Reporting a Confidence Interval for a Proportion

Why Reporting Needs More Than Two Endpoints

A confidence interval is useful only if readers can tell which population it concerns, how the sample was obtained, and what the interval estimates. The endpoints alone do not provide that information. For example, “the interval is 54.2% to 63.8%” leaves unanswered whether the proportion concerns city residents, clinic patients, or survey respondents—and whether those people were selected randomly.

A concise report connects the sample result to its intended population without claiming more than the sampling design supports. It states the sample estimate, the interval and confidence level, the population, and the sampling method. It also notes important limitations, such as incomplete coverage or nonresponse.

Definition: A report of a one-proportion confidence interval gives the sample proportion \(\hat{p}\), the interval estimate for the population proportion \(p\), its confidence level, the population the estimate is intended to describe, and the sampling method. It also identifies limitations that may affect whether the sample represents that population.

A Reporting Framework

Before writing, identify the quantity being estimated. For a binary outcome, such as “supports” versus “does not support,” \(p\) is the proportion in the target population with the outcome of interest. The sample proportion \(\hat{p}=x/n\) is the estimate, where \(x\) is the number of sampled individuals with that outcome and \(n\) is the sample size.

For a simple random sample, one familiar approximate interval is the Wald interval. It is appropriate only when the sampling and approximation conditions are reasonably satisfied: observations are independent (or the sample is a small fraction of the population when sampling without replacement), and there are enough observed successes and failures for the normal approximation to be reasonable. A common check is that \(n\hat{p}\) and \(n(1-\hat{p})\) are each at least 10. The Wald interval can perform poorly near 0 or 1; in such cases a different interval method may be preferable.

Formula: When the Wald method and its conditions are appropriate, a confidence interval for \(p\) is

\hat{p}\ \pm\ z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}},

where \(z^*\) is the standard-normal critical value for the chosen confidence level. The square-root term is the estimated standard error, and the product of \(z^*\) and that standard error is the margin of error.

The reporting logic follows from the interval construction. Under the model and approximation conditions, the sample proportion varies across repeated samples, with estimated standard error \(\sqrt{\hat{p}(1-\hat{p})/n}\). A central confidence level corresponds to keeping the standardized estimate within \(z^*\) standard errors of the population proportion. Rearranging that condition to isolate \(p\) gives the interval:

\left|\frac{\hat{p}-p}{\sqrt{\hat{p}(1-\hat{p})/n}}\right|\le z^* \quad\Longrightarrow\quad \hat{p}-z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \le p \le \hat{p}+z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.

This derivation explains why the endpoints are estimates, not observed boundaries of the population. It also highlights the role of the assumptions: if the sample does not support the model or normal approximation, a neatly worded report cannot repair an unsuitable interval.

Name the outcome and population.
State what counts as a success and the group whose proportion is of interest.

Identify the estimate and interval.
Report \(\hat{p}\), the endpoints, and the confidence level. Use percentages or proportions consistently.

Describe how the sample was obtained.
Name the sampling method and any important departures from the intended design.

State the main limitation.
Explain whether coverage, nonresponse, measurement, or another issue restricts how far the result can be generalized.

Worked Reports

Worked Example: A Random Sample of City Residents

A hypothetical city has a population of 80,000 adult residents. A simple random sample of 400 adults is asked whether they support a proposed park renovation. Of those sampled, 236 say yes. Construct and report a 95% Wald interval, assuming the responses are recorded accurately.

The outcome is “supports the renovation,” and the target population is all adult residents of the city. The estimate is

\hat{p}=\frac{236}{400}=0.590.

The sample was selected at random, and \(400\) is less than 10% of \(80{,}000\), so the usual independence condition for sampling without replacement is reasonable. There are 236 successes and \(400-236=164\) failures, both well above 10. These checks support using the Wald approximation.

For a 95% confidence level, use \(z^*=1.96\). The estimated standard error and margin of error are

SE=\sqrt{\frac{0.590(1-0.590)}{400}} =\sqrt{\frac{0.2419}{400}} =\sqrt{0.00060475} \approx 0.02459,

ME=1.96(0.02459)\approx 0.04820.

Thus the interval is

0.590\pm 0.04820=(0.5418,\ 0.6382).

Report: In a hypothetical simple random sample of 400 adult residents of the city, 59.0% supported the proposed park renovation. The 95% confidence interval for the proportion of all adult city residents who support it is 54.2% to 63.8%. The interval reflects sampling uncertainty under the stated conditions; it does not account for inaccurate responses or any problem with how the question was understood.

The report gives the population and sampling method rather than saying only “adults.” Its final sentence identifies a limitation that the interval itself cannot assess.

Worked Example: Reporting a 99% Interval

In a hypothetical survey, researchers take a simple random sample of 500 people from a registry of 12,000 eligible voters in a district. A total of 315 sampled voters say they intend to vote in an upcoming local election. Find and report a 99% Wald interval for the registry population.

