2b:null 2c:["$","title",null,{"children":"Hypothesis Testing | William Pang | Observable"}] 31:T1f19,{ let crit_distribution = null; let alternative_distribution_true = null; let crit_distribution_2 = null; let alternative_distribution_non_overlap = null; const svg = d3.select(DOM.svg(width, height)); const margin = {top: 30, right: 20, bottom: 20, left: 40}; const x_values = d3.scaleLinear() .domain([-2, 2]) .range([margin.left, width - margin.right]); const sample_null_distribution = _.range(-2, 2, increments).map(x => ({x: x, y: normalDistributionPDF(x, x_mean, (stdev/Math.sqrt(n)))})); const sample_alternate_distribution = _.range(-2, 2, increments).map(x => ({x: x, y: normalDistributionPDF(x, x_test, (stdev/Math.sqrt(n)))})); const y_values = d3.scaleLinear() .domain([0, d3.max(sample_alternate_distribution.map(d => d.y))]) .range([height - margin.bottom, margin.top]); const z_left_null = norminv(alpha, x_mean, (stdev/Math.sqrt(n))) const z_right_null = norminv(1-alpha, x_mean, (stdev/Math.sqrt(n))) const z_left_alternative = norminv(alpha, x_test, (stdev/Math.sqrt(n))) const z_right_alternative = norminv(1-alpha, x_test, (stdev/Math.sqrt(n))) const z_left_null_half = norminv(alpha/2, x_mean, (stdev/Math.sqrt(n))) const z_right_null_half = norminv(1-alpha/2, x_mean, (stdev/Math.sqrt(n))) const z_left_alternative_half = norminv(alpha/2, x_test, (stdev/Math.sqrt(n))) const z_right_alternative_half = norminv(1-alpha/2, x_test, (stdev/Math.sqrt(n))) if (hypothesis == "One Sided, Greater than"){ crit_distribution = _.range(z_right_null, 2, increments).map(x => ({x: x, y: normalDistributionPDF(x, x_mean, (stdev/Math.sqrt(n)))})); crit_distribution.unshift({x: z_right_null, y: 0 }) if (x_mean < x_test){ var z_diff = z_left_alternative - z_right_null if (z_diff < 0){ alternative_distribution_true = _.range(z_right_null, 2, increments).map(x => ({x: x, y0: normalDistributionPDF(x, x_mean, (stdev/Math.sqrt(n))), y1: normalDistributionPDF(x, x_test, (stdev/Math.sqrt(n)))})); } else{ // z_diff > 0 alternative_distribution_true = _.range(z_right_null + z_diff/2, 2, increments).map(x => ({x: x, y0: normalDistributionPDF(x, x_mean, (stdev/Math.sqrt(n))), y1: normalDistributionPDF(x, x_test, (stdev/Math.sqrt(n)))})); } } else { // x_mean > x_test alternative_distribution_true = _.range(z_right_null, 2, increments).map(x => ({x: x, y: normalDistributionPDF(x, x_mean, (stdev/Math.sqrt(n)))})); alternative_distribution_true.unshift({x: z_left_null, y: 0 }) } } else if (hypothesis == "One Sided, Less than"){ crit_distribution = _.range(-2, z_left_null, increments).map(x => ({x: x, y: normalDistributionPDF(x, x_mean, (stdev/Math.sqrt(n)))})); crit_distribution.unshift({x: z_left_null, y: 0 }) if (x_mean > x_test){ var z_diff = z_left_null - z_right_alternative if (z_diff < 0){ alternative_distribution_true = _.range(-2, z_left_null, increments).map(x => ({x: x, y0: normalDistributionPDF(x, x_mean, (stdev/Math.sqrt(n))), y1: normalDistributionPDF(x, x_test, (stdev/Math.sqrt(n)))})); } else { // (z_diff > 0) alternative_distribution_true = _.range(-2, z_right_alternative + z_diff/2, increments).map(x => ({x: x, y0: normalDistributionPDF(x, x_mean, (stdev/Math.sqrt(n))), y1: normalDistributionPDF(x, x_test, (stdev/Math.sqrt(n)))})); } } else { //(x_mean < x_test) alternative_distribution_true = _.range(-2, z_left_null, increments).map(x => ({x: x, y: normalDistributionPDF(x, x_mean, (stdev/Math.sqrt(n)))})); alternative_distribution_true.unshift({x: z_left_null, y: 0 }) } } else { // Hypothesis == "Two Sided" crit_distribution = _.range(-2, z_left_null_half, increments).map(x => ({x: x, y: normalDistributionPDF(x, x_mean, (stdev/Math.sqrt(n)))})); crit_distribution.unshift({x: z_left_null_half, y: 0 }) crit_distribution_2 = _.range(z_right_null_half, 2, increments).map(x => ({x: x, y: normalDistributionPDF(x, x_mean, (stdev/Math.sqrt(n)))})); crit_distribution_2.unshift({x: z_right_null_half, y: 0 }) if (x_mean < x_test){ // Analogous to "One Sided, Greater than" var z_diff = z_left_alternative_half - z_right_null_half if (z_diff < 0){ alternative_distribution_true = _.range(z_right_null_half, 2, increments).map(x => ({x: x, y0: normalDistributionPDF(x, x_mean, (stdev/Math.sqrt(n))), y1: normalDistributionPDF(x, x_test, (stdev/Math.sqrt(n)))})); } else{ // z_diff > 0 alternative_distribution_true = _.range(z_right_null_half + z_diff/2, 2, increments).map(x => ({x: x, y0: normalDistributionPDF(x, x_mean, (stdev/Math.sqrt(n))), y1: normalDistributionPDF(x, x_test, (stdev/Math.sqrt(n)))})); } } else { // x_mean > x_test // Analogous to "One Sided, Less than" var z_diff = z_left_null_half - z_right_alternative_half if (z_diff < 0){ alternative_distribution_true = _.range(-2, z_left_null_half, increments).map(x => ({x: x, y0: normalDistributionPDF(x, x_mean, (stdev/Math.sqrt(n))), y1: normalDistributionPDF(x, x_test, (stdev/Math.sqrt(n)))})); } else { // (z_diff > 0) alternative_distribution_true = _.range(-2, z_right_alternative_half + z_diff/2, increments).map(x => ({x: x, y0: normalDistributionPDF(x, x_mean, (stdev/Math.sqrt(n))), y1: normalDistributionPDF(x, x_test, (stdev/Math.sqrt(n)))})); } } } const crit_region = svg.append("path") .datum(crit_distribution) .attr("fill", "steelblue") .attr("stroke", "steelblue") .attr("stroke-width", 1.5) .attr("stroke_linejoin", "round") .attr("d", d3.line() .curve(d3.curveBasis) .x((d) => x_values(d.x)) .y((d) => y_values(d.y))) if (crit_distribution_2 != null){ const crit_region_2 = svg.append("path") .datum(crit_distribution_2) .attr("fill", "steelblue") .attr("stroke", "steelblue") .attr("stroke-width", 1.5) .attr("stroke_linejoin", "round") .attr("d", d3.line() .curve(d3.curveBasis) .x((d) => x_values(d.x)) .y((d) => y_values(d.y))) } const null_curve = svg.append("path") .datum(sample_null_distribution) .attr("fill", "none") .attr("stroke", "steelblue") .attr("stroke-width", 1.5) .attr("stroke_linejoin", "round") .attr("d", d3.line() .curve(d3.curveBasis) .x((d) => x_values(d.x)) .y((d) => y_values(d.y))) const alternative_curve = svg.append("path") .datum(sample_alternate_distribution) .attr("fill", "none") .attr("stroke", "#FFAC1C") .attr("opacity", 0.5) .attr("stroke-width", 1.5) .attr("stroke_linejoin", "round") .attr("d", d3.line() .curve(d3.curveBasis) .x((d) => x_values(d.x)) .y((d) => y_values(d.y))) const alternative_true_curve = svg.append("path") .datum(alternative_distribution_true) .attr("fill", "#FFAC1C") .attr("d", d3.area() .x((d) => x_values(d.x)) .y0((d) => y_values(d.y0)) .y1((d) => y_values(d.y1))) svg.append("g") .attr("class", "axis axis--x") .attr("transform", `translate(${0},${height - margin.bottom})`) .call(d3.axisBottom(x_values)); svg.append('line') .attr('x1', x_values(x_test)) .attr('x2', x_values(x_test)) .attr('y1', margin.top - 10) .attr('y2', height - margin.bottom) .attr('stroke', reject ? "green": "red") svg.append('text') .attr('x', x_values(x_test)) .attr('y', margin.top - 15) .attr('text-anchor', 'middle') .attr('fill', reject ? "green" : 'red') .text("Sample mean") return svg.node() }32:T520,md` In hypothesis testing, we're often interested in tackling the question: is a particular belief or viewpoint true?

