Two-Proportion z Interval: Difference in Proportions

AP Statistics Topic Guide

Two-Proportion z Interval: Difference in Proportions

Estimate the difference between two population proportions while preserving group order, using separate sample proportions in the standard error and interpreting the interval in context.

StatusCurrent AP Statistics inference method

Main keywordtwo proportion z interval

Worked analysis24 cases

Focused practice20 MCQs + 8 FRQs

Study progress0 completed

Two-Proportion z Interval: Difference in Proportions: direct answer

In a two proportion z interval problem, begin by identifying the population target and the inferential role of the data. Estimate the difference between two population proportions while preserving group order, using separate sample proportions in the standard error and interpreting the interval in context.

This page uses two proportion z interval as its single primary search focus. The lesson, numerical cases, and retained questions are restricted to that intent so the page does not function as a generic inference question bank.

Quick reference for Two-Proportion z Interval: Difference in Proportions

Two-Proportion z Interval: Difference in Proportions quick reference
Parameter	p₁−p₂
Point estimate	p̂₁−p̂₂
Standard error	√[p̂₁(1−p̂₁)/n₁ + p̂₂(1−p̂₂)/n₂]
Interval	(p̂₁−p̂₂) ± z* × SE
Core count check	Successes and failures in both groups
Interpretation target	Difference in population proportions, preserving group order

Concept mastery: two proportion z interval

Define the parameter as an ordered difference

The parameter is p1-p2, not merely “the difference.” Define which population is group 1 before computing anything, because reversing the group order reverses the sign of the estimate and both confidence-interval endpoints.

Compute two sample proportions separately

Calculate p-hat1=x1/n1 and p-hat2=x2/n2. The point estimate is p-hat1-p-hat2. Keep the groups labeled through the arithmetic so the final contextual interpretation uses the same direction.

Use an unpooled standard error for an interval

For a confidence interval, the standard error uses each group’s own sample proportion: sqrt[p-hat1(1-p-hat1)/n1 + p-hat2(1-p-hat2)/n2]. Pooling the two samples belongs to a two-proportion z test under the null hypothesis of equal proportions, not to an interval.

Check success-failure counts in both groups

Each sample needs enough observed successes and failures for the normal approximation. That means four counts must be inspected: x1, n1-x1, x2, and n2-x2. A large combined sample does not rescue one sparse group.

Check independence within and between groups

The two samples or randomized groups should be independent of each other, and observations within each group should be approximately independent. For sampling without replacement, apply the finite-population check separately when relevant.

Interpret zero correctly

If an interval for p1-p2 contains zero, the data do not establish a directional difference at the corresponding two-sided confidence level. If every value is positive, the evidence supports p1>p2; if every value is negative, it supports p1<p2.

Explain the sign before explaining significance

A negative interval is not an error. It means the first population proportion is estimated to be lower than the second under the chosen ordering. Reversing the group labels would produce the same substantive comparison with positive endpoints.

Separate statistical difference from practical importance

An interval can exclude zero yet describe a difference too small to matter in practice. Conversely, a wide interval may contain both practically important differences and zero. Use the endpoints to discuss the range of plausible effect sizes, not only whether zero appears.

Do not interpret the interval as individual outcomes

The interval estimates a difference between population proportions. It does not say that a percentage of individuals fall inside the interval, nor does it predict the response of a randomly selected person.

Preserve the study design when stating scope

Random sampling supports generalization to populations; random assignment supports causal comparison between treatments. The two-proportion calculation alone supplies neither. The conclusion must respect how the two groups were obtained.

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Worked analysis for two proportion z interval

Difference case 1: County Library

Group 1 has 40 of 105 observations meeting the criterion for visitors who attended a weekly program, while group 2 has 40 of 120. Thus p̂₁=0.381, p̂₂=0.333, and the ordered estimate p̂₁−p̂₂=0.048. A 90% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0640. With z*=1.645, the interval is (-0.058, 0.153).

Difference case 2: Public High School

Group 1 has 48 of 112 observations meeting the criterion for seniors who submitted the financial-aid form, while group 2 has 45 of 126. Thus p̂₁=0.429, p̂₂=0.357, and the ordered estimate p̂₁−p̂₂=0.071. A 95% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0633. With z*=1.960, the interval is (-0.053, 0.196).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero lies inside this interval, so the data allow no difference as well as differences in the directions represented by the endpoints. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 2: Public High School context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 3: Municipal Water Office

Group 1 has 57 of 119 observations meeting the criterion for households reporting no service interruption, while group 2 has 51 of 132. Thus p̂₁=0.479, p̂₂=0.386, and the ordered estimate p̂₁−p̂₂=0.093. A 99% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0624. With z*=2.576, the interval is (-0.068, 0.253).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero lies inside this interval, so the data allow no difference as well as differences in the directions represented by the endpoints. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 3: Municipal Water Office context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 4: State Park

Group 1 has 67 of 126 observations meeting the criterion for visitors who used the marked trail system, while group 2 has 58 of 138. Thus p̂₁=0.532, p̂₂=0.420, and the ordered estimate p̂₁−p̂₂=0.111. A 90% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0612. With z*=1.645, the interval is (0.011, 0.212).

