What a One-Proportion Test Output Tells You
Statistical software can calculate a one-proportion test quickly, but its output is useful only if you can tell what question it answered. A line labeled “p-value” does not, by itself, tell you whether the test was one-sided or two-sided, whether the calculation used a normal approximation, or which proportion was treated as the null value. These details determine how the result should be interpreted.
A reliable reading starts with the hypotheses and the observed count. Then identify the reported sample proportion, the test method, the test statistic if present, and the p-value. Check that these pieces agree before drawing a conclusion. The examples below use hypothetical data.
Core Quantities in the Output
Suppose \(n\) independent trials each have a success or failure outcome, and let \(x\) be the number of successes observed. The sample proportion is the observed fraction of successes. The null hypothesis specifies a value \(p_0\) for the population success proportion \(p\). The alternative hypothesis specifies the direction or directions of difference being tested.
The word “extreme” depends on the alternative. For \(H_a:p>p_0\), the p-value is an upper-tail probability; for \(H_a:p<p_0\), it is a lower-tail probability. For \(H_a:p\ne p_0\), the p-value is two-sided, with the precise calculation depending on the test method. The p-value is not the probability that \(H_0\) is true.
Recognizing the Test Method
A one-proportion z test uses a normal approximation to the sampling distribution of \(\hat{p}\) under the null. For the usual z procedure, the Bernoulli outcomes should be independent, the null proportion should be strictly between zero and one, and the expected numbers of successes and failures under \(H_0\) should be sufficiently large for the normal approximation to be appropriate. A common diagnostic is \(np_0\) and \(n(1-p_0)\); software cannot make an inadequate approximation reliable merely by printing a z statistic.
When those conditions are not suitable for the approximation, an exact binomial test calculates probabilities from the binomial distribution under the null. Its basic model is a fixed number \(n\) of independent trials, each with the same success probability \(p_0\) under \(H_0\). For a two-sided exact test, software packages can use different standard definitions of which outcomes count as “at least as extreme.” The output or documentation should identify the convention when it matters.
This relationship is also a useful consistency check. Since \(\hat{p}=x/n\), an output that reports \(n\), \(x\), \(p_0\), \(\hat{p}\), and \(z\) should satisfy
The displayed statistic may differ slightly from a hand check because software rounds the printed values. A mismatch larger than rounding suggests that a different null value, alternative, or method may have been used.
Worked Examples
Worked Example: Reading an Upper-Tail Z Test
A hypothetical poll records 118 successes among 200 independent trials. The question is whether the success proportion exceeds 0.50. The output reports \(p_0=0.50\), \(\hat{p}=0.5900\), \(z=2.546\), and a p-value of 0.0055, using a one-proportion z test. Check the reported values and interpret them.
The hypotheses are \(H_0:p=0.50\) and \(H_a:p>0.50\). The observed sample proportion is
Under the null, the expected success and failure counts are \(200(0.50)=100\) each, so both are sufficiently large for the normal approximation. Assuming the trials are independent, the z method is appropriate for this example.
The null standard error is \(\sqrt{0.50(1-0.50)/200}=0.03536\). Substituting gives
This agrees with the reported statistic. Because the alternative is upper-tailed, the p-value is the standard normal probability above 2.546, about 0.0055. If the null proportion were 0.50, the probability of obtaining a sample proportion at least as high as 0.5900 in 200 trials would be about 0.55%. This is evidence against \(H_0\) in favor of a larger proportion. The p-value is not a 0.55% probability that the null hypothesis is true.
Worked Example: Reading a Two-Sided Exact Binomial Test
In a hypothetical set of 20 independent trials, 15 are successes. Software reports \(\hat{p}=0.7500\), a null proportion of 0.50, a two-sided exact binomial p-value of 0.0414, and no z statistic. Interpret the output, using the convention that outcomes with null probabilities no greater than the observed outcome’s null probability are included in the two-sided p-value.
