viewof C = Inputs.range(
[1, 99.99],
{value: 95, step: 0.01, label: "Confidence Level (% in the middle):"}
)
function standard_z(from, to) {
var x = d3.range(from, to + 0.005, 0.01);
var normal = [];
x.forEach((val, i) => {
normal.push({"x": val, "y": jStat.normal.pdf(val, 0, 1), "lab": "Standard Normal"});
})
return normal;
}
function critical_val(C) {
var val = jStat.normal.inv((100 - C) / 2 / 100, 0, 1);
return [val, -1 * val]
}
function conf_region(C) {
var cutoffs = critical_val(C);
var x = d3.range(cutoffs[0], cutoffs[1] + 0.005, 0.01);
var result = [];
x.forEach((val, i) => {
result.push({"x": val, "y": jStat.normal.pdf(val, 0, 1), "lab": "Standard Normal"});
});
return result;
}
function neg_region(C) {
var cutoffs = critical_val(C);
var x = d3.range(-4, cutoffs[0] + 0.005, 0.01);
var result = [];
x.forEach((val, i) => {
result.push({"x": val, "y": jStat.normal.pdf(val, 0, 1), "lab": "Standard Normal"});
});
return result;
}
function pos_region(C) {
var cutoffs = critical_val(C);
var x = d3.range(-4, cutoffs[1] + 0.005, 0.01);
var result = [];
x.forEach((val, i) => {
result.push({"x": val, "y": jStat.normal.pdf(val, 0, 1), "lab": "Standard Normal"});
});
return result;
}
Plot.plot({
marks: [
Plot.areaY(conf_region(C),
{
x: "x",
y: "y",
fillOpacity: 0.3
}
),
Plot.line(standard_z(-4, 4),
{
x: "x",
y: "y",
stroke: "lab",
strokeWidth: 3
}
),
Plot.ruleX([critical_val(C)[0]]),
Plot.ruleX([critical_val(C)[1]]),
Plot.text([{"x": critical_val(C)[0] + 0.5, "y": 0.35}, {"x": critical_val(C)[1] - 0.5, "y": 0.35}], {x: "x", y:"y", text: ["-|z*| = " + critical_val(C)[0].toFixed(2), "|z*| = " + critical_val(C)[1].toFixed(2)]})
],
x: { label: "z" },
y: {domain: [0, .41],
label: "Density"},
color: {
legend: true
},
caption: "Standard Normal Distribution"
})Appendix C — \(z^\ast\) Critical Values
When constructing confidence intervals, we need the value \(z^\ast\) such that \(C\)% of the estimated sampling distribution is between \(-|z^\ast|\) and \(|z^\ast|\). In other words, imagine:
As we learned in Chapter 2, we can find this cut-off by using the symmetry of the Normal Distribution. One way of finding this is the find the negative cutoff. Find the area corresponding to the left of the middle region. Once you find it, you can plug this into invNorm with a mean of 0 and standard deviation of 1.
Another way of finding this is the find the positive cutoff. Find the area corresponding to the left of the middle region plus the middle region itself. Once you find it, you can plug this into invNorm with a mean of 0 and standard deviation of 1.
So if we want to \(z\) that corresponds to the confidence level, we have to determine the area of the left/right tail that corresponds to the confidence level. We know that if there’s \(C\)% in the middle, then there’s \(100\% - C\) left in the two tails. For one tail, that’ll be \(\frac{100\% - C}{2}\).
Look up that proportion in Table A in the middle of the table, take the larger value if it’s not on there specifically, and back track to the margins to get your desired value of \(z^\ast\).
The sign of any critical value does not matter, so just take the absolute value as your critical value.
C.1 Examples
- 90% confidence
Find that the left area that we want is: \(\frac{100\% - 90%}{2} = 5\% = 0.05\)
Try to find 0.05 in the middle of Table A and you notice that it is between 0.0505 and 0.0495.
Take the larger value, 0.0505 and look at the margins for the \(z\) that corresponds to it, -1.64.
Just take the positive value, \(z^\ast = 1.64\).
Alternatively, do the calculator command invNorm(0.05) to get a value of around -1.64. Take the positive value as your critical value, \(z^\ast = 1.64\).
- 99% confidence
Find that the left area that we want is: \(\frac{100\% - 99%}{2} = 0.5\% = 0.005\)
Try to find 0.005 in the middle of Table A and you notice that it is between 0.0051 and 0.0049.
Take the larger value, 0.0051 and look at the margins for the \(z\) that corresponds to it, -2.57.
Just take the positive value, \(z^\ast = 2.57\).
Alternatively, do the calculator command invNorm(0.005) to get a value of around -2.5758. Take the positive value as your critical value, \(z^\ast = 2.58\).