\n\noch \n\n${tex`s = \\sqrt{\\frac{\\sum_{i=1}^n (x_i-\\bar x)^2}{n-1}}=`} ${s.toFixed(3)}\n\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":923,"value":"hypotestext = md`\n### Hypoteser\n\n${tex`\nH_0:\\mu = \\mu_0 \\\\\nH_A:\\mu < \\mu_0\n`}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":780,"value":"med teststatistikan\n\n${tex`T = \\frac{\\bar X -\\mu_0}{s/\\sqrt n}\\sim t_{n-1}`}\n\nunder ${tex`H_0`}.\n\n","pinned":false,"hidden":false,"mode":"md"},{"id":817,"value":"I figuren nedan ritar vi upp samplingfördelningen för ${tex`T`} under ${tex`H_0`}:","pinned":false,"hidden":false,"mode":"md"},{"id":118,"value":"viewof inputs = Inputs.form([\n Inputs.range([5, 50], {value: 20, step: 0.1, label: tex`\\mu_0`}),\n Inputs.range([0.0, 0.2], {value: 0.05, step: 0.01, label: tex`\\alpha`}),\n ])","pinned":false,"hidden":false,"mode":"ojs"},{"id":529,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":305,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHär följer bara kod och detaljer som du antagligen inte vill läsa.","pinned":false,"hidden":false,"mode":"md"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":849,"value":"n = x.length","pinned":true,"hidden":false,"mode":"ojs"},{"id":891,"value":"xbar = d3.mean(x)","pinned":true,"hidden":false,"mode":"ojs"},{"id":855,"value":"s = jstat.stdev(x, true)","pinned":true,"hidden":false,"mode":"ojs"},{"id":889,"value":"tobs = (xbar - mu0)/(s/Math.sqrt(n))","pinned":true,"hidden":false,"mode":"ojs"},{"id":699,"value":"mu0 = inputs[0]","pinned":true,"hidden":false,"mode":"ojs"},{"id":736,"value":"alpha = inputs[1]","pinned":true,"hidden":false,"mode":"ojs"},{"id":17,"value":"pdfdata = {\n const x = d3.range(-5, 5, 0.01);\n const tpdfH0 = x.map(x => ({x: x, pdf: jstat.studentt.pdf(x, n-1), type: \"H0\"}));\n return tpdfH0\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":806,"value":"tcrit_left = -jstat.studentt.inv(1-alpha, n-1)","pinned":true,"hidden":false,"mode":"ojs"},{"id":887,"value":"pvalue = jstat.studentt.cdf(tobs, n-1)","pinned":true,"hidden":false,"mode":"ojs"},{"id":621,"value":"maxpdf = jstat.studentt.pdf(0,n-1)","pinned":false,"hidden":false,"mode":"ojs"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}