Hypothesis Testing for Population Mean

hypotext = 
{
  var hyposign;
  if (alternative == 'equal to'){
    hyposign = "\\neq"
  }else if (alternative == 'less than'){
    hyposign = "<"
  }else{
    hyposign = ">"
  }
   return tex`\textbf{Hypotheses }\\
              H_0: \mu = \mu_0 = ${mu0}\\
              H_A: \mu ${hyposign} \mu_0 = ${mu0}`
}

tobstext = 
{
   return tex`\textbf{Test statistic }\\
    t_{obs} = \frac{\bar x -\mu_0}{\frac{s}{\sqrt{n}}} = \frac{${xbar.toFixed(3)} - ${mu0.toFixed(3)}}{\frac{${s.toFixed(3)}}{\sqrt{${n}}}} = ${tobs.toFixed(3)}`
}

tcrittext = 
{
   return tex`\textbf{Critical value }\\
t_{crit} = t_{${alphaside}, \,${n-1}} = ${tcrit.toPrecision(4)}`
}

pvaluetext = {
  var rejecttext;
  var sign;
  if (pvaluestudentt < alpha){
    rejecttext = " \\textbf{we reject} ";
    sign = "<";
  }else{
    rejecttext = "we \\textbf{do not reject} ";
    sign = "\\geq";
  }
   return tex`\textbf{p-value}\\
p\text{-value}=${pvaluestudentt.toPrecision(4)}`
}

pconclusion = {
  var rejecttext;
  var sign;
  if (pvaluestudentt < alpha){
    rejecttext = "We\\textcolor{red}{ \\textbf{reject}}\ ";
    sign = "<";
  }else{
    rejecttext = "We\\textcolor{steelblue}{ \\textbf{do not reject}} ";
    sign = "\\geq";
  }
   return tex`\text{${rejecttext}} H_0 \text{ at } ${(alpha*100).toFixed(1)}\% \text{ significance level.}  `
}

viewof inputs = Inputs.form([
      Inputs.range([2, 1000], {value: 5, step: 1, label: tex`\text{sample }n`}), 
      Inputs.range([-10, 10], {value: 1.7, step: 0.01, label: tex`\text{sample } \bar x`}),
      Inputs.range([Number.EPSILON, 10], {value: 0.5, step: 0.1, label: tex`\text{sample }s`}),
      Inputs.range([0.001, 1], {value: 0.05, step: 0.001, label: tex`\text{sig. lvl }\alpha`}),
      Inputs.range([-10, 10], {value: 1, step: 0.1, label: tex`\text{null }\mu_0`}),
      Inputs.select(["equal to","greater than", "less than"], {label: tex`\text{alternative}`})
    ])

viewof toggles = Inputs.form([
  Inputs.toggle({value: true, label: "show critical regions"}),
  Inputs.toggle({value: false, label: "show p-values"})
  ])

jstat = require('jstat')

textfontsize = 14

showcrit = toggles[0]

showpvalue = toggles[1]

n = inputs[0]
xbar = inputs[1]
s = inputs[2]
alpha = inputs[3]
mu0 = inputs[4]
alternative = inputs[5]

alphaside = {
  if (alternative == 'equal to'){
    return alpha/2
  }else{
    return alpha
  }
}

tobs = (xbar-mu0)/(s/Math.sqrt(n))

tcrit = jstat.studentt.inv(1-alphaside, n-1)

zcrit = jstat.normal.inv(1-alphaside, 0, 1)

pvaluestudentt = {
  if (alternative == 'equal to'){
      return 2*jstat.studentt.cdf(-Math.abs(tobs), n-1)
  }else if(alternative == 'less than'){
    return jstat.studentt.cdf(tobs, n-1)
  }else{
    return 1-jstat.studentt.cdf(tobs, n-1)
  }
}

pdfdata = {
  const x = d3.range(-10, 10, 0.01);
  var data = x.map(x => ({x: x, pdf: jstat.normal.pdf(x, 0, 1), dist: "normal", type: "pdf"}));
  const studentpdf = x.map(x => ({x: x, pdf: jstat.studentt.pdf(x, n-1), dist: "student-t", type: "pdf"}));
  data = data.concat(studentpdf)
  data = data.concat(tcritdata)
   data = data.concat(tobsdata)
  return data
}

