Two Proportions: Estimate Separately, Test a Common Null

These lessons and problems are newly written for the revised course. Worked examples show complete reasoning; the application questions ask you to make and defend your own choices.

Define the comparison before calculating

Two schools want to compare breakfast preference. The same binary response is measured in independent SRSs: school A selects 140 students from 4,000 and finds 98 supporters; school B selects 180 from 5,000 and finds 90. Define p_A and p_B as the population proportions preferring the proposed breakfast menu in the respective enrolled-student populations. The comparison p_A−p_B estimates A minus B. Naming the order prevents an unexplained positive or negative answer.

For estimating the difference, choose a two-sample z-interval for a difference between population proportions. For testing equality, choose a two-sample z-test for that difference. Both use a binary response in two independent groups, but estimation and an equality-null test make different variance assumptions. The population parameters are p_A and p_B, while 0.70 and 0.50 are observed sample proportions.

Randomized experiments can also compare two binary responses under assigned treatments. Then state the treatment-response proportions and the scope supported by the experiment. Random assignment addresses causal comparison; it does not by itself make volunteers representative of every school student. If the same student answers twice in two dependent conditions, the observations are not two independent samples; the ordinary independent two-proportion procedure needs a different design.

The full condition set uses both groups

For an interval, randomization comes from two independent random samples or a randomized experiment. For sampling without replacement, each sample must satisfy its own bound: n_A≤0.10N_A and n_B≤0.10N_B. This population-sampling condition is unnecessary for a randomized experiment’s treatment assignment. Do not invent an unprovided population size for volunteers merely to check a box.

The normality check uses four observed counts for an interval: successes and failures in A, then successes and failures in B. All four should be at least 10. For the worked schools, SRSs are independent; 140≤400 and 180≤500; observed counts are A:98 successes and 42 failures, B:90 and 90. Every required count passes.

An equality-null test shares the randomization and population-fraction requirements, including the experiment exception, but models a common population success rate. Pool successes to estimate that common null value:

The test checks four expected null counts: n_A p̂_c, n_A(1−p̂_c), n_B p̂_c and n_B(1−p̂_c), all at least 10. Here they are 82.25, 57.75, 105.75 and 74.25. Pooling is weighted by sample size, not the unweighted mean of the two observed percentages. A count failure in either group can undermine the normal approximation; good counts in the other group do not repair it.

Keep separate variance estimates for estimation

The point estimate p̂_A−p̂_B is 0.70−0.50=0.20. With no assumption that the unknown population proportions are equal, each sample estimates its own variance contribution. Add these contributions for independent groups and take the square root to obtain the standard error of their difference.

SE=√[0.70(0.30)/140+0.50(0.50)/180]≈0.0537484. With 95% z*=1.96, MOE≈0.1053468 and interval≈(0.0946532,0.3053468). The margin is critical value times SE; it is half the full width. We are 95% confident that A’s population breakfast-preference proportion exceeds B’s by about 9.47 to 30.53 percentage points. A difference of proportions is expressed in percentage points when multiplied by 100; it is not the relative percentage change from one proportion to the other.

Approximately 95% of intervals formed from repeated independent samples of these same sizes from these same populations would capture p_A−p_B. This particular interval may or may not capture the fixed difference. Since zero is excluded and all values are positive, the interval supports a population difference with A higher. A separate interval (−0.08,0.04) would include zero and provide insufficient evidence of a difference; it would not prove equality. Reversing the order B−A negates the point estimate and gives endpoints (−0.3053468,−0.0946532), preserving information and zero exclusion.

Write every hypothesis about the population

For “do the schools differ?”, the null can be H₀:p_A=p_B or equivalently H₀:p_A−p_B=0. The two-sided alternative is Hₐ:p_A≠p_B or Hₐ:p_A−p_B≠0. A question prespecified as “is A greater?” uses p_A>p_B or p_A−p_B>0. A question prespecified as “is A smaller?” uses p_A<p_B or p_A−p_B<0. Those are the less, greater and not-equal alternatives; choosing one after seeing the sample sign changes the study’s intended question.

Equal population proportions do not require equal observed sample proportions. Sampling variation produces differences even under the null. Pooling successes models that common unknown population rate under the null, which is why the test SE differs from the interval SE. Its standardized statistic compares the observed difference with zero:

For the worked data, pooled null SE≈0.0554741 and z≈3.6052830. Under the equality null and verified conditions, z is approximately standard normal. A normal table or technology gives upper-tail P(Z≥3.6052830)≈0.000155906. “Differ” uses both tails, so p≈0.000311813. A higher-A alternative would instead use the upper tail alone; a lower-A alternative uses the lower tail, which is large for this positive statistic. The alternative determines extremeness.

