Constructing a Confidence Interval for a Difference in Two Proportions: Study Guide (Statistics 3.10)

From one proportion to a difference

In Topic 3.3 you estimated one population proportion with an interval of the form point estimate ± margin of error. Now the question compares two groups: how much larger (or smaller) is the proportion in one population than in another? The parameter is the difference p1 − p2.

The point estimate is the difference in sample proportions, p̂1 − p̂2. In Topic 3.9 you saw that, with independent samples, p̂1 − p̂2 is unbiased for p1 − p2 and has standard deviation √[p1(1 − p1)/n1 + p2(1 − p2)/n2]. You cannot calculate that standard deviation now, because p1 and p2 are unknown. So, as in Topic 3.3, you replace each p with its sample estimate. This gives the two-sample z-interval for a difference between population proportions:

(p̂1 − p̂2) ± z √[ p̂1(1 − p̂1)/n1 + p̂2(1 − p̂2)/n2 ]*

This guide builds the interval in four steps: identify the procedure and the parameter, check the conditions, calculate, and check the result. What the interval and the confidence level mean, and how to use them to judge a claim, is Topic 3.11.

Step 1: name the procedure and define the parameter

Write the procedure by name: “two-sample z-interval for a difference between population proportions” (or for p1 − p2).

Then define the parameter. A full definition has four parts:

• that it is a difference in proportions, with the order stated;
• the response variable (what counts as a success);
• population 1, named in context;
• population 2, named in context.

Weak: “p1 − p2 = difference in the proportions who said yes.” Strong: “p1 − p2 = the proportion of all workers in Fenwick who work from home at least one day a week, minus the proportion of all workers in Harrowgate who do.”

Use subscripts that remind you of the groups (pF − pH) if that helps. The parameter is always about the populations (or, in an experiment, the treatments), never the samples.

Step 2: check the three conditions

Condition	What to check	Why it matters
Randomization	Two independent random samples, one from each population, or a randomized experiment (treatments randomly assigned).	Makes p̂1 − p̂2 unbiased and the two groups independent.
10%	When sampling without replacement: n1 ≤ 10% of N1 and n2 ≤ 10% of N2. Not needed for a randomized experiment.	Keeps observations within each sample close enough to independent for the standard-error formula.
Normality (large counts)	The observed successes and failures in both groups are at least 10: n1p̂1, n1(1 − p̂1), n2p̂2, n2(1 − p̂2).	Makes the sampling distribution of p̂1 − p̂2 approximately normal, so z* is the right multiplier.

Two reminders:

• The large-counts check uses the observed counts, because p1 and p2 are unknown. In Topic 3.9 the proportions were given, so you used expected counts.
• There are four counts. One count below 10 is enough to make the z-interval unreliable.

Step 3: standard error, margin of error and the interval

Standard error. Replacing p1 and p2 by p̂1 and p̂2 in the standard deviation formula gives the standard error of the difference:

SE(p̂1 − p̂2) = √[ p̂1(1 − p̂1)/n1 + p̂2(1 − p̂2)/n2 ]

It estimates how far p̂1 − p̂2 typically lands from p1 − p2 in repeated sampling. Add the two variance terms, then take one square root.

Critical value. z* comes from the standard normal distribution exactly as in Topic 3.3: the area to the left of z* is (1 + C)/2. Common values are 1.645 (90%), 1.960 (95%), 2.326 (98%) and 2.576 (99%).

Margin of error. MOE = z* × SE. It is half the width of the interval.

Interval. (p̂1 − p̂2) − MOE to (p̂1 − p̂2) + MOE.

Worked example 1: two independent random samples, 95%

Question. Two fictional cities want to compare home working. In Fenwick (52,000 workers), a random sample of 400 workers found 172 who work from home at least one day a week. In Harrowgate (41,000 workers), an independent random sample of 350 workers found 119. Construct a 95% confidence interval for the difference in the proportions of all workers in the two cities who work from home at least one day a week.

Identify. Two-sample z-interval for a difference between population proportions. Let pF − pH = the proportion of all Fenwick workers who work from home at least one day a week, minus the proportion of all Harrowgate workers who do.

Check conditions.

1. Randomization: independent random samples were taken from the workers of each city.
2. 10%: 400 ≤ 10% of 52,000 = 5,200 and 350 ≤ 10% of 41,000 = 4,100. ✓
3. Normality: Fenwick has 172 successes and 228 failures; Harrowgate has 119 successes and 231 failures. All four are at least 10. ✓

Calculate.

1. p̂F = 172 ÷ 400 = 0.43 and p̂H = 119 ÷ 350 = 0.34.
2. Point estimate: p̂F − p̂H = 0.09.
3. SE = √(0.43 × 0.57 ÷ 400 + 0.34 × 0.66 ÷ 350) = √(0.0006128 + 0.0006411) ≈ 0.03541.
4. For 95% confidence, z* = 1.960.
5. MOE = 1.960 × 0.03541 ≈ 0.0694.
6. Interval: 0.09 ± 0.0694 = (0.021, 0.159).