Difference case 5: Food Cooperative

Group 1 has 77 of 133 observations meeting the criterion for members who renewed before the deadline, while group 2 has 65 of 144. Thus p̂₁=0.579, p̂₂=0.451, and the ordered estimate p̂₁−p̂₂=0.128. A 95% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0596. With z*=1.960, the interval is (0.011, 0.244).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero is outside this interval, so the plausible differences all have the same sign under this ordering. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 5: Food Cooperative context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 6: University Advising Center

Group 1 has 88 of 140 observations meeting the criterion for appointments starting within ten minutes, while group 2 has 72 of 150. Thus p̂₁=0.629, p̂₂=0.480, and the ordered estimate p̂₁−p̂₂=0.149. A 99% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0577. With z*=2.576, the interval is (-0.000, 0.297).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero lies inside this interval, so the data allow no difference as well as differences in the directions represented by the endpoints. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 6: University Advising Center context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 7: Regional Manufacturer

Group 1 has 100 of 147 observations meeting the criterion for parts meeting the diameter specification, while group 2 has 80 of 156. Thus p̂₁=0.680, p̂₂=0.513, and the ordered estimate p̂₁−p̂₂=0.167. A 90% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0555. With z*=1.645, the interval is (0.076, 0.259).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero is outside this interval, so the plausible differences all have the same sign under this ordering. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 7: Regional Manufacturer context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 8: Public Health Clinic

Group 1 has 65 of 154 observations meeting the criterion for clients returning for the scheduled checkup, while group 2 has 87 of 162. Thus p̂₁=0.422, p̂₂=0.537, and the ordered estimate p̂₁−p̂₂=-0.115. A 95% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0558. With z*=1.960, the interval is (-0.224, -0.006).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero is outside this interval, so the plausible differences all have the same sign under this ordering. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 8: Public Health Clinic context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 9: Urban Recreation Program

Group 1 has 76 of 161 observations meeting the criterion for participants completing the eight-week session, while group 2 has 96 of 168. Thus p̂₁=0.472, p̂₂=0.571, and the ordered estimate p̂₁−p̂₂=-0.099. A 99% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0548. With z*=2.576, the interval is (-0.241, 0.042).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero lies inside this interval, so the data allow no difference as well as differences in the directions represented by the endpoints. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 9: Urban Recreation Program context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 10: School District

Group 1 has 87 of 168 observations meeting the criterion for families responding to the annual survey, while group 2 has 104 of 174. Thus p̂₁=0.518, p̂₂=0.598, and the ordered estimate p̂₁−p̂₂=-0.080. A 90% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0536. With z*=1.645, the interval is (-0.168, 0.008).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero lies inside this interval, so the data allow no difference as well as differences in the directions represented by the endpoints. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 10: School District context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 11: Local Election Office

Group 1 has 100 of 175 observations meeting the criterion for mailed ballots returned before election day, while group 2 has 61 of 180. Thus p̂₁=0.571, p̂₂=0.339, and the ordered estimate p̂₁−p̂₂=0.233. A 95% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0514. With z*=1.960, the interval is (0.132, 0.333).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero is outside this interval, so the plausible differences all have the same sign under this ordering. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 11: Local Election Office context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 12: Energy-Efficiency Pilot

Group 1 has 113 of 182 observations meeting the criterion for homes meeting the target reduction, while group 2 has 69 of 186. Thus p̂₁=0.621, p̂₂=0.371, and the ordered estimate p̂₁−p̂₂=0.250. A 99% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0505. With z*=2.576, the interval is (0.120, 0.380).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero is outside this interval, so the plausible differences all have the same sign under this ordering. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 12: Energy-Efficiency Pilot context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 13: Community Broadband Project

Group 1 has 127 of 189 observations meeting the criterion for households achieving the advertised speed, while group 2 has 77 of 192. Thus p̂₁=0.672, p̂₂=0.401, and the ordered estimate p̂₁−p̂₂=0.271. A 90% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0492. With z*=1.645, the interval is (0.190, 0.352).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero is outside this interval, so the plausible differences all have the same sign under this ordering. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 13: Community Broadband Project context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 14: Regional Bus Network