Figure 1: Source: https://observablehq.com/@mattiasvillani/hypothesis-test-mean

max_y = 1.05*jstat.normal.pdf(0, 0, 1)

twosided = alternative == 'equal to'

side = {
  if (alternative == 'less than'){
    return "left"
  }else if(alternative == 'greater than'){
    return "right"
  }else{
    return "both"
  }
}

tcritdata = {
  if (twosided){
    return [{x: tcrit, y: max_y/4, dist: "student-t", type: "crit"}, {x: -tcrit, y: max_y/4, dist: "student-t", type: "crit"}]
  }else if (alternative == 'less than'){
    return [{x: -Math.abs(tcrit), y: max_y/4, dist: "student-t", type: "crit"}]
  }else
  {
    return [{x: Math.abs(tcrit), y: max_y/4, dist: "student-t", type: "crit"}]
  }
}

tobsdata = {
  if (twosided){
    return [{x: tobs, y: 0, dist: "student-t", type: "obs"}, {x: -tobs, y: 0, dist: "student-t", type: "obs"}]
  }else if (alternative == 'less than'){
    return [{x: -Math.abs(tobs), y: 0, dist: "student-t", type: "obs"}]
  }else
  {
    return [{x: Math.abs(tobs), y: 0, dist: "student-t", type: "obs"}]
  }
}

conclusioncolor = {
  if(pvaluestudentt < alpha){
    return "red"
  }else{
    return "steelblue"
  }
}

nudge = 1.1
bignudge = 1.3

plt = Plot.plot({
    width: 800, // or a dynamic value based on `width` variable
    height: 600,
    style: {fontSize: "16px"},
    color: {
      legend: false
    },
    x: {
      label: "t-statistic",
      axis: true,
      domain: [-10,10]
    },
    y: {
      axis: false,
      domain: [0,max_y]
    },
    marks: [
      Plot.ruleY([0]),
      Plot.line(pdfdata, {filter: d => d.dist == "student-t" && d.type == "pdf", x: "x", y: "pdf", stroke : "dist", strokeWidth: 2}),
      Plot.areaY(pdfdata, {filter: d => showpvalue && side == "left" && d.x <= tobs && 
          d.dist == "student-t" && d.type == "pdf", x: "x", y: "pdf", fill: "orange", opacity: 0.5}),
      Plot.areaY(pdfdata, {filter: d => showpvalue && side == "right" && d.x >= tobs && 
        d.dist == "student-t" && d.type == "pdf", x: "x", y: "pdf", fill: "orange", opacity: 0.5}),
      Plot.areaY(pdfdata, {filter: d => showpvalue && side == "both" && d.x <= -Math.abs(tobs) && 
          d.dist == "student-t" && d.type == "pdf", x: "x", y: "pdf", fill: "orange", opacity: 0.5}),
      Plot.areaY(pdfdata, {filter: d => showpvalue && side == "both" && d.x >= Math.abs(tobs) && 
        d.dist == "student-t" && d.type == "pdf", x: "x", y: "pdf", fill: "orange", opacity: 0.5}),
      
      Plot.dot([{x: tobs, y: 0}], {x: "x", y: "y", fill: conclusioncolor, symbol: "circle", r: 5}),
      Plot.arrow([{x1: tobs, y1: max_y/12, x2:tobs, y2: 0.01}], {x1: "x1", y1: "y1", x2: "x2", y2: "y2", stroke: conclusioncolor}),
      Plot.text([{x1: tobs, y1: bignudge*max_y/12, texter: "t(obs)"}], {x: "x1", y: "y1", text: "texter", fontSize: textfontsize}),
      
