Check Conditions for Proportion Procedures

Why Check Conditions Before Calculating?

A one- or two-proportion procedure uses a model for how sample proportions vary from sample to sample. The formulas do not make that model appropriate by themselves. Before calculating a test statistic or confidence interval, check how the data were collected, whether observations can reasonably be treated as independent, and whether the success and failure counts are large enough for a normal approximation.

These checks answer different questions. A suitable sampling or assignment design supports the kind of inference being made. Independence concerns how observations relate to one another. Success-failure counts concern whether the sampling distribution is adequately approximated by a normal distribution. Passing one check does not compensate for failing another.

Design and Independence

Begin by identifying the population or groups named by the question and how the individuals entered the data. A random sample can support inference to the population from which it was selected, provided the other conditions are reasonable. Random assignment in an experiment supports a comparison of assigned treatments for the experimental units in that study; by itself, it does not make those units representative of a wider population. A convenience sample or voluntary-response sample does not become representative merely because it is large.

Independence also depends on design. For a one-proportion procedure, observations should be independent, or close enough to independent for the intended approximation. For a two-proportion procedure, the observations within each group should be independent, and the two groups should be independent of one another. Random sampling or random assignment can support these assumptions, but the researcher should also look for repeated measurements, clustering, or other connections between observations.

When sampling without replacement from a finite population, selecting one individual changes who remains available. A common condition for treating observations as approximately independent is that the sample size is no more than 10% of the population size: \(n\leq 0.10N\). This is called the 10% condition. It is a practical approximation, not a claim that the draws are literally independent.

For a two-group procedure, check the design as well as the sample sizes. If the same person supplies an outcome in each group, or people are deliberately matched into pairs, the two sets of outcomes are not independent groups. A two-proportion procedure for independent groups would ignore that pairing and is not appropriate.

Why the 10% Condition Helps

The effect of sampling without replacement can be seen from the variance of a sample proportion. Suppose a finite population has size \(N\), with population proportion \(p_N\) of successes, and a simple random sample of size \(n\) is taken without replacement. Let \(I_j\) equal 1 if the \(j\)th sampled individual is a success and 0 otherwise. Then \(\hat{p}=(I_1+\cdots+I_n)/n\).

Each indicator has variance \(p_N(1-p_N)\). For two distinct draws, the indicators have covariance \(-p_N(1-p_N)/(N-1)\): drawing a success slightly reduces the chance that the next draw is a success. Adding the variances and covariances gives the variance of the sample proportion:

The factor \((N-n)/(N-1)\) reduces the variance compared with the independent-draw expression \(p_N(1-p_N)/n\). At the 10% limit, the square root of this factor is close to \(\sqrt{0.9}\), or about 0.949. Thus the standard deviation is reduced by only about 5% or less when the sample is at most 10% of the population. The condition is a useful rule for deciding when that dependence is small enough to ignore in routine procedures.

Success-Failure Conditions Depend on the Procedure

The normal approximation is checked using different counts for tests and confidence intervals. Use the condition before proceeding with a routine normal-based procedure. A common introductory rule is that each relevant success and failure count should be at least 10.

• One-proportion confidence interval: check the observed counts \(x\) and \(n-x\). Both should be at least 10.
• One-proportion test: use the null proportion \(p_0\) in \(H_0\). Check the expected counts \(np_0\) and \(n(1-p_0)\), not just the observed counts.
• Two-proportion confidence interval: check observed successes and failures separately in each group: \(x_1,\ n_1-x_1,\ x_2,\) and \(n_2-x_2\).
• Two-proportion test of equal proportions: under \(H_0:p_1=p_2\), pool the observed successes to estimate the common proportion, \(\hat{p}_{\text{pool}}=(x_1+x_2)/(n_1+n_2)\). Check the four expected counts based on that pooled estimate.

For a test, the null hypothesis specifies the model used to assess expected counts. For a confidence interval, there is no single null value to use, so the check uses the observed counts in the sample or samples. These checks are not interchangeable. When the relevant counts are too small, the usual normal-based procedure is not justified by this rule; the analyst should not proceed as if the condition had passed.

A Practical Checking Sequence

Common Mistakes and Why the Checks Matter

• Using observed counts for a test. A large observed success count does not guarantee large expected counts under \(H_0\). Calculate test expectations from the null model.
• Using pooled counts for a confidence interval. The pooled proportion belongs to the null model for a test of equality. For a two-proportion interval, check successes and failures observed in each group separately.
• Calling any large sample representative. A large convenience sample can still be biased. Sample size does not replace an appropriate sampling design.
• Treating paired data as independent groups. If the same individuals contribute both outcomes, a procedure for two independent proportions does not match the design.
• Checking only the count condition. Adequate counts do not fix dependence, poor sampling, or an unsuitable design. Each condition addresses a different source of difficulty.

The purpose of condition checking is not to certify that a study is flawless. It is to make the assumptions behind a procedure visible and to determine whether its standard approximation is credible for the data at hand. A clear report names the design, explains the independence assessment, shows the relevant counts, and identifies any condition that is not met.