31:T6ed,## Estimator and its sampling distribution A natural **estimator** of the population proportion ${tex`p`} is the **sample proportion** ${tex`\hat p =\frac{1}{n}\sum_{i=1} X_i,`}
The **sampling distribution** describes how the estimator ${tex`\hat p`} can vary *from sample to sample*. While we typically only have one sample ${tex`x_1,x_2,\ldots,x_n`} available, but the idea is to consider how ${tex`\hat p`} could have been different *if* we would have obtained another sample. Since ${tex`\hat p =\frac{1}{n}\sum_{i=1} X_i,`}, the sampling distribution of the sample proportion is that of a Binomial variable ${tex`\sum_{i=1} X_i`} (recall that a sum of independent Bernoulli variables follows a [Binomial distribution](binomial-distribution)) scaled by ${tex`\frac{1}{n}`}.

The mean of this distribution is ${tex`\mathbb{E}(\hat p)=\frac{1}{n}\mathbb{E}(\sum_{i=1}^n X_i) = \frac{1}{n}(\sum_{i=1}^n \mathbb{E} (X_i) ) = \frac{1}{n}(\sum_{i=1}^n p ) = \frac{1}{n}np = p`}
The variance in the sampling distribution for ${tex`\hat p`} is ${tex`\mathbb{V}(\hat p)=\mathbb{V}(\frac{1}{n}\sum_{i=1} X_i) = \frac{1}{n^2} \mathbb{V}(\sum_{i=1} X_i) = \frac{1}{n^2} \sum_{i=1} \mathbb{V}(X_i) = \frac{1}{n^2} \sum_{i=1} pq = \frac{1}{n^2} npq = \frac{pq}{n}`}
where ${tex`q=1-p`} is the proportion of non-democrats, and we have used the following results in the derivation: - ${tex`\mathbb{V}(cX)=c^2\mathbb{V}(X)`} - ${tex`\mathbb{V}(\sum_{i=1} X_i) = \sum_{i=1} \mathbb{V}(X_i)`}, when the random variables are independent - ${tex`\mathbb{V}(X)=pq`} for a Bernoulli variable with success probability *p* So the standard deviation of the sample proportion ${tex`\hat p`} is ${tex`\mathbb{S}(\hat p)=\sqrt{\frac{pq}{n}}`}32:[] 2f:{"title":"Normal approximation of the sampling distribution for a proportion","cells":[{"id":620,"value":"# Normal approximation of the sampling distribution for a proportion","pinned":false,"hidden":false,"mode":"md"},{"id":1200,"value":"## Model of the population\n\nConsider a random sample of ${tex`n`} binary [Bernoulli](bernoulli-distribution) variables from a population with success probability ${tex`p`}\n\n${tex.block`X_1,X_2,\\ldots,X_n \\overset{\\mathrm{iid}}{\\sim}\\mathrm{Bernoulli}(p)`}\n\nAn example is to estimate the proportion ${tex`p`} of democratic votes in the US:\n\n${tex.block`\nX_i=\n \\begin{cases}\n 1 \\text{ (success)} & \\text{if person }i \\text{ votes for the democratic party} \\\\\n 0 \\text{ (failure)} & \\text{otherwise}\n \\end{cases}`\n}","pinned":false,"hidden":false,"mode":"md"},{"id":1209,"value":"$31","pinned":false,"hidden":false,"mode":"md"},{"id":1007,"value":"## Normal approximation of the sampling distribution\n\nIt is common to use the above mean and standard deviation to approximate the sampling distribution of ${tex`\\hat p`} by a [normal distribution](normal-gaussian-distribution):\n\n${tex`\\hat p \\sim N\\Big(\\mu = p,\\sigma = \\sqrt{\\frac{pq}{n}}\\Big)`}\n\nIn the interactive graph below you can investigate how well this approximates the sampling distribution for a given *n* and *p*. The normal approximation is known to be accurate only when certain conditions on *n*, *p* and *q* are fulfilled. The graph also warns when any of the conditions are violated. You can also investigate the **law of large numbers** by increasing *n* and observing how the sampling distribution becomes more and more concentrated around *p*.","pinned":false,"hidden":false,"mode":"md"},{"id":930,"value":"viewof userinputs = Inputs.form([\n Inputs.range([1,1000], {transform: Math.log, value: 30, step: 1, label: tex`n`}),\n Inputs.range([0.01, 0.99], {value: 0.5, step: 0.01, label: tex`p`})\n])","pinned":false,"hidden":false,"mode":"ojs"},{"id":965,"value":"condition_sample_size = {\n if (sample_size_condition)\n {\n return md`☑️ Sample size ${tex`n=${userinputs[0]}`} is larger than 30. `\n }\n else\n {\n return md`❌ Sample size ${tex`n=${userinputs[0]}`} is smaller than 30. `\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":985,"value":"condition_successes = {\n if (number_success_condition)\n {\n return md`☑️ Expected number of successes ${tex`np=${(userinputs[0]*userinputs[1]).toPrecision(3)}`} is larger than 10.