Group 1 has 80 of 196 observations meeting the criterion for trips arriving within the on-time window, while group 2 has 85 of 198. Thus p̂₁=0.408, p̂₂=0.429, and the ordered estimate p̂₁−p̂₂=-0.021. A 95% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0497. With z*=1.960, the interval is (-0.119, 0.076).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero lies inside this interval, so the data allow no difference as well as differences in the directions represented by the endpoints. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 14: Regional Bus Network context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 15: Campus Dining Service

Group 1 has 93 of 203 observations meeting the criterion for transactions using reusable containers, while group 2 has 94 of 204. Thus p̂₁=0.458, p̂₂=0.461, and the ordered estimate p̂₁−p̂₂=-0.003. A 99% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0494. With z*=2.576, the interval is (-0.130, 0.125).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero lies inside this interval, so the data allow no difference as well as differences in the directions represented by the endpoints. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 15: Campus Dining Service context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 16: Workforce Training Program

Group 1 has 107 of 210 observations meeting the criterion for participants earning the credential, while group 2 has 103 of 210. Thus p̂₁=0.510, p̂₂=0.490, and the ordered estimate p̂₁−p̂₂=0.019. A 90% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0488. With z*=1.645, the interval is (-0.061, 0.099).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero lies inside this interval, so the data allow no difference as well as differences in the directions represented by the endpoints. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 16: Workforce Training Program context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 17: County Recycling Audit

Group 1 has 122 of 217 observations meeting the criterion for sampled loads meeting contamination limits, while group 2 has 112 of 216. Thus p̂₁=0.562, p̂₂=0.519, and the ordered estimate p̂₁−p̂₂=0.044. A 95% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0479. With z*=1.960, the interval is (-0.050, 0.137).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero lies inside this interval, so the data allow no difference as well as differences in the directions represented by the endpoints. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 17: County Recycling Audit context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 18: University Residence Halls

Group 1 has 137 of 224 observations meeting the criterion for rooms passing the first safety inspection, while group 2 has 122 of 222. Thus p̂₁=0.612, p̂₂=0.550, and the ordered estimate p̂₁−p̂₂=0.062. A 99% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0466. With z*=2.576, the interval is (-0.058, 0.182).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero lies inside this interval, so the data allow no difference as well as differences in the directions represented by the endpoints. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 18: University Residence Halls context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 19: Telehealth Pilot

Group 1 has 152 of 231 observations meeting the criterion for appointments completed without rescheduling, while group 2 has 132 of 228. Thus p̂₁=0.658, p̂₂=0.579, and the ordered estimate p̂₁−p̂₂=0.079. A 90% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0452. With z*=1.645, the interval is (0.005, 0.153).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero is outside this interval, so the plausible differences all have the same sign under this ordering. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 19: Telehealth Pilot context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 20: Public Museum

Group 1 has 95 of 238 observations meeting the criterion for visitors using the audio guide, while group 2 has 143 of 234. Thus p̂₁=0.399, p̂₂=0.611, and the ordered estimate p̂₁−p̂₂=-0.212. A 95% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0450. With z*=1.960, the interval is (-0.300, -0.124).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero is outside this interval, so the plausible differences all have the same sign under this ordering. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 20: Public Museum context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 21: Youth Sports League

Group 1 has 110 of 245 observations meeting the criterion for players completing concussion training, while group 2 has 84 of 240. Thus p̂₁=0.449, p̂₂=0.350, and the ordered estimate p̂₁−p̂₂=0.099. A 99% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0442. With z*=2.576, the interval is (-0.015, 0.213).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero lies inside this interval, so the data allow no difference as well as differences in the directions represented by the endpoints. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 21: Youth Sports League context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 22: Rural Pharmacy Network

Group 1 has 126 of 252 observations meeting the criterion for prescriptions filled within the service target, while group 2 has 93 of 246. Thus p̂₁=0.500, p̂₂=0.378, and the ordered estimate p̂₁−p̂₂=0.122. A 90% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0441. With z*=1.645, the interval is (0.049, 0.195).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero is outside this interval, so the plausible differences all have the same sign under this ordering. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 22: Rural Pharmacy Network context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 23: City Tree Program

Group 1 has 142 of 259 observations meeting the criterion for new plantings surviving the first year, while group 2 has 103 of 252. Thus p̂₁=0.548, p̂₂=0.409, and the ordered estimate p̂₁−p̂₂=0.140. A 95% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0438. With z*=1.960, the interval is (0.054, 0.225).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero is outside this interval, so the plausible differences all have the same sign under this ordering. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 23: City Tree Program context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Difference case 24: District Tutoring Initiative

Group 1 has 160 of 266 observations meeting the criterion for students attending at least six sessions, while group 2 has 114 of 258. Thus p̂₁=0.602, p̂₂=0.442, and the ordered estimate p̂₁−p̂₂=0.160. A 99% two proportion z interval uses the unpooled standard error √[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂]=0.0431. With z*=2.576, the interval is (0.049, 0.271).