The hypotheses are \(H_0:p=0.50\) and \(H_a:p\ne 0.50\), and the reported sample proportion checks out:
Under \(H_0\), the number of successes \(X\) follows a binomial distribution with \(n=20\) and \(p_0=0.50\). With this symmetric null distribution, the observed outcome \(X=15\) has the same probability as \(X=5\). The outcomes at least as extreme under the stated probability-ordering convention are \(X\ge15\) and \(X\le5\). Therefore,
This matches the reported value after rounding. The method label “exact binomial” explains why there is no z statistic to interpret: the p-value was calculated from binomial probabilities, not from a normal approximation. Under the null model, outcomes at least as extreme as 15 successes out of 20 have probability about 4.14% using this two-sided convention. The output provides evidence against \(p=0.50\) at the 5% significance level, but does not establish that the population proportion is exactly 0.75.
Worked Example: A Large P-Value From a Lower-Tail Test
A hypothetical study records 52 successes in 80 independent trials. The research question is whether the success proportion is less than 0.60. Software reports a one-proportion z test with \(p_0=0.60\), \(\hat{p}=0.6500\), \(z=0.9129\), and p-value 0.8193. Explain why the p-value is large even though the statistic is positive.
The hypotheses are \(H_0:p=0.60\) and \(H_a:p<0.60\). The sample proportion is \(52/80=0.6500\). Under the null, the expected counts are \(80(0.60)=48\) successes and \(80(0.40)=32\) failures. With independent trials and both expected counts large, the z approximation is reasonable.
The null standard error and statistic are
Because the alternative is lower-tailed, the p-value is the standard normal probability below 0.9129, approximately 0.8193. The positive statistic indicates that the observed proportion is above the null value, opposite the direction of the alternative. A result that far below \(p_0\) would be unusual under the null; this result is not. The output is internally consistent, and it provides little evidence for the claim that \(p<0.60\).
A Practical Reading Sequence
When examining a software report, read it in an order that keeps the statistical question attached to the numbers. Software labels vary, so do not rely on a single field name.
Identify the null value and whether the alternative is greater than, less than, or different from that value.
Check that the reported number of successes and sample size correspond to the data, and verify that \(\hat{p}=x/n\).
Look for labels such as “one-proportion z test,” “normal approximation,” or “exact binomial.” Check that the method’s assumptions are reasonable.
For a z test, verify the sign and approximate size of \(z\), then confirm that the p-value uses the tail specified by \(H_a\). An exact test may report no z statistic.
Describe the strength of evidence against \(H_0\) and relate it to the research question. Do not describe the p-value as a probability that \(H_0\) is true.
Common Mistakes and Why the Details Matter
A common error is to treat every printed p-value as though it answers the same question. For example, if \(z\) is positive but the alternative is \(p<p_0\), using the upper-tail probability would answer the wrong question. In the third example, the correct lower-tail p-value is 0.8193, not the much smaller upper-tail probability. Always read the alternative before interpreting the p-value.
Another mistake is to call \(\hat{p}\) the null proportion. The sample proportion is calculated from the observed data; \(p_0\) is specified by the null hypothesis. They play different roles in the test statistic. Likewise, a reported method is substantive information, not a software detail to skip: a z test uses an approximation, while an exact binomial test uses the binomial model. When two programs give different exact two-sided p-values, check their definitions and settings before assuming one result is an error.
Check Your Understanding
For each question, identify the output detail that supports your answer.
- A report gives \(x=36\), \(n=60\), and \(\hat{p}=0.60\). Verify the sample proportion and explain how it differs from a null value of 0.50.
- A z test reports \(z=-1.8\) for \(H_a:p>p_0\). Is the observed proportion above or below the null value? Which tail determines the p-value?
- What does an “exact binomial” method label tell you about how the p-value was computed? Should you expect a z statistic to be essential to its interpretation?
- Why should you check a two-sided exact test’s definition before comparing its p-value across software packages?
- In context, what does a p-value of 0.03 mean under the null model, and what does it not mean?