Before the method: the idea behind it
From a single estimate to a range
You now know that a sample proportion is one draw from a distribution centred on the true value. That fact can be turned around. Instead of reporting a single number and hoping, you can report a range wide enough that it usually captures the truth, and state exactly how often “usually” means. That is a confidence interval, and it is the final idea in the course.
Part one
Running the sampling distribution backwards
Start from what you know: \(\hat{p}\) is approximately normal, centred on p, with standard deviation \(\sqrt{\frac{p(1-p)}{n}}\).
So about 95% of all samples produce a \(\hat{p}\) within 1.96 standard deviations of p. Written as a statement about distance, that says \(\hat{p}\) lands within 1.96 standard deviations of p about 95% of the time.
Now notice the symmetry of that statement. If \(\hat{p}\) is within a certain distance of p, then p is within the same distance of \(\hat{p}\). Distance does not care which end you measure from. So an interval built around your sample proportion, reaching out 1.96 standard deviations in each direction, will contain p about 95% of the time.
That single swap is the whole idea. Everything else is arithmetic.
The shaded central region holds 95% of all sample proportions. Its edges sit 1.96 standard deviations either side of p.
The z-value depends only on the confidence level you choose, so these few numbers cover almost every question you will meet.
| Confidence level | Area in each tail | z-value |
|---|---|---|
| 90% | 5% | 1.645 |
| 95% | 2.5% | 1.960 |
| 98% | 1% | 2.326 |
| 99% | 0.5% | 2.576 |
One practical wrinkle remains. The formula needs the standard deviation, which depends on p, the unknown. As in the last lesson, you substitute the point estimate \(\hat{p}\) in its place. That substitution is why the result is called an approximate confidence interval.
The quantity added and subtracted is called the margin of error, written E. So every confidence interval has the same shape: the point estimate, plus or minus the margin of error. The next lesson is entirely about E and what controls it.
Part two
What “95% confident” actually means
This is the part that is most often stated wrongly, including in newspapers, and it is examined directly.
Below, the true proportion is fixed at 0.4 and marked by the dashed line. Each horizontal bar is a confidence interval built from one random sample. Generate a batch and see what happens.
Each interval comes from a fresh sample of 200, at 95% confidence. Blue intervals contain the true value; red ones miss it entirely.
About 95 in every 100 intervals capture the true value, and some do not. That long-run frequency is what the confidence level describes. It is a property of the method, not of any single interval you happen to have calculated.
The textbook sets out four conclusions that follow, and each of them appears in exams as a written-answer question.
Conclusion 3 is the one that trips people up. Saying “there is a 95% probability that p lies between 0.42 and 0.48” is wrong, because p is not random: it is a fixed number that either is or isn’t in that range. The randomness was in the sampling, which has already happened. The correct phrasing is that 95% of intervals constructed this way contain p.
Conclusion 4 has a direct exam consequence. If you are shown two intervals and asked which is more likely to contain p, the answer is neither: that cannot be determined from the intervals themselves. The textbook flags this explicitly as a trick question.
Part three
Reading an interval backwards
Because an interval is symmetric about the point estimate, you can recover everything from the two endpoints.
The centre is the average of the endpoints, giving \(\hat{p}\). Half the width is the margin of error E. And once you have both, together with the confidence level’s z-value, you can solve for the sample size n that produced it.
Questions that give you an interval and ask for the sample size, the confidence level, or the number of successes observed are common, and they all start with these first two steps.
Where this shows up in real life
Every poll you have ever read
When a poll reports 52% support “with a margin of error of 3 percentage points”, it is quoting a 95% confidence interval from 49% to 55%. Since that range includes 50%, the poll cannot actually distinguish a majority from a minority, which is precisely the kind of conclusion reporting tends to skip.
Drug approval
Regulators require confidence intervals for a treatment’s effect, not just a point estimate. If the interval includes zero improvement, the trial has not demonstrated the drug works, however promising the central figure looks.
Before you start
What to reach for
| The question asks for… | Do this |
|---|---|
| a confidence interval from sample data | Find p, look up z, compute p ± z√(p(1−p)/n) |
| the point estimate from an interval | Average the two endpoints |
| the sample size from an interval | E is half the width, then solve n = z²p(1−p)/E² |
| an interpretation of the confidence level | Talk about the long-run proportion of intervals, never the probability for one interval |
| which interval is more likely to contain p | Neither: it cannot be determined |
| whether a claimed value is plausible | Check whether it lies inside the interval |
Next: the working method, twenty practice questions, ten harder ones, and the ClassPad guide.