Because the parameter was defined as group 1 minus group 2, positive values favor group 1 and negative values favor group 2. Zero is outside this interval, so the plausible differences all have the same sign under this ordering. The four observed success/failure counts should be checked individually; a large combined total is not enough if one group is sparse. In the Difference case 24: District Tutoring Initiative context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Ordering stress test 25: Service comparison

Suppose group 1 records 126 successes among 220 observations and group 2 records 101 among 205. Then p̂₁=0.573, p̂₂=0.493, so the ordered difference is 0.080. The unpooled standard error is 0.0483; for 95% confidence, z*=1.960 and the interval is (-0.015, 0.175). Each group has enough observed successes and failures for the normal approximation in this case.

This example is designed to test ordering rather than just arithmetic. Reversing the parameter to p₂−p₁ would produce (-0.175, 0.015), which carries the same substantive information with signs reversed. Because the current interval contains zero, the evidence about a difference should be described from these endpoints rather than from the sign of the point estimate alone.

Ordering stress test 26: Service comparison

Suppose group 1 records 91 successes among 160 observations and group 2 records 82 among 175. Then p̂₁=0.569, p̂₂=0.469, so the ordered difference is 0.100. The unpooled standard error is 0.0544; for 99% confidence, z*=2.576 and the interval is (-0.040, 0.240). Each group has enough observed successes and failures for the normal approximation in this case.

This example is designed to test ordering rather than just arithmetic. Reversing the parameter to p₂−p₁ would produce (-0.240, 0.040), which carries the same substantive information with signs reversed. Because the current interval contains zero, the evidence about a difference should be described from these endpoints rather than from the sign of the point estimate alone.

Ordering stress test 27: Service comparison

Suppose group 1 records 194 successes among 310 observations and group 2 records 159 among 295. Then p̂₁=0.626, p̂₂=0.539, so the ordered difference is 0.087. The unpooled standard error is 0.0400; for 90% confidence, z*=1.645 and the interval is (0.021, 0.153). Each group has enough observed successes and failures for the normal approximation in this case.

This example is designed to test ordering rather than just arithmetic. Reversing the parameter to p₂−p₁ would produce (-0.153, -0.021), which carries the same substantive information with signs reversed. Because the current interval does not contain zero, the evidence about a difference should be described from these endpoints rather than from the sign of the point estimate alone.

Two Proportion Z Interval multiple-choice practice

Question 1. Two-Proportion z Interval

At a farm cooperative in Atlantic Corridor during a multiweek validation study, Group 1 has 94140 meeting a criterion and Group 2 has 77150. Construct a 95% interval for p₁−p₂.

A. (0.101, 0.215); omit 1.96.
B. Use a pooled standard error because all two-proportion procedures pool.
C. (-0.27, -0.047) while still labeling it p₁−p₂.
D. (0.047, 0.27) for p₁−p₂.

Answer: D

p̂₁=0.671, p̂₂=0.513, difference=0.158. The unpooled SE is √0.671(0.329)140+0.513(0.487)150=0.0569. The 95% interval is 0.158±1.96(0.0569)=(0.047, 0.27).

Question 2. Two-Proportion z Interval

At a housing authority in Great Lakes during a multiweek validation study, Group 1 has 72110 meeting a criterion and Group 2 has 68160. Construct a 95% interval for p₁−p₂.

A. (-0.347, -0.112) while still labeling it p₁−p₂.
B. Use a pooled standard error because all two-proportion procedures pool.
C. (0.112, 0.347) for p₁−p₂.
D. (0.17, 0.289); omit 1.96.

Answer: C

p̂₁=0.655, p̂₂=0.425, difference=0.23. The unpooled SE is √0.655(0.345)110+0.425(0.575)160=0.0599. The 95% interval is 0.23±1.96(0.0599)=(0.112, 0.347).

Question 3. Two-Proportion z Interval

At a solar installer in Sunbelt district during a quarterly performance study, Group 1 has 99220 meeting a criterion and Group 2 has 132220. Construct a 95% interval for p₁−p₂.

A. (-0.197, -0.103); omit 1.96.
B. (-0.242, -0.058) for p₁−p₂.
C. (0.058, 0.242) while still labeling it p₁−p₂.
D. Use a pooled standard error because all two-proportion procedures pool.