Here \(n=500\), \(x=315\), and \(\hat{p}=315/500=0.630\). The sample is under 10% of the registry population, and there are 315 successes and 185 failures, so the independence and large-count checks are satisfied. Assume responses are accurate and the registry covers the population the researchers intend to describe.

For 99% confidence, \(z^*=2.576\). Then

SE=\sqrt{\frac{0.630(1-0.630)}{500}} =\sqrt{\frac{0.2331}{500}} =\sqrt{0.0004662} \approx 0.02159,

ME=2.576(0.02159)\approx 0.05562, \qquad 0.630\pm 0.05562=(0.5744,\ 0.6856).

Report: In a hypothetical simple random sample of 500 people from the district’s registry of eligible voters, 63.0% said they intended to vote. The 99% confidence interval for the proportion of eligible voters on that registry who would give that response is 57.4% to 68.6%. The inference depends on accurate responses and on the registry adequately covering the intended population; stated intention may also differ from later voting behavior.

The last sentence matters because the measured outcome is reported intention, not verified turnout. The interval estimates the proportion giving the stated response at the time of the survey; it does not estimate a different outcome automatically.

Worked Example: Reporting a Survey With Nonresponse

A hypothetical organization randomly selects 400 adults from a town directory and invites them to complete a survey about using a new public library service. Only 320 respond, and 142 respondents say they would use the service. Calculate a nominal 95% Wald interval for the respondents and explain the limitation.

For the 320 respondents, \(\hat{p}=142/320=0.44375\). There are 142 successes and \(320-142=178\) failures, so the large-count check is satisfied. Using the respondent count,

SE=\sqrt{\frac{0.44375(1-0.44375)}{320}} =\sqrt{\frac{0.24684}{320}} \approx 0.02777,

ME=1.96(0.02777)\approx 0.05443, \qquad 0.44375\pm 0.05443=(0.3893,\ 0.4982).

This calculation treats the 320 responses as if they provide an appropriate sample for the stated inference. However, the people who respond may differ from those who do not. The reported interval therefore does not, by itself, account for nonresponse bias.

Careful report: Among the 320 adults who responded to a hypothetical survey invitation sent to 400 adults randomly selected from the town directory, 44.4% said they would use the new library service. A nominal 95% Wald interval for the respondent proportion is 38.9% to 49.8%. Because one fifth of those invited did not respond, and their views may differ from respondents’ views, this interval should not be treated as a reliable estimate for all adults in the town without further support for that generalization.

The phrase “respondent proportion” matches what was actually observed. A random invitation does not guarantee that the final group of respondents remains representative after nonresponse.

What the Report Can—and Cannot—Claim

A confidence interval describes uncertainty arising from sampling under a specified procedure and model. It does not automatically capture every source of error. A report should distinguish that sampling uncertainty from limitations that can shift the estimate systematically or change the population to which it applies.

Reporting element	What to state	Why it matters
Estimate	The sample proportion, preferably as a percentage in context	Shows the observed result around which the interval is built
Interval and level	Both endpoints and the confidence level	Conveys the range of plausible values under the procedure and its uncertainty level
Population	The precise group the proportion concerns	Prevents the result from being generalized to an unspecified group
Sampling method	For example, a simple random sample or a survey of volunteers	Helps readers judge whether the method supports the intended inference
Limitation	Relevant concerns such as nonresponse, coverage, or measurement	Clarifies uncertainty or bias not represented by the interval endpoints

Use units consistently: endpoints such as 54.2% and 63.8% are percentages, while 0.542 and 0.638 are proportions. If you give the sample estimate as a percentage, report interval endpoints in percentage terms as well. Round enough to communicate the precision without suggesting that the endpoints are exact.

The confidence level should be attached to the interval, not to a probability claim about the fixed population proportion after the interval has been calculated. A suitable interpretation is that the procedure, when repeated under its conditions, captures the population proportion at the stated rate in the long run. For an individual report, say that the interval estimates the population proportion; do not claim there is a 95% probability that the fixed proportion lies in the particular calculated interval.

Common Mistake

Wrong reasoning: “The survey was random, so the 95% interval applies to everyone in the region.” Random selection does not make the population unlimited. It supports inference to the population represented by the sampling frame, subject to conditions and limitations.

Correction: Name the population actually sampled, such as “adults listed in the town directory.” Then explain whether the directory covers the intended population and whether nonresponse or other problems could limit generalization. A careful report is often more informative than a broader but unsupported claim.

Key takeaway: A strong report links the estimate and interval to a clearly named population and sampling method. State what the interval estimates, and disclose limitations that the interval calculation does not resolve.

Check Your Understanding

For each question, focus on what a reader needs in order to interpret a proportion interval responsibly.

A report gives an estimate of 42% and a 95% interval from 36% to 48%. What additional information is needed to understand what population the interval describes?
Why should a report name the sampling method rather than give only the sample size?
A random sample is drawn from a directory, but many selected people do not respond. What limitation should the report mention?
In a report, what does the confidence level describe, and what does it not mean about the particular calculated interval?
A sample contains 18 successes and 182 failures. Does the usual large-count check for a Wald interval pass? What else should be checked before reporting that interval?