Build the method
The usual two-proportion test of equality estimates a common null proportion by pooling successes and sample sizes. A confidence interval generally keeps separate observed proportions in its estimated standard error. These constructions answer different questions and should not be mixed without justification.
Worked example
With eleven successes among twenty-one in group A and twenty-six among fifty-one in B, the pooled estimate is 37/72. An unweighted average of 11/21 and 26/51 gives each group equal influence despite their unequal numbers of observations. Pooling people gives the weighted estimate.
Compare the alternatives
For observed proportions 0.6 and 0.4 with sample sizes 100 and 200, the estimated variance of the difference is 0.24/100+0.24/200=0.0036. Its standard error is 0.06. Variances add for independent differences; subtracting the components can even produce an impossible negative variance.
Avoid a specific trap
An interval of 0.02 to 0.08 for pA−pB means A is estimated to exceed B by two to eight percentage points. It does not mean A is two to eight percent larger relative to B. The latter would divide the difference by B and requires different information.
Transfer check with explanation
Transfer: increasing independent sample sizes generally raises power for a fixed true effect and significance level. Reducing the significance level instead makes rejection harder and generally lowers power. Before promising improvement, check that added observations do not introduce selection bias or dependence.
Two independent samples have 11/21 and 26/51 successes. For H0:p1=p2, what pooled estimate of the common proportion is used in the usual two-proportion test?
- Pool people, adding both success counts and both sample sizes.
- Combined successes 37; combined n 72; pooled rate 37/72.
- Why an alternative fails: Average the sample proportions equally despite unequal sample sizes.
Compare the other answer choices
Average the sample proportions equally despite unequal sample sizes.
Use only the first sample and discard the second sample successes and trials.
Use only the second sample and discard the first sample successes and trials.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Statistics Course and Exam Description, effective Fall 2026