      Plot.ruleX([{x: tcrit, y: max_y/4}], {filter: d => showcrit && side == "both", x: "x", y: "y", fill: "green"}),
      Plot.ruleX([{x: -tcrit, y: max_y/4}], {filter: d => showcrit && side == "both", x: "x", y: "y", fill: "green"}),
      Plot.text([{x1: tcrit, y1: nudge*max_y/4, texter: "t(crit)"}], 
                {filter: d => showcrit && side == "both",x: "x1", y: "y1", text: "texter", fontSize: textfontsize}),
      Plot.text([{x1: -tcrit, y1: nudge*max_y/4, texter: "-t(crit)"}], 
                {filter: d => showcrit && side == "both",x: "x1", y: "y1", text: "texter", fontSize: textfontsize}),

      Plot.ruleX([{x: tcrit, y: max_y/4}], {filter: d => showcrit && side == "right", x: "x", y: "y", fill: "green"}),
      Plot.text([{x1: tcrit, y1: nudge*max_y/4, texter: "t(crit)"}], 
                {filter: d => showcrit && side == "right",x: "x1", y: "y1", text: "texter", fontSize: textfontsize}),
      
      Plot.ruleX([{x: -tcrit, y: max_y/4}], {filter: d => showcrit && side == "left", x: "x", y: "y", fill: "green"}),
      Plot.text([{x1: -tcrit, y1: nudge*max_y/4, texter: "-t(crit)"}], 
                {filter: d => showcrit && side == "left",x: "x1", y: "y1", text: "texter", fontSize: textfontsize}),

      Plot.arrow([{x1: tcrit*bignudge, y1: max_y/6, x2:10, y2: max_y/6}], {filter: d => showcrit && side == "right", 
                                                                  x1: "x1", y1: "y1", x2: "x2", y2: "y2", stroke: "red"}),
      Plot.arrow([{x1: -tcrit*bignudge, y1: max_y/6, x2:-10, y2: max_y/6}], {filter: d => showcrit && side == "left", 
                                                                    x1: "x1", y1: "y1", x2: "x2", y2: "y2", stroke: "red"}),
      Plot.arrow([{x1: tcrit*bignudge, y1: max_y/6, x2:10, y2: max_y/6}], {filter: d => showcrit && side == "both", 
                                                                  x1: "x1", y1: "y1", x2: "x2", y2: "y2", stroke: "red"}),
      Plot.arrow([{x1: -tcrit*bignudge, y1: max_y/6, x2:-10, y2: max_y/6}], {filter: d => showcrit && side == "both", 
                                                                    x1: "x1", y1: "y1", x2: "x2", y2: "y2", stroke: "red"}),
      Plot.text([{x1: (-10 - tcrit*bignudge)/2, y1: max_y/4, texter: "Reject H₀"}], 
                {filter: d => showcrit && side == "left", x: "x1", y: "y1", text: "texter", fontSize: textfontsize}),
      Plot.text([{x1: (10 + tcrit*bignudge)/2, y1: max_y/4, texter: "Reject H₀"}], 
                {filter: d => showcrit && side == "right", x: "x1", y: "y1", text: "texter", fontSize: textfontsize}),
      Plot.text([{x1: (-10 - tcrit*bignudge)/2, y1: max_y/4, texter: "Reject H₀"}], 
                {filter: d => showcrit && side == "both", x: "x1", y: "y1", text: "texter", fontSize: textfontsize}),
      Plot.text([{x1: (10 + tcrit*bignudge)/2, y1: max_y/4, texter: "Reject H₀"}], 
                {filter: d => showcrit && side == "both", x: "x1", y: "y1", text: "texter", fontSize: textfontsize})
    ]
  })

Figure 2: https://observablehq.com/@mattiasvillani/hypothesis-test-mean?collection=@mattiasvillani/statistics