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:

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;
}

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] + 0.4, "y": 0.35}, {"x": critical_val(C)[1] - 0.4, "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"
})

viewof C = Inputs.range(
[1, 99.99],
{value: 95, step: 0.01, label: "Confidence Level:"}
)

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 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 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, \(z^\ast = 2.58\).