One-proportion confidence interval is one of 2 confidence intervals formulas in the statistics section of this library, and it is used at ap · university level.
Why one-proportion confidence interval works
The unknown true proportion is replaced by the one you observed, which is good enough once the sample is reasonably large. A normal critical value is used because the count of successes follows a binomial distribution that takes on a bell shape when there are plenty of results on both sides.
What each symbol means
$\hat p$ is sample proportion and $z^*$ sets confidence level.
One-proportion confidence interval: when it holds
Use random independent data and enough expected successes and failures for the normal approximation.
When it stops applying
The normal approximation assumes roughly ten successes and ten failures. With 2 successes out of 20, the 95 percent interval runs from about -0.031 to 0.231, and a negative proportion is impossible. That is a clear signal the approximation has broken down and an exact or adjusted method is needed.
One-proportion confidence interval: a worked example
If $\hat p=.60,n=100,z^*=1.96$, margin is about $.096$.
The mistake to avoid
What people do: Putting the number of successes into the slot meant for the number of trials.
Why it goes wrong: For 60 successes out of 100 the correct standard error is about 0.049 and the 95 percent margin is about 0.096. Using 60 as the sample size gives 0.063 and a margin nearly 30 percent too wide.
Do this instead: Label the two numbers before substituting: the count of successes builds the proportion, and the count of trials goes underneath in the standard error.
One-proportion confidence interval: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $\hat p$ is sample proportion and $z^*$ sets confidence level.
- Check the conditions before substituting. Use random independent data and enough expected successes and failures for the normal approximation.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most statistics slips.
Where this formula fits
- Subject
- Statistics formulas — 21 entries in this library
- Topic
- Confidence intervals
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where one-proportion confidence interval comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Standard Deviation — the lesson behind this formula: measure typical distance from the mean.
- Statistics Calculator — check your substitution and the value it produces.
- Study statistics — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 21 statistics formulas — the full grouped reference, or the complete formula library.
Questions about one-proportion confidence interval
What critical values go with the common confidence levels?
They are 1.645 for 90 percent, 1.960 for 95 percent, and 2.576 for 99 percent. Each one is the point leaving half the remaining probability in each tail of the standard normal curve.
What happens if my sample proportion is 0 or 1?
The formula gives an interval of zero width, which is obviously wrong. Use the plus-four adjustment or a Wilson interval, both of which pull the estimate away from the boundary before building the range.
Can I turn this into an interval for a count instead?
Yes, multiply both endpoints by the population size. An interval for the share of a city that recycles becomes an interval for the number of residents who do.
Why is a normal critical value used here when means need t?
Because no separate spread parameter is being estimated. For a proportion the spread is determined by p itself, so there is no extra uncertainty for a t distribution to account for.