Check. A calculator’s two-proportion z-interval with x1 = 172, n1 = 400, x2 = 119, n2 = 350 and C-level 0.95 gives the same interval. It is centred on 0.09 and its half-width is the margin of error.

Conclude (preview of Topic 3.11). We are 95% confident that the interval from 0.021 to 0.159 captures the difference (Fenwick − Harrowgate) in the proportions of all workers who work from home at least one day a week.

Worked example 2: a randomized experiment, 90%

Question. A fictional garden centre tests whether soaking seeds overnight helps them germinate. It randomly assigns 300 seeds to two treatments: 150 are soaked and 150 are planted dry. After two weeks, 126 soaked seeds and 108 dry seeds have germinated. (a) Construct a 90% confidence interval for the difference in germination proportions (soaked − dry). (b) What interval do you get for dry − soaked? (c) Would a 95% interval be wider or narrower?

(a) Identify. Two-sample z-interval for p1 − p2. Let pS − pD = the proportion of seeds like these that would germinate if soaked, minus the proportion that would germinate if planted dry.

Check conditions.

1. Randomization: the two treatments were randomly assigned to the seeds.
2. 10%: not needed, because this is a randomized experiment and the seeds were not sampled from a population.
3. Normality: 126 and 24 (soaked), 108 and 42 (dry). All four are at least 10. ✓

Calculate.

1. p̂S = 126 ÷ 150 = 0.84 and p̂D = 108 ÷ 150 = 0.72. Point estimate 0.12.
2. SE = √(0.84 × 0.16 ÷ 150 + 0.72 × 0.28 ÷ 150) = √(0.000896 + 0.001344) ≈ 0.04733.
3. For 90% confidence, z* = 1.645.
4. MOE = 1.645 × 0.04733 ≈ 0.0778.
5. Interval: 0.12 ± 0.0778 = (0.042, 0.198).

(b) Reversing the order gives point estimate −0.12 and interval (−0.198, −0.042). Both intervals say the same thing: soaked seeds germinated at a higher rate in this experiment. Choose one order, state it, and keep it.

(c) Wider. A 95% interval uses z* = 1.960 instead of 1.645, so the margin of error grows to about 0.0928 and the interval becomes (0.027, 0.213).

Worked example 3: reading an interval, and when not to build one

Question. (a) A report gives a 95% confidence interval of (−0.018, 0.142) for p1 − p2. Find the point estimate, the margin of error and the standard error. (b) In a small trial, 40 volunteers are randomly assigned to each of two exercise plans; 33 and 24 complete their plans. Should you construct a two-sample z-interval?

(a) The interval is symmetric about the point estimate.

1. Point estimate: (−0.018 + 0.142) ÷ 2 = 0.062.
2. Margin of error: (0.142 − (−0.018)) ÷ 2 = 0.160 ÷ 2 = 0.080.
3. Standard error: 0.080 ÷ 1.960 ≈ 0.0408.

The interval contains negative and positive values. Topic 3.11 explains what that tells you about the claim “the proportions differ”.

(b) Count successes and failures in both groups: 33 and 7, 24 and 16. The first group has only 7 failures, which is less than 10. The normality condition fails, so the sampling distribution of p̂1 − p̂2 may not be approximately normal and a z-interval is not appropriate. Say this clearly. A larger experiment would fix the problem.

Common misconceptions

• Pooling the proportions. For a confidence interval, each sample keeps its own p̂ in the standard error. Pooling is for a significance test that assumes p1 = p2.
• Defining the parameter with the samples. “The difference between the 400 Fenwick workers and the 350 Harrowgate workers” is p̂F − p̂H, a statistic. The parameter is about all workers in each city.
• No order of subtraction. Without it, a reader cannot tell what a positive or negative endpoint means.
• Checking only two counts, or using n ≥ 30. Check all four observed counts against 10.
• Checking the 10% condition in an experiment. It is not needed; random assignment is what matters.
• Adding the two standard errors or the two margins of error. Add the variance terms under one square root.
• “A negative endpoint is impossible.” The parameter is a difference, so negative values are possible.
• Using a t critical value. Intervals for proportions use z*.

Where this leads

Next, Topic 3.11 shows how to interpret the interval and the confidence level in context, and how to use the interval (for example, whether it contains 0) to justify a claim about two populations: see Justifying a Claim Based on a Confidence Interval for Two Proportions. Try the practice questions now, then use the revision notes and the checklist. For the sampling distribution behind the interval, revisit Topic 3.9; for critical values, revisit Topic 3.3.