From Confidence Level to a Normal-Curve Area
In Standard Error of \(\hat{p}\) Versus Standard Deviation, you distinguished the estimated standard error used in a one-proportion confidence interval from the theoretical standard deviation of the sampling distribution. In Structure of a One-Proportion z-Interval, you saw that the interval’s margin of error is \(z^*\) times the estimated standard error. This tutorial focuses on how to find that critical value \(z^*\) for 90%, 95%, and 99% confidence.
A critical value is a cutoff on a probability distribution. For a two-sided confidence interval, \(z^*\) is the positive cutoff on the standard Normal curve that leaves the chosen confidence level’s area in the middle. The matching negative cutoff is \(-z^*\). The area not in the middle is split evenly between the two tails.
Here, \(C\) is written as a decimal: for example, 95% confidence means \(C=0.95\). The remaining area, called \(\alpha\), is \(1-C\). Since this is a two-sided interval, each tail contains \(\alpha/2\). The calculator function invNorm finds a value from the area to its left, so the input is the left-tail cumulative area at the positive cutoff, \(1-\alpha/2\).
The last two arguments specify the standard Normal distribution’s mean, 0, and standard deviation, 1. On a TI-84 or similar calculator, use the invNorm function with the left-tail area followed by 0 and 1. A frequent error is to enter the area in one tail, \(\alpha/2\), as if it were the area to the left of the positive cutoff. That instead points to a negative value in the left tail.
Sketch the Middle Area and Both Tails
A quick sketch helps make the calculator input clear. Draw a symmetric, bell-shaped standard Normal curve centered at 0. Mark \(-z^*\) to the left of 0 and \(z^*\) the same distance to the right. The confidence level is the area between the marks; the two outside areas are equal.
The marks are symmetric because the standard Normal curve is symmetric about 0. The area from the far left up to \(z^*\) includes both the left tail and the central area. That is why invNorm receives \(1-\alpha/2\), not \(C\) alone. For example, with 95% confidence, the area to the left of the positive cutoff is \(0.025+0.95=0.975\).
The Three Common Critical Values
The same sequence works for each confidence level: calculate \(\alpha=1-C\), divide \(\alpha\) by 2, subtract that tail area from 1, and use invNorm. The table summarizes the results. The critical values are rounded to three decimal places, as is standard for these common values.
| Confidence level \(C\) | \(\alpha=1-C\) | Area in each tail | invNorm left area | Critical value \(z^*\) |
|---|---|---|---|---|
| 90% | 0.10 | 0.05 | 0.95 | 1.645 |
| 95% | 0.05 | 0.025 | 0.975 | 1.960 |
| 99% | 0.01 | 0.005 | 0.995 | 2.576 |
These are positive cutoffs. On the curve, the corresponding central intervals are from \(-1.645\) to \(1.645\), from \(-1.960\) to \(1.960\), and from \(-2.576\) to \(2.576\). The exact invNorm values before rounding are approximately 1.644854, 1.959964, and 2.575829, respectively.
Worked Example: Deriving the 90% Critical Value
A student is preparing a 90% confidence interval for a population proportion. Find \(z^*\) from the standard Normal distribution and explain the areas shown on a sketch.
Find the tail areas. Write the confidence level as \(C=0.90\). The total area outside the central region is:
On the sketch, mark \(-z^*\) and \(z^*\) around 0. Label the middle area 0.90 and each tail area 0.05. To use invNorm for the positive cutoff, add the central area and the left-tail area, or equivalently subtract the right-tail area from 1:
Use invNorm. Enter the cumulative area to the left of the positive cutoff, with standard Normal mean and standard deviation:
Check the result on the curve. The matching cutoffs are approximately \(-1.645\) and \(1.645\). The area between them is approximately \(0.9000\) when rounded to four decimal places. In particular, using the rounded cutoffs gives \(\operatorname{normalcdf}(-1.645,1.645,0,1)\approx0.9000\), not 0.9001.
Interpret the value. The positive critical value for a 90% central confidence interval is \(z^*=1.645\). In the one-proportion interval formula, it will multiply the estimated standard error to set the margin of error; the matching lower cutoff on the Normal curve is \(-1.645\).
Worked Example: Deriving the 95% Critical Value
A community group plans to report a 95% confidence interval for the proportion of local households that compost food scraps. Find the critical value and show how the central area determines the invNorm input. No sample results are needed to find \(z^*\).