`\n }\n else\n {\n return md`❌ Expected number of successes ${tex`np=${(userinputs[0]*userinputs[1]).toPrecision(3)}`} is smaller than 10.`\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":990,"value":"condition_failure = {\n if (number_failure_condition)\n {\n return md`☑️ Expected number of failures ${tex`np=${(userinputs[0]*(1-userinputs[1])).toPrecision(3)}`} is larger than 10.`\n }\n else\n {\n return md`❌ Expected number of failures ${tex`nq=${(userinputs[0]*(1-userinputs[1])).toPrecision(3)}`} is smaller than 10.`\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":934,"value":"plt = Plot.plot({\n color: {\n legend: true,\n domain: [\"Normal approximation\",\"Exact scaled binomial distribution\"],\n range: [colors[2], d3.rgb(192, 163, 77, 0.3)]\n },\n x: {\n label: \"sample proportion\",\n axis: true,\n domain: [0,1]\n },\n y: {\n axis: false,\n domain: [0,1.05*maxpdf]\n },\n marks: [\n Plot.areaY(pdfbinom.map(d => ({x: d.x, pdf: d.pdf*(maxpdf/maxbinompdf)})),\n {x: \"x\", y: \"pdf\", curve: \"step\", strokeWidth: 0, stroke: colors[7], fill: colors[1], opacity: 0.3}),\n Plot.ruleY([0]),\n Plot.line(datapdf, {x: \"x\", y: \"pdf\", stroke : colors[2], strokeWidth: 3}),\n ]\n })","pinned":false,"hidden":false,"mode":"ojs"},{"id":566,"value":"
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n\n---\n## 🙈 Appendix 🙈\n\nHär följer bara kod och detaljer som du antagligen inte vill läsa.","pinned":false,"hidden":false,"mode":"md"},{"id":657,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs"},{"id":936,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":1307,"value":"colors = [\"#6C8EBF\", \"#c0a34d\", \"#780000\", \"#007878\", \"#bf9d6c\", \"#3A6B35\", \"#b5c6df\",\"#eadaaa\", \"#bb989a\", \"#98bbb9\", \"#bf8d6c\", \"#CBD18F\"]","pinned":true,"hidden":false,"mode":"ojs"},{"id":938,"value":"datapdf = {\n const n = userinputs[0];\n const p = userinputs[1];\n const x = d3.range(0,1, 0.001);\n const normpdf = x.map(x => ({x: x, pdf: jstat.normal.pdf( x, p, Math.sqrt(p*(1-p)/n)), dist: \"samplingfördelningen för andelen\"}));\n return normpdf\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":947,"value":"maxpdf = jstat.normal.pdf( userinputs[1], userinputs[1], Math.sqrt(userinputs[1]*(1-userinputs[1])/userinputs[0]))","pinned":false,"hidden":false,"mode":"ojs"},{"id":1043,"value":"pdfbinom = d3.range(userinputs[0] + 1).map(d => ({x:d/userinputs[0], pdf: jstat.binomial.pdf(d, userinputs[0], userinputs[1])}))","pinned":false,"hidden":false,"mode":"ojs"},{"id":1110,"value":"maxbinompdf = d3.max(pdfbinom.map(d => d.pdf))","pinned":true,"hidden":false,"mode":"ojs"},{"id":957,"value":"sample_size_condition = userinputs[0] >= 30","pinned":true,"hidden":false,"mode":"ojs"},{"id":960,"value":"number_success_condition = userinputs[0]*userinputs[1] >= 10","pinned":true,"hidden":false,"mode":"ojs"},{"id":962,"value":"number_failure_condition = userinputs[0]*(1-userinputs[1]) >= 10","pinned":true,"hidden":false,"mode":"ojs"},{"id":591,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":true,"hidden":false,"mode":"ojs"}],"files":"$W32","stdlib":"1","titling":"auto"}