How to Calculate a Confidence Interval (Mean, Proportion)

The one formula behind every confidence interval

Every confidence interval you build in AP Statistics uses the same structure from the official formula sheet:

The statistic is your point estimate from the sample. That is the sample proportion $\hat{p}$ (p-hat) when you study a categorical variable, or the sample mean $\bar{x}$ (x-bar) when you study a quantitative variable. Throughout, $n$ is the sample size.

The critical value sets how confident you want to be. A larger critical value gives a wider interval and higher confidence. You read it from a $z$ table for a proportion or a $t$ table for a mean.

The standard error measures how much the statistic moves from one random sample to the next. The product (critical value)(standard error) is the margin of error, the plus-or-minus part that you add and subtract.

Once you have those three pieces, every interval is the same short job: find the statistic, look up the critical value, compute the standard error, then add and subtract the margin of error. The two recipes below differ only in which standard error formula and which table you use.

How to calculate a confidence interval for a proportion

Use this when your data is counts or percentages, such as the share of voters who back a measure. Call the true population proportion $p$ and the sample proportion $\hat{p}$, found by dividing the number of successes by the sample size $n$. A success just means the outcome you are counting, like a voter who says yes.

The recipe:

1. Confirm the conditions (random, 10% condition, large counts). See the conditions section below.
2. Compute $\hat{p} = \frac{\text{successes}}{n}$.
3. Find the critical value $z^*$ for your confidence level from the table below.
4. Compute the standard error $SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$.
5. Build the interval: $\hat{p} \pm z^* \cdot SE$.

For a proportion the standard deviation of the sampling distribution depends only on $p$, and under the large-counts condition the distribution of $\hat{p}$ is approximately normal, so you use a $z$ critical value. The College Board calls this a one-sample z-interval for a population proportion, and you should name it that way on the exam.

How to calculate a confidence interval for a mean

Use this when your data is measurements, such as reaction times or battery life. Call the true population mean $\mu$ (mu) and estimate it with the sample mean $\bar{x}$.

The recipe:

1. Confirm the conditions (random, 10% condition, normal or large sample). See the conditions section below.
2. Compute $\bar{x}$ and the sample standard deviation $s$.
3. Find the critical value $t^*$ using degrees of freedom $df = n - 1$ for your confidence level.
4. Compute the standard error $SE = \frac{s}{\sqrt{n}}$.
5. Build the interval: $\bar{x} \pm t^* \cdot SE$.

You use $t$, not $z$, because you estimate the unknown population standard deviation with the sample value $s$. That extra uncertainty makes the $t$ distribution wider than $z$, and the gap shrinks as $n$ grows. The degrees of freedom $df = n - 1$ tell you which row of the $t$ table to read.

This is a one-sample t-interval for a population mean. The same recipe works on the differences when you have paired data.

Critical values for 90, 95, and 99 percent confidence

For a proportion the critical value $z^*$ depends only on the confidence level:

A 95 percent confidence interval for a proportion therefore uses $z^* = 1.960$. You can confirm any of these values on the z table.

For a mean the critical value $t^*$ depends on both the confidence level and the degrees of freedom, so there is no single number. At 95% confidence, for example, $t^* = 2.776$ when $df = 4$ but $t^* = 2.262$ when $df = 9$. Always look up $t^*$ on the t table using $df = n - 1$. As $n$ grows large, $t^*$ moves toward the matching $z^*$, which is why the two procedures agree for big samples.

Conditions you must check first

An interval is valid only if the sampling distribution of your statistic is roughly normal and the standard error formula holds. The AP course lists three conditions to verify before you compute.

Random. The data must come from a random sample or a randomized experiment. Without randomness the interval does not estimate the population.

10% condition. When you sample without replacement, the population must be at least 10 times the sample size, so $n \leq 0.10 N$, where $N$ is the population size. This keeps the standard error formula accurate.

Large counts or normal. For a proportion, check that the number of successes and the number of failures are both at least 10, that is $n\hat{p} \geq 10$ and $n(1 - \hat{p}) \geq 10$, so the sampling distribution is roughly normal. For a mean, the sampling distribution is roughly normal if the sample size is large (a common guideline is $n \geq 30$) or if the population is roughly normal with no strong skew or outliers when $n$ is small. State each check in the context of the problem rather than just writing the symbols.

How to read the interval you get

Report the interval as two numbers and interpret it in context. For a 95% interval, write: you are 95% confident that the true population value lies between the lower bound and the upper bound.

The confidence level describes the long-run method, not one particular interval. If you repeated the sampling and built an interval each time, about 95% of those intervals would capture the true parameter. For the exact wording graders reward, see what 95 percent confidence means.

To check your arithmetic on either recipe, use the confidence interval calculator. Enter the count and sample size for a proportion, or the summary statistics for a mean, and confirm the bounds match your hand calculation.

Common mistakes when building a confidence interval

**Using $z$ for a mean.** When you estimate the population standard deviation with the sample value $s$, use $t$, not $z$. Reaching for $z$ makes the interval too narrow.

Skipping the conditions. An interval with no condition check may not be valid and loses credit on the exam. State random, the 10% condition, and large counts or normality every time.

Plugging in the wrong count. For a proportion, divide successes by $n$ to get $\hat{p}$ first. Do not put the raw number of successes into the standard error formula.

Rounding the critical value too early. Keep $z^*$ or $t^*$ to at least three decimals until the final step, or your bounds can drift.

Reading the interval as a probability. A 95% interval does not mean there is a 95% chance the parameter is inside it. The parameter is fixed, and the confidence describes the method across many samples.