2c:null 2d:["$","title",null,{"children":"Konfidensintervall för en andel | Mattias Villani | Observable"}] 32:T4cc,md` Ett approximativt konfidensinterval för en populationsandel *p* med sannolikhet ${tex`1-\alpha`} baserat på normalapproximation är ${tex`\hat p \pm z_{\alpha/2} \sqrt{\hat p \hat q/n}`} där ${tex`\hat p`} är andelen i stickprovet och ${tex`z_{\alpha/2}`} hämtas från Z-tabell som det värde som har ${tex`\alpha/2`} sannolikhet till **höger** om värdet. Ett konfidensintervall med t ex konfidensnivå *95*% har tolkningen att intervallet kommer innehålla det sanna populationsandelen *p* **i *95*% av alla möjliga stickprov** från populationen. För ett givet stickprov kommer intervallet antingen att innehålla populationsandelen *p* eller inte göra det, vilket av dessa fall som är gäller vet vi normalt inte, eftersom vi inte känner till *p*. I den interaktiva grafen undersöker vi detta i en kontrollerad situation där vi bestämmer populationsandelen *p* (visas som en röd linje) och sen simulerar en massa datamaterial, alla med *n* st observationer. För varje simulerat datamaterial ritar vi upp konfidensintervallet för *p* som en vertikal linje. De intervall som är orangea är för datamaterial där intervallet missade *p*. De blåa intervallen är intervall som innehåller *p*. `33:[] 30:{"id":"fa3ac3f819ff567e","slug":"konfidensintervall-for-en-andel","title":"Konfidensintervall för en andel","thumbnail":"b4fdad539444a6842a2a82733d8fdf97dfe3fec526a0d67575ca5779d014c080","access_level":"public","create_time":"$D2023-02-20T11:22:07.044Z","update_time":"$D2023-10-30T13:32:40.670Z","view_time":null,"first_public_time":"$D2023-02-20T11:22:07.044Z","has_unpublished_edits":null,"fork":false,"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":266,"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":1,"collection":{"id":"209e13808f44ed11","slug":"sda1","title":"SDA1","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"titling":"auto","title":"Konfidensintervall för en andel","cells":[{"id":0,"value":"# Konfidensintervall för en andel","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":144,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":5,"value":"viewof settings = Inputs.form([\n Inputs.range([0.1, 0.9], {value: 0.5, label: \"Populationsandel, p\", step: 0.01}),\n Inputs.range([0, 100], {value: 90, label: \"Konfidensnivå\", step: 1}),\n Inputs.range([0, 200], {value: 50, label: \"Stickprovsstorlek, n\", step: 1})\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":204,"value":"viewof nrep = Scrubber(d3.ticks(1, 1000, 1000), {\n autoplay: false,\n loop: false,\n initial: 1,\n format: x => `antal stickprov = ${x.toFixed(0)}`\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":117,"value":"textinfo = md`Av totalt ${nrep} stickprov innehöll ${d3.sum(data.map(d => d.rep <= nrep && (p >= d.lower && p <= d.upper)))} st (**${(d3.sum(data.map(d => d.rep <= nrep && (p >= d.lower && p <= d.upper)))*100/nrep).toFixed(3)}%**) av de ${confidence}%-iga konfidensintervallen den sanna populationsandelen ${tex`p= `} ${p}.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":67,"value":"plt = Plot.plot({\n style: {fontSize: \"12px\"},\n width: 960,\n y: {\n label: \"stickprovsandel\",\n domain: [0,1]\n },\n x: {\n label: \"stickprovsnummer\",\n domain: [0, nrep]\n },\n marks: [\n Plot.ruleX([0]), \n Plot.ruleX(data, {\n filter: d => (d.rep <= nrep && p >= d.lower && p <= d.upper),\n x: \"rep\",\n y1: \"lower\",\n y2: \"upper\", \n stroke: \"steelblue\",\n strokeWidth: 1.5\n }),\n Plot.dot(data, {filter: d => (d.rep <= nrep && p >= d.lower && p <= d.upper), x: \"rep\", y: \"phat\", fill: \"steelblue\", r: 3}),\n\n Plot.ruleX(data, {\n filter: d => d.rep <= nrep && (p < d.lower || p > d.upper),\n x: \"rep\",\n y1: \"lower\",\n y2: \"upper\", \n stroke: \"orange\",\n strokeWidth: 1.5\n }),\n Plot.dot(data, {filter: d => d.rep <= nrep && (p < d.lower || p > d.upper), x: \"rep\", y: \"phat\", fill: \"orange\", r: 3}),\n Plot.ruleY([p], {stroke: \"#D22B2B\", strokeWidth: 1.5})\n\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":12,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHär följer bara kod och detaljer som du antagligen inte vill läsa.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":246,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":15,"value":"jstat = require('jstat')","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":206,"value":"import {Scrubber} from \"@mbostock/scrubber\"","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":248,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":39,"value":"function simulate_proportions(p, nobs, nrep){\n const zvalue = jstat.normal.inv(1-(1-(confidence/100))/2,0,1);\n var data = [];\n for (let j = 1; j <= nrep; j++){\n let phat = d3.mean(d3.range(nobs).map(d => d3.randomUniform(0,1)()<=p))\n let lower = phat - zvalue*Math.sqrt(phat*(1-phat)/nobs)\n let upper = phat + zvalue*Math.sqrt(phat*(1-phat)/nobs)\n data.push({rep: j, phat: phat, lower: lower, upper: upper})\n }\n return data\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":250,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":36,"value":"p = settings[0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":28,"value":"confidence = settings[1]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":26,"value":"nobs = settings[2]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":65,"value":"data = {\n nobs;\n p;\n const data = simulate_proportions(p, nobs, 1000)\n return data\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W33","stdlib":"1"}} 31:"$30:body"