2a:T703,md` An approximate (Wald) confidence interval with coverage probability ${tex`1-\alpha`} for a population proportion *p* based on normal approximation is ${tex`\hat p \pm z_{\alpha/2} \sqrt{\hat p \hat q/n}`} where ${tex`\hat p`} is the proportion in the sample and ${tex`z_{\alpha/2}`} is obtained from the Z-table as the value that has ${tex`\alpha/2`} probability to the right of the value. A confidence interval with, for example, a confidence level of **95%** has the interpretation that the interval will contain the true population proportion *p* **in *95%* of all possible samples** from the population. For a given sample, the interval will either contain the population proportion *p* or not, which of these cases applies we do not know, because we do not know *p*. In the interactive graph, we investigate this in a controlled situation where we determine the population proportion *p* ourselves (shown as a red line) and then simulate a lot of datasets, all with *n* observations. For each simulated dataset, we draw the confidence interval for *p* as a vertical line with a dot for the sample proportion. The intervals that are orange are for data where the interval missed *p*. The blue intervals are intervals that contain *p*. Drag the slider to the right of the Pause button to speed up the animation. The Wald confidence interval is simple to understand, but has some drawbacks, particularly when the true proportion is close to 0 or 1 (see this [article](https://doi.org/10.1080/00031305.2023.2183257)). An alternative interval, the Wilson interval, has better properties; for example, the [Wilson interval has a better coverage property](https://observablehq.com/@mattiasvillani/ci-proportion-wald-vs-wilson). Try it by clicking the radio button labelled *Wilson*. `2b:[] 28:{"title":"Confidence interval for a proportion","cells":[{"id":0,"value":"# Confidence interval for a proportion\n","pinned":false,"hidden":false,"mode":"md"},{"id":144,"value":"$2a","pinned":false,"hidden":false,"mode":"ojs"},{"id":286,"value":"viewof intervaltype = Inputs.radio([\"Wald\", \"Wilson\"], {value: \"Wald\", label: \"interval type:\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":5,"value":"viewof settings = Inputs.form([\n Inputs.range([0.01, 0.99], {value: 0.5, label: \"Population proportion, p\", step: 0.01}),\n Inputs.range([0, 100], {value: 90, label: \"Confidence level\", step: 1}),\n Inputs.range([0, 200], {value: 50, label: \"Sample size, n\", step: 1})\n ])","pinned":false,"hidden":false,"mode":"ojs"},{"id":204,"value":"viewof nrep = Scrubber(d3.ticks(1, 1000, 1000), {\n autoplay: false,\n loop: false,\n initial: 1,\n delay: 500,\n format: x => `number of datasets = ${x.toFixed(0)}`\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":117,"value":"textinfo = md`\nOut of a total of ${nrep} samples, ${d3.sum(data.map(d => d.rep <= nrep && (p >= d.lower && p <= d.upper)))} (**${(d3.sum(data.map(d => d.rep <= nrep && (p >= d.lower && p <= d.upper)))*100/nrep).toFixed(3)}%**) of the ${confidence}% confidence intervals contained the true population proportion ${tex`p= `} ${p}.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":67,"value":"plt = Plot.plot({\n style: {fontSize: \"12px\"},\n width: 960,\n y: {\n label: \"sample proportion\",\n domain: [-0.1,1]\n },\n x: {\n label: \"sample number\",\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\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":12,"value":"
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n\n---\n## 🙈 Appendix 🙈\n\nHere is just code and details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md"},{"id":246,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":15,"value":"jstat = require('jstat')","pinned":true,"hidden":false,"mode":"ojs"},{"id":206,"value":"import {Scrubber} from \"@mbostock/scrubber\"","pinned":true,"hidden":false,"mode":"ojs"},{"id":248,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md"},{"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 if (intervaltype == \"Wald\"){\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 }\n if (intervaltype == \"Wilson\"){\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 midpoint = phat + (zvalue**2/(2*nobs))*((1-2*phat)/(1+zvalue**2/nobs))\n let margin = zvalue*Math.sqrt( phat*(1 - phat)/nobs + (zvalue**2/(4*nobs**2)) )/(1+zvalue**2/nobs)\n let lower = midpoint - margin\n let upper = midpoint + margin\n data.push({rep: j, phat: midpoint, lower: lower, upper: upper})\n }\n return data\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":250,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":291,"value":"intervaltype","pinned":false,"hidden":false,"mode":"ojs"},{"id":36,"value":"p = settings[0]","pinned":false,"hidden":false,"mode":"ojs"},{"id":28,"value":"confidence = settings[1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":26,"value":"nobs = settings[2]","pinned":false,"hidden":false,"mode":"ojs"},{"id":65,"value":"data = {\n nobs;\n p;\n const data = simulate_proportions(p, nobs, 1000)\n return data\n}","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W2b","stdlib":"1","titling":"auto"}