Answer: B

p̂₁=0.45, p̂₂=0.6, difference=-0.15. The unpooled SE is √0.45(0.55)220+0.6(0.4)220=0.0471. The 95% interval is -0.15±1.96(0.0471)=(-0.242, -0.058).

Question 4. Two-Proportion z Interval

At a state park in Midwest consortium during a semester-long cohort study, Group 1 has 149240 meeting a criterion and Group 2 has 107180. Construct a 95% interval for p₁−p₂.

A. (-0.068, 0.121) for p₁−p₂.
B. (-0.022, 0.075); omit 1.96.
C. Use a pooled standard error because all two-proportion procedures pool.
D. (-0.121, 0.068) while still labeling it p₁−p₂.

Answer: A

p̂₁=0.621, p̂₂=0.594, difference=0.026. The unpooled SE is √0.621(0.379)240+0.594(0.406)180=0.0482. The 95% interval is 0.026±1.96(0.0482)=(-0.068, 0.121).

Question 5. Two-Proportion z Interval

At a city recreation department in Great Lakes during a quarterly performance study, Group 1 has 48110 meeting a criterion and Group 2 has 86210. Construct a 95% interval for p₁−p₂.

A. Use a pooled standard error because all two-proportion procedures pool.
B. (-0.141, 0.087) while still labeling it p₁−p₂.
C. (-0.087, 0.141) for p₁−p₂.
D. (-0.031, 0.085); omit 1.96.

Answer: C

p̂₁=0.436, p̂₂=0.41, difference=0.027. The unpooled SE is √0.436(0.564)110+0.41(0.59)210=0.0582. The 95% interval is 0.027±1.96(0.0582)=(-0.087, 0.141).

Question 6. Two-Proportion z Interval

At a regional hospital in Coastal Plains during a service-improvement study, Group 1 has 165270 meeting a criterion and Group 2 has 81210. Construct a 95% interval for p₁−p₂.

A. (0.181, 0.27); omit 1.96.
B. (-0.313, -0.138) while still labeling it p₁−p₂.
C. (0.138, 0.313) for p₁−p₂.
D. Use a pooled standard error because all two-proportion procedures pool.

Answer: C

p̂₁=0.611, p̂₂=0.386, difference=0.225. The unpooled SE is √0.611(0.389)270+0.386(0.614)210=0.0448. The 95% interval is 0.225±1.96(0.0448)=(0.138, 0.313).

Question 7. Two-Proportion z Interval

At a county election office in Atlantic Corridor during a semester-long cohort study, Group 1 has 108150 meeting a criterion and Group 2 has 112290. Construct a 95% interval for p₁−p₂.

A. (-0.425, -0.243) while still labeling it p₁−p₂.
B. Use a pooled standard error because all two-proportion procedures pool.
C. (0.243, 0.425) for p₁−p₂.
D. (0.287, 0.38); omit 1.96.

Answer: C

p̂₁=0.72, p̂₂=0.386, difference=0.334. The unpooled SE is √0.72(0.28)150+0.386(0.614)290=0.0465. The 95% interval is 0.334±1.96(0.0465)=(0.243, 0.425).

Question 8. Two-Proportion z Interval

At a public high school in South Harbor during a six-week field trial, Group 1 has 60130 meeting a criterion and Group 2 has 70220. Construct a 95% interval for p₁−p₂.

A. (0.09, 0.197); omit 1.96.
B. Use a pooled standard error because all two-proportion procedures pool.
C. (-0.249, -0.038) while still labeling it p₁−p₂.
D. (0.038, 0.249) for p₁−p₂.

Answer: D

p̂₁=0.462, p̂₂=0.318, difference=0.143. The unpooled SE is √0.462(0.538)130+0.318(0.682)220=0.0538. The 95% interval is 0.143±1.96(0.0538)=(0.038, 0.249).

Question 9. Two-Proportion z Interval

At a grocery cooperative in Prairie District during a regional benchmarking study, Group 1 has 203280 meeting a criterion and Group 2 has 77170. Construct a 95% interval for p₁−p₂.

A. Use a pooled standard error because all two-proportion procedures pool.
B. (0.181, 0.363) for p₁−p₂.
C. (-0.363, -0.181) while still labeling it p₁−p₂.
D. (0.225, 0.319); omit 1.96.

Answer: B

p̂₁=0.725, p̂₂=0.453, difference=0.272. The unpooled SE is √0.725(0.275)280+0.453(0.547)170=0.0466. The 95% interval is 0.272±1.96(0.0466)=(0.181, 0.363).

Question 10. Two-Proportion z Interval

At a farm cooperative in Capital Region during a semester-long cohort study, Group 1 has 68150 meeting a criterion and Group 2 has 105200. Construct a 95% interval for p₁−p₂.