A probability and a decision answer different questions

The two-sided p-value means: assuming the population breakfast-preference proportions in schools A and B are equal, the probability of a standardized sample difference at least this extreme in either direction is about 0.0003118. The equality assumption and both school populations belong in the interpretation. This does not assign a probability that equality itself is true.

At a prespecified α=0.05, p≤α gives reject H₀. There is convincing statistical evidence that the enrolled-student population breakfast-preference proportions differ between schools A and B. The conclusion names both populations, follows the not-equal alternative and uses qualified evidence language. Independent random samples support population comparison, but this observational school comparison does not establish that school membership causes preference.

For another valid equality test with p=0.18 and α=0.05, p>α gives fail to reject. The correct conclusion is insufficient evidence of a population-proportion difference, not that the schools have identical preferences. A complete test conclusion compares p with α explicitly and answers the investigative question; reporting only a calculator probability leaves the question unanswered.

Independent Application

Work through each investigation before opening the separate solutions. State parameters, show calculations and connect your reasoning to the actual study.

Original generated practice for the Fall 2026 course.

Do two towns differ in library support?

Independent SRSs are selected without replacement: 250 adults from town A’s 20,000 adults and 200 from town B’s 15,000. The same question asks whether they favor a library expansion. In A, 150 favor it; in B, 90 do. Estimate p_A−p_B with 95% confidence, using z*=1.96.

Part a

Define both population proportions and their ordered difference. Choose the interval procedure and justify every condition, giving all four observed success/failure counts and each population fraction.

Your reasoning

Part b

Calculate the point estimate, SE, MOE and interval. Explain why each group contributes its own observed variance term and why the order of subtraction must be retained.

Your reasoning

Part c

Interpret the interval and confidence level, then justify a claim about the difference in town support. Distinguish a proportion-point difference from a relative percentage increase.

Your reasoning

Worked answer and rubric

Test a common support proportion

Two independent SRSs of 240 adults each are selected without replacement from two town populations of 40,000 adults each. In town A, 144 favor a housing proposal; in town B, 96 favor it. Test whether the population support proportions differ at α=0.05. Standard normal upper-tail output from the full-precision calculated statistic z≈4.381780460 is approximately 0.0000058856695488.

Part a

Define the parameters and select the test. State both equivalent null forms. Give the two-sided alternative in both equivalent forms, and the less-than and greater-than alternatives in both forms for different investigative questions.

Your reasoning

Part b

Compute the pooled proportion and justify all three conditions using it. Give all four expected success/failure counts. Explain why these differ from the conditions used to construct an interval.

Your reasoning

Part c

Calculate z and the two-sided p-value, naming the null reference distribution and how the probability is found. Show the combined SE rather than substituting an unpooled interval SE.

Your reasoning

Part d

Interpret the p-value in context, make the formal decision and answer the housing-support question with a qualified population conclusion.

Your reasoning

Worked answer and rubric

Compare two randomly assigned recycling messages

Two hundred volunteers are randomly assigned, 100 each, to recycling message A or B. Each then sorts one item; 65 in A and 45 in B sort it correctly. Let p_A,p_B describe the success proportions for comparable volunteers under the respective messages. This is a randomized experiment, not an SRS of all city residents. Use a 95% interval with z*=1.96 and a prespecified upper alternative p_A>p_B with α=0.05. Normal upper-tail output at z=2.8426762 is 0.00223682.

Part a

Choose the interval and test procedures, define the ordered parameter difference, and justify all conditions for each. Explain whether the 10% population-sampling condition applies. Calculate both the four observed counts and four pooled null expected counts.

Your reasoning

Part b

Calculate the 95% interval with its SE/MOE, then calculate the upper-tail z-test using its own SE. Explain why the two SEs need not match.

Your reasoning

Part c

Interpret the interval, confidence level and p-value. Make the formal test decision and a defensible claim about the messages, addressing both causal evidence and population generalization.

Your reasoning

Worked answer and rubric

Reading signs and zero honestly

Independent representative samples compare the proportions of adults choosing electric transport in towns X and Y. The parameter order is p_X−p_Y. A 95% interval is (−0.11,0.03). A valid two-sided equality test from those same observations has p=0.18. In a separate study with the same named parameter order, a 95% interval is (0.04,0.20); that study’s valid two-sided equality-test p=0.008. Each study used α=0.05.

Part a

Interpret each interval in its town context, explain its confidence meaning and make a claim about population differences. Describe the new interval for p_Y−p_X and explain why reversing the subtraction does not create stronger evidence.

Your reasoning

Part b

Interpret each p-value under the equal-proportion null, make both formal decisions and give qualified investigative conclusions. Explain why p=0.18 does not show the two populations are identical.

Your reasoning

Worked answer and rubric