More concretely, consider repeatedly sampling the palm length of ${n} random strangers in New York, finding that their average handsbreadth is ${tex`\bar X=${x_test}`}m. Naturally, we might be interested in asking: Is the average handsbreth of New Yorkers ${(hypothesis == "Two Sided") ? "no different than" : "One Sided, Greater than" ? "greater than": "less than"} the average American handsbreth of ${tex`\mu=${x_mean}`}m? Using the notation of hypothesis testing, we're testing the ${(hypothesis == "Two Sided") ? "two sided" : "one-sided"} hypothesis of: ${hypothesis == "Two Sided" ? tex.block`H_0: \mu = ${x_mean} \text{ versus } H_a: \mu \neq ${x_mean}` : hypothesis == "One Sided, Greater than" ? tex.block`H_0: \mu = ${x_mean} \text{ versus } H_a: \mu > ${x_mean}` : tex.block`H_0: \mu = ${x_mean} \text{ versus } H_a: \mu < ${x_mean}`} Note when the ${textcolor('orange curve', 'orange')} becomes more shaded, and when it becomes un-shaded. Our conclusion is that at a significance level of ${tex`\alpha = ${alpha}`}, we ${reject ? `reject the null hypothesis`: `fail to reject the null hypothesis`}. ` 33:[] 2f:{"id":"20ea3799eb3a6604","slug":"hypothesis-testing","title":"Hypothesis Testing","thumbnail":"bffa6c88c55ca7951249de63c68871e519d8a5cba731f67babe599c89bb834a1","access_level":"public","create_time":"$D2023-12-06T19:31:46.822Z","update_time":"$D2023-12-28T16:54:13.632Z","view_time":null,"first_public_time":"$D2023-12-06T19:31:46.822Z","fork":false,"stars":1,"starred":false,"owner":{"type":"team","id":"88df4b53b2fce8d2","login":"will-pang","name":"William Pang","avatar_url":"https://avatars-next.observableusercontent.com/4e5603477ef111cf1fc74cb0aab0dd58ddab37219e1b2a4e5e9c3d082c5532a0","tier":"starter_2024"},"version":776,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"30424cd7adf510bb","login":"williampangbest1","name":"William Pang","avatar_url":"https://avatars-next.observableusercontent.com/4e5603477ef111cf1fc74cb0aab0dd58ddab37219e1b2a4e5e9c3d082c5532a0","tier":null}],"members":[{"type":"individual","id":"30424cd7adf510bb","login":"williampangbest1","name":"William Pang","avatar_url":"https://avatars-next.observableusercontent.com/4e5603477ef111cf1fc74cb0aab0dd58ddab37219e1b2a4e5e9c3d082c5532a0","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":0,"collection":{"id":"60a184aedd1dcb3b","slug":"statistics","title":"Statistics","owner":{"id":"88df4b53b2fce8d2","login":"will-pang","name":"William Pang"}},"type":"notebook","body":{"title":"Hypothesis Testing","cells":[{"id":0,"value":"# Hypothesis Testing","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":18,"value":"viewof x_mean = slider({\n min: -2,\n max: 2,\n value: 0.33,\n title: \"Mean of Population\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":33,"value":"viewof stdev= slider({\n min: 0.02,\n max: 2,\n value: 1,\n title: \"Standard Deviation of Population\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":49,"value":"viewof x_test = slider({\n min: -2,\n max: 2,\n value: 0,\n title: \"Mean of Sample\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":87,"value":"viewof hypothesis = select({ options: [\"Two Sided\", \"One Sided, Greater than\", \"One Sided, Less than\"], title: \"Hypothesis\" })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":7,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":451,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":65,"value":"## Notes\n[1] The idea behind this notebook draws heavy inspiration from Herb Sussman's [work](https://observablehq.com/@herbps10/sampling-distribution-of-the-mean-from-a-normal-distributi?collection=@herbps10/probability-distributions), which has been a great source of on-going inspiration. Although I built much of my code based on his work, I've made heavy edits to properly delineate between the null-hypothesis distribution and the alterantive-hypothesis distribution, which I believe makes it easier to understand the simple hypothesis test.
\n
[2] My notebook also broadly draws upon visualizations created by [Professor Mattias Villani](https://observablehq.com/@mattiasvillani/hypothesis-test-for-a-mean-type-i-and-ii-errors), which also provides an excellent visualization of Type I and Type II errors.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":8,"value":"## Appendix","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":11,"value":"height = 200","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":40,"value":"n = 50","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":756,"value":"increments = 0.005","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":718,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":224,"value":"alpha = 0.05","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":573,"value":"z_alpha = norminv(((hypothesis == \"Two Sided\") ? (1-alpha/2) : (hypothesis == \"One Sided, Greater than\") ? (1-alpha): (alpha)), 0, 1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":564,"value":"z_statistic = ((hypothesis == \"Two Sided\") ? Math.abs((x_test - x_mean)/(stdev/Math.sqrt(n))):(x_test - x_mean)/(stdev/Math.sqrt(n)))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":695,"value":"reject = {\n if(hypothesis == \"Two Sided\"){\n if (z_statistic > z_alpha)\n return true\n else {\n return false\n }\n } else if (hypothesis == \"One Sided, Greater than\"){\n if (z_statistic > z_alpha)\n return true\n else {\n return false\n }\n }\n else\n if (z_statistic < z_alpha)\n return true\n else {\n return false\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":36,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":38,"value":"// Written by ChatGPT\nfunction normalDistributionPDF(x, mean, stdDev) {\n const coefficient = 1 / (stdDev * Math.sqrt(2 * Math.PI));\n const exponent = -((x - mean) ** 2) / (2 * (stdDev ** 2));\n return coefficient * Math.exp(exponent);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":20,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":24,"value":"import {slider, select} from \"@jashkenas/inputs\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":209,"value":"import {norminv} from \"@williampangbest1/sampling-distribution\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":522,"value":"import {textcolor} from \"@observablehq/text-color-annotations-in-markdown\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W33","stdlib":"1","titling":"auto"}} 30:"$2f:body"