A. Use a pooled standard error because all two-proportion procedures pool.
B. (-0.126, -0.018); omit 1.96.
C. (-0.177, 0.034) for p₁−p₂.
D. (-0.034, 0.177) while still labeling it p₁−p₂.

Answer: C

p̂₁=0.453, p̂₂=0.525, difference=-0.072. The unpooled SE is √0.453(0.547)150+0.525(0.475)200=0.0538. The 95% interval is -0.072±1.96(0.0538)=(-0.177, 0.034).

Question 11. Two-Proportion z Interval

At a regional hospital in Central County during a monthly quality review, Group 1 has 88240 meeting a criterion and Group 2 has 126200. Construct a 95% interval for p₁−p₂.

A. Use a pooled standard error because all two-proportion procedures pool.
B. (0.173, 0.354) while still labeling it p₁−p₂.
C. (-0.31, -0.217); omit 1.96.
D. (-0.354, -0.173) for p₁−p₂.

Answer: D

p̂₁=0.367, p̂₂=0.63, difference=-0.263. The unpooled SE is √0.367(0.633)240+0.63(0.37)200=0.0462. The 95% interval is -0.263±1.96(0.0462)=(-0.354, -0.173).

Question 12. Two-Proportion z Interval

At a school district in Westview during a spring 2027 pilot, Group 1 has 90140 meeting a criterion and Group 2 has 76210. Construct a 95% interval for p₁−p₂.

A. (-0.384, -0.178) while still labeling it p₁−p₂.
B. (0.229, 0.333); omit 1.96.
C. (0.178, 0.384) for p₁−p₂.
D. Use a pooled standard error because all two-proportion procedures pool.

Answer: C

p̂₁=0.643, p̂₂=0.362, difference=0.281. The unpooled SE is √0.643(0.357)140+0.362(0.638)210=0.0523. The 95% interval is 0.281±1.96(0.0523)=(0.178, 0.384).

Question 13. Two-Proportion z Interval

At a state park in Cedar Grove during a weekday operations study, Group 1 has 133270 meeting a criterion and Group 2 has 135280. Construct a 95% interval for p₁−p₂.

A. (-0.032, 0.053); omit 1.96.
B. (-0.094, 0.073) while still labeling it p₁−p₂.
C. (-0.073, 0.094) for p₁−p₂.
D. Use a pooled standard error because all two-proportion procedures pool.

Answer: C

p̂₁=0.493, p̂₂=0.482, difference=0.01. The unpooled SE is √0.493(0.507)270+0.482(0.518)280=0.0426. The 95% interval is 0.01±1.96(0.0426)=(-0.073, 0.094).

Question 14. Two-Proportion z Interval

At a regional hospital in Mountain Region during a community outreach cycle, Group 1 has 176250 meeting a criterion and Group 2 has 135210. Construct a 95% interval for p₁−p₂.

A. Use a pooled standard error because all two-proportion procedures pool.
B. (-0.025, 0.147) for p₁−p₂.
C. (0.017, 0.105); omit 1.96.
D. (-0.147, 0.025) while still labeling it p₁−p₂.

Answer: B

p̂₁=0.704, p̂₂=0.643, difference=0.061. The unpooled SE is √0.704(0.296)250+0.643(0.357)210=0.0439. The 95% interval is 0.061±1.96(0.0439)=(-0.025, 0.147).

Question 15. Two-Proportion z Interval

At a public health department in Riverbend during a six-week field trial, Group 1 has 85160 meeting a criterion and Group 2 has 79160. Construct a 95% interval for p₁−p₂.

A. (-0.147, 0.072) while still labeling it p₁−p₂.
B. (-0.018, 0.093); omit 1.96.
C. (-0.072, 0.147) for p₁−p₂.
D. Use a pooled standard error because all two-proportion procedures pool.

Answer: C

p̂₁=0.531, p̂₂=0.494, difference=0.037. The unpooled SE is √0.531(0.469)160+0.494(0.506)160=0.0558. The 95% interval is 0.037±1.96(0.0558)=(-0.072, 0.147).

Question 16. Two-Proportion z Interval

At a food safety laboratory in New England network during a quarterly performance study, Group 1 has 114250 meeting a criterion and Group 2 has 77130. Construct a 95% interval for p₁−p₂.

A. (-0.241, -0.032) for p₁−p₂.
B. Use a pooled standard error because all two-proportion procedures pool.
C. (0.032, 0.241) while still labeling it p₁−p₂.
D. (-0.19, -0.083); omit 1.96.

Answer: A

p̂₁=0.456, p̂₂=0.592, difference=-0.136. The unpooled SE is √0.456(0.544)250+0.592(0.408)130=0.0534. The 95% interval is -0.136±1.96(0.0534)=(-0.241, -0.032).

