33:T672,md` Consider the population model ${tex`X_1,\ldots,X_n \overset{iid}{\sim}N(\mu,\sigma)`} with both the population mean ${tex`\mu`} and standard deviation ${tex`\sigma`} unknown. A confidence interval with coverage probability ${tex`1-\alpha`} for the mean μ is ${tex`\bar x \pm t_{\alpha/2,n-1} \frac{s}{\sqrt{n}}`} where ${tex`\bar x`} is the sample mean, ${tex`\bar s = \sum_{i=1}^n (x_i- \bar x)^2/(n-1)`} the sample standard deviation and ${tex`t_{\alpha/2,n-1}`} is obtained from the student-t 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 mean μ **in *95%* of all possible samples** from the population. For a given sample, the interval will either contain the population mean μ or not. In practice, we will never know if the interval covers μ or not, since we don't know μ. In the interactive graph, we investigate this in a controlled situation where we determine the population mean μ 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 μ as a vertical line with a dot for the sample mean. The intervals that are orange are for data where the interval missed μ. The blue intervals are intervals that contain μ. Drag the slider to the right of the Pause button to speed up the animation. To clearly see the effect of changing sample size and confidence level of the intervals, click *Lock vertical axis* to keep the same scale. `34:T408,plt = Plot.plot({ style: {fontSize: "12px"}, width: 960, y: { label: "mean", domain: lockvertical ? [mu- 3, mu + 3] : [mu - 5*sigma/Math.sqrt(nobs),mu + 5*sigma/Math.sqrt(nobs)] }, x: { label: "sample number", domain: [0, nrep] }, marks: [ Plot.ruleX([0]), Plot.ruleX(data, { filter: d => (d.rep <= nrep && mu >= d.lower && mu <= d.upper), x: "rep", y1: "lower", y2: "upper", stroke: "steelblue", strokeWidth: 1.5 }), Plot.dot(data, {filter: d => (d.rep <= nrep && mu >= d.lower && mu <= d.upper), x: "rep", y: "xbar", fill: "steelblue", r: 3}), Plot.ruleX(data, { filter: d => d.rep <= nrep && (mu < d.lower || mu > d.upper), x: "rep", y1: "lower", y2: "upper", stroke: "orange", strokeWidth: 1.5 }), Plot.dot(data, {filter: d => d.rep <= nrep && (mu < d.lower || mu > d.upper), x: "rep", y: "xbar", fill: "orange", r: 3}), Plot.ruleY([mu], {stroke: "#D22B2B", strokeWidth: 1.5}) ] })35:[] 31:{"title":"Confidence interval for a mean","cells":[{"id":0,"value":"# Confidence interval for a mean\n","pinned":false,"hidden":false,"mode":"md"},{"id":144,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":5,"value":"viewof settings = Inputs.form([\n Inputs.range([-5, 5], {value: 1, label: tex`\\text{population mean, }\\mu`, step: 0.01}),\n Inputs.range([0.01, 5], {value: 1, label: tex`\\text{pop stdev, }\\sigma`, step: 0.01}),\n Inputs.range([0, 99], {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, ${d3.sum(data.map(d => d.rep <= nrep && (mu >= d.lower && mu <= d.upper)))} (**${(d3.sum(data.map(d => d.rep <= nrep && (mu >= d.lower && mu <= d.upper)))*100/nrep).toFixed(3)}%**) of the ${confidence}% confidence intervals contained the true population mean ${tex`\\mu= `} ${mu}.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":371,"value":"viewof lockvertical = Inputs.toggle({label: \"Lock vertical axis\", value: false})","pinned":false,"hidden":false,"mode":"ojs"},{"id":67,"value":"$34","pinned":false,"hidden":false,"mode":"ojs"},{"id":12,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHere is just code and details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md"},{"id":246,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":15,"value":"jstat = require('jstat')","pinned":true,"hidden":false,"mode":"ojs"},{"id":206,"value":"import {Scrubber} from \"@mbostock/scrubber\"","pinned":true,"hidden":false,"mode":"ojs"},{"id":248,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md"},{"id":39,"value":"function simulate_means(mu, sigma, nobs, nrep){\n \n const tvalue = jstat.studentt.inv(1-(1-(confidence/100))/2, nobs-1);\n var data = [];\n for (let j = 1; j <= nrep; j++){\n let sample = d3.range(nobs).map(d => d3.randomNormal(mu,sigma)())\n let xbar = d3.mean(sample)\n let s = jstat.stdev(sample, true) \n let lower = xbar - tvalue*s/Math.sqrt(nobs)\n let upper = xbar + tvalue*s/Math.sqrt(nobs)\n data.push({rep: j, xbar: xbar, lower: lower, upper: upper})\n }\n return data\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":250,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":36,"value":"mu = settings[0]","pinned":false,"hidden":false,"mode":"ojs"},{"id":298,"value":"sigma = settings[1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":28,"value":"confidence = settings[2]","pinned":false,"hidden":false,"mode":"ojs"},{"id":26,"value":"nobs = settings[3]","pinned":false,"hidden":false,"mode":"ojs"},{"id":65,"value":"data = {\n nobs;\n mu;\n sigma;\n const d = new Date();\n let time = d.getTime();\n const data = simulate_means(mu, sigma, nobs, 1000)\n return data\n}","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W35","stdlib":"1","titling":"auto"}