Find the total and individual tail areas. With \(C=0.95\), the area outside the central region is:
A sketch has 0.025 in each tail and 0.95 between the two cutoffs. Therefore, the area to the left of the positive cutoff is \(1-0.025=0.975\). Enter that left area in invNorm:
The endpoints on the standard Normal curve are \(-1.960\) and \(1.960\). Checking the rounded values gives \(\operatorname{normalcdf}(-1.960,1.960,0,1)\approx0.9500\) to four decimal places. Thus \(z^*=1.960\) is the positive critical value used for this 95% confidence level. The household context describes what the future interval estimates; it does not change the standard Normal area calculation.
Worked Example: Deriving and Comparing the 99% Critical Value
A school technology team wants to use a 99% confidence level when estimating the proportion of students who can access a learning platform from home. Find \(z^*\), verify the approximate central area, and compare the value with the 95% critical value.
Find the tail areas. For \(C=0.99\), the total area outside the central region is \(1-0.99=0.01\), so each tail has area \(0.01/2=0.005\). The left area up to the positive cutoff is \(1-0.005=0.995\).
Use invNorm and check. With standard Normal mean 0 and standard deviation 1:
The symmetric marks are \(-2.576\) and \(2.576\), with 0.005 in each tail. Using the rounded cutoffs, \(\operatorname{normalcdf}(-2.576,2.576,0,1)\approx0.9900\) to four decimal places.
Compare the values. The 99% critical value, 2.576, is greater than the 95% critical value, 1.960. The 99% interval must extend farther on each side of its point estimate, if the point estimate and standard error are held fixed. This gives the interval a larger margin of error. A higher confidence level asks for more area in the center, so its cutoffs must move farther from 0.
Why the Critical Value Changes with Confidence
Imagine keeping a one-proportion interval’s sample proportion and estimated standard error fixed. Raising the confidence level from 90% to 95% to 99% increases the desired central area. The two tail areas consequently shrink, and the positive cutoff moves farther to the right. By symmetry, the negative cutoff moves just as far to the left.
This gives an important tradeoff: at the same sample size and with the same sample proportion, greater confidence produces a larger \(z^*\) and a wider interval. A wider interval reflects the choice to capture the population proportion in a larger fraction of repeated-sampling intervals, under the method’s conditions. Critical values are not chosen by the observed data; they are chosen from the confidence level.
Keep this calculation separate from checking whether a one-proportion \(z\)-interval is appropriate. The random-sampling, 10%, and Large Counts conditions discussed in earlier tutorials concern whether the interval method is suitable for the data. They do not change the standard Normal cutoff for a specified confidence level. The next step, constructing the interval by hand, combines the chosen \(z^*\) with the point estimate and estimated standard error.
Common Mistakes and AP Exam Tips
- Entering the confidence level directly in invNorm. For a 95% central interval, invNorm needs 0.975, the area left of the positive cutoff, not 0.95.
- Putting all of \(\alpha\) in one tail. A two-sided interval splits the area outside the center equally, so each tail has area \(\alpha/2\).
- Reporting a negative \(z^*\). The critical value in the interval formula is the positive distance \(z^*\). The two curve cutoffs are \(-z^*\) and \(z^*\).
- Confusing \(z^*\) with an observed z-score. A z-score standardizes a particular observation. A critical value is selected from the desired confidence level before the interval is calculated.
- Assuming more confidence makes an interval narrower. With the same estimate and standard error, a higher confidence level uses a larger critical value and therefore a wider interval.
- Rounding the area check incorrectly. If checking with rounded cutoffs, report the Normal area to an appropriate number of decimals. For example, \(\operatorname{normalcdf}(-1.645,1.645,0,1)\) is about 0.90003, which rounds to 0.9000 to four decimal places.
Key Takeaway
For a central confidence level \(C\), find the total outside area \(1-C\), divide it equally between the tails, and use invNorm with the area left of the positive cutoff. The common values are 1.645 for 90%, 1.960 for 95%, and 2.576 for 99%. Higher confidence requires a larger critical value and, all else equal, a wider interval.
Check Your Understanding
For each question, identify the tail area and the left cumulative area needed for invNorm.
- For an 80% central confidence level, find \(\alpha\), the area in each tail, and the invNorm left area for the positive cutoff.
- For a 90% confidence level, explain why the invNorm input is 0.95 rather than 0.90 or 0.05.
- For a 95% confidence level, state both standard Normal cutoffs and the area in each tail.
- For a 99% confidence level, write the invNorm expression and report \(z^*\) to three decimal places.
- If the sample proportion and estimated standard error stay the same, explain how changing confidence from 90% to 99% affects the margin of error and interval width.