Question 17. Two-Proportion z Interval

At a solar installer in Midwest consortium during a follow-up evaluation period, Group 1 has 96180 meeting a criterion and Group 2 has 35140. Construct a 95% interval for p₁−p₂.

A. (0.231, 0.336); omit 1.96.
B. (0.181, 0.386) for p₁−p₂.
C. (-0.386, -0.181) while still labeling it p₁−p₂.
D. Use a pooled standard error because all two-proportion procedures pool.

Answer: B

p̂₁=0.533, p̂₂=0.25, difference=0.283. The unpooled SE is √0.533(0.467)180+0.25(0.75)140=0.0522. The 95% interval is 0.283±1.96(0.0522)=(0.181, 0.386).

Question 18. Two-Proportion z Interval

At a community college in Lakeside district during a weekday operations study, Group 1 has 115210 meeting a criterion and Group 2 has 103260. Construct a 95% interval for p₁−p₂.

A. (0.062, 0.241) for p₁−p₂.
B. (0.106, 0.197); omit 1.96.
C. Use a pooled standard error because all two-proportion procedures pool.
D. (-0.241, -0.062) while still labeling it p₁−p₂.

Answer: A

p̂₁=0.548, p̂₂=0.396, difference=0.151. The unpooled SE is √0.548(0.452)210+0.396(0.604)260=0.0458. The 95% interval is 0.151±1.96(0.0458)=(0.062, 0.241).

Question 19. Two-Proportion z Interval

At a public high school in Desert County during a weekday operations study, Group 1 has 134260 meeting a criterion and Group 2 has 119190. Construct a 95% interval for p₁−p₂.

A. Use a pooled standard error because all two-proportion procedures pool.
B. (-0.203, -0.019) for p₁−p₂.
C. (-0.158, -0.064); omit 1.96.
D. (0.019, 0.203) while still labeling it p₁−p₂.

Answer: B

p̂₁=0.515, p̂₂=0.626, difference=-0.111. The unpooled SE is √0.515(0.485)260+0.626(0.374)190=0.0468. The 95% interval is -0.111±1.96(0.0468)=(-0.203, -0.019).

Question 20. Two-Proportion z Interval

At a housing authority in Cedar Grove during a six-week field trial, Group 1 has 198270 meeting a criterion and Group 2 has 41130. Construct a 95% interval for p₁−p₂.

A. (-0.514, -0.322) while still labeling it p₁−p₂.
B. (0.322, 0.514) for p₁−p₂.
C. Use a pooled standard error because all two-proportion procedures pool.
D. (0.369, 0.467); omit 1.96.

Answer: B

p̂₁=0.733, p̂₂=0.315, difference=0.418. The unpooled SE is √0.733(0.267)270+0.315(0.685)130=0.0488. The 95% interval is 0.418±1.96(0.0488)=(0.322, 0.514).

Two Proportion Z Interval free-response practice

FRQ set 1: Two-Proportion z Interval

Scenario. At a state park in Prairie District during a six-week field trial, Group 1 has 182260 meeting a criterion and Group 2 has 36140. Construct a 95% interval for p₁−p₂.

Define the population parameter and preserve the stated group order.
Verify randomization, independence, and procedure-specific success/failure or shape conditions.
Construct the interval from the statistic, critical value, and standard error.
Interpret the confidence statement for the population parameter and explain one factor affecting margin of error.

Model response

p̂₁=0.7, p̂₂=0.257, difference=0.443. The unpooled SE is √0.7(0.3)260+0.257(0.743)140=0.0466. The 95% interval is 0.443±1.96(0.0466)=(0.352, 0.534).

FRQ set 2: Two-Proportion z Interval

Scenario. At a farm cooperative in Sunbelt district during a follow-up evaluation period, Group 1 has 101180 meeting a criterion and Group 2 has 128290. Construct a 95% interval for p₁−p₂.

Define the population parameter and preserve the stated group order.
Verify randomization, independence, and procedure-specific success/failure or shape conditions.
Construct the interval from the statistic, critical value, and standard error.
Interpret the confidence statement for the population parameter and explain one factor affecting margin of error.

Model response

p̂₁=0.561, p̂₂=0.441, difference=0.12. The unpooled SE is √0.561(0.439)180+0.441(0.559)290=0.0471. The 95% interval is 0.12±1.96(0.0471)=(0.027, 0.212).

FRQ set 3: Two-Proportion z Interval

Scenario. At a public high school in Central County during a fall 2026 audit, Group 1 has 63130 meeting a criterion and Group 2 has 78120. Construct a 95% interval for p₁−p₂.

Define the population parameter and preserve the stated group order.
Verify randomization, independence, and procedure-specific success/failure or shape conditions.
Construct the interval from the statistic, critical value, and standard error.
Interpret the confidence statement for the population parameter and explain one factor affecting margin of error.

Model response

p̂₁=0.485, p̂₂=0.65, difference=-0.165. The unpooled SE is √0.485(0.515)130+0.65(0.35)120=0.0618. The 95% interval is -0.165±1.96(0.0618)=(-0.286, -0.044).

FRQ set 4: Two-Proportion z Interval

Scenario. At a community college in Prairie District during a school-year data collection, Group 1 has 171290 meeting a criterion and Group 2 has 73150. Construct a 95% interval for p₁−p₂.

Define the population parameter and preserve the stated group order.
Verify randomization, independence, and procedure-specific success/failure or shape conditions.
Construct the interval from the statistic, critical value, and standard error.
Interpret the confidence statement for the population parameter and explain one factor affecting margin of error.

Model response

p̂₁=0.59, p̂₂=0.487, difference=0.103. The unpooled SE is √0.59(0.41)290+0.487(0.513)150=0.05. The 95% interval is 0.103±1.96(0.05)=(0.005, 0.201).

FRQ set 5: Two-Proportion z Interval

Scenario. At a housing authority in Mountain Region during a regional benchmarking study, Group 1 has 110250 meeting a criterion and Group 2 has 116200. Construct a 95% interval for p₁−p₂.

Define the population parameter and preserve the stated group order.
Verify randomization, independence, and procedure-specific success/failure or shape conditions.
Construct the interval from the statistic, critical value, and standard error.
Interpret the confidence statement for the population parameter and explain one factor affecting margin of error.

Model response

p̂₁=0.44, p̂₂=0.58, difference=-0.14. The unpooled SE is √0.44(0.56)250+0.58(0.42)200=0.0469. The 95% interval is -0.14±1.96(0.0469)=(-0.232, -0.048).

FRQ set 6: Two-Proportion z Interval

Scenario. At a farm cooperative in Desert County during a regional benchmarking study, Group 1 has 63130 meeting a criterion and Group 2 has 99230. Construct a 95% interval for p₁−p₂.

Define the population parameter and preserve the stated group order.
Verify randomization, independence, and procedure-specific success/failure or shape conditions.
Construct the interval from the statistic, critical value, and standard error.
Interpret the confidence statement for the population parameter and explain one factor affecting margin of error.

Model response

p̂₁=0.485, p̂₂=0.43, difference=0.054. The unpooled SE is √0.485(0.515)130+0.43(0.57)230=0.0547. The 95% interval is 0.054±1.96(0.0547)=(-0.053, 0.161).

FRQ set 7: Two-Proportion z Interval

Scenario. At a county library in Desert County during a follow-up evaluation period, Group 1 has 149220 meeting a criterion and Group 2 has 103250. Construct a 95% interval for p₁−p₂.

Define the population parameter and preserve the stated group order.
Verify randomization, independence, and procedure-specific success/failure or shape conditions.
Construct the interval from the statistic, critical value, and standard error.
Interpret the confidence statement for the population parameter and explain one factor affecting margin of error.

Model response

p̂₁=0.677, p̂₂=0.412, difference=0.265. The unpooled SE is √0.677(0.323)220+0.412(0.588)250=0.0443. The 95% interval is 0.265±1.96(0.0443)=(0.178, 0.352).

FRQ set 8: Two-Proportion z Interval

Scenario. At a wildlife clinic in Westview during a pre-exam training cycle, Group 1 has 92240 meeting a criterion and Group 2 has 114280. Construct a 95% interval for p₁−p₂.

Define the population parameter and preserve the stated group order.
Verify randomization, independence, and procedure-specific success/failure or shape conditions.
Construct the interval from the statistic, critical value, and standard error.
Interpret the confidence statement for the population parameter and explain one factor affecting margin of error.

Model response

p̂₁=0.383, p̂₂=0.407, difference=-0.024. The unpooled SE is √0.383(0.617)240+0.407(0.593)280=0.043. The 95% interval is -0.024±1.96(0.043)=(-0.108, 0.06).

Next steps after mastering two proportion z interval

Next, contrast this interval with a two-proportion z test. Keep the same group order and write the interval standard error beside the pooled test standard error. That side-by-side comparison makes clear why pooling is appropriate under H₀:p₁=p₂ but inappropriate when estimating the unknown difference p₁−p₂.

Review mistakes by checking group labels before arithmetic. Recompute one case as p₂−p₁ and verify that both endpoints change sign but the substantive conclusion does not. Then inspect each of the four observed success/failure counts separately; this prevents a large total sample from hiding a sparse group.