---
title: "CI for difference of two proportions"
subject: "Probability and Statistics"
chapter: 7
key: "semester_5/probability_and_statistics:ci-difference-proportions-speed-limit"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/ci-difference-proportions-speed-limit
site: BCT Engineering
---

# CI for difference of two proportions

Probability and Statistics · Chapter 7: Inference concerning proportion

- Asked in: 2072 Kartik, 5 marks
- Video: [Confidence intervals for the difference between two proportions | AP Statistics | Khan Academy](https://www.youtube.com/watch?v=hcUxFwUjCxM) — Khan Academy
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/ci-difference-proportions-speed-limit
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-7

## Question

A random sample of 500 adult residents of Maricopa County found that 385 were in favor of increasing the highway speed limit to 75 mph, while another sample of 400 adult residents of Pima County found that 267 were in favor of the increased speed limit. Construct 95% confidence interval on the difference in the two proportions.

## Short answer

Compute p̂₁=0.77, p̂₂=0.6675. Difference 0.1025. SE = √(p̂₁q̂₁/n₁ + p̂₂q̂₂/n₂) = 0.03015. Using z=1.96, margin=0.0591. 95% CI is (0.043, 0.162). Since interval excludes zero, the proportion favoring higher limit is significantly greater in Maricopa County.

## Answer

For two large independent samples, the confidence interval for p₁ − p₂ is (p̂₁ − p̂₂) ± z·√(p̂₁q̂₁/n₁ + p̂₂q̂₂/n₂).

```text
Maricopa: n₁ = 500, p̂₁ = 385/500 = 0.7700, q̂₁ = 0.2300
Pima:     n₂ = 400, p̂₂ = 267/400 = 0.6675, q̂₂ = 0.3325

p̂₁ − p̂₂ = 0.7700 − 0.6675 = 0.1025
SE = √(0.77×0.23/500 + 0.6675×0.3325/400)
   = √(0.0003542 + 0.0005549) = 0.03015
z (95%) = 1.960
margin = 1.960 × 0.03015 = 0.0591
limits = 0.1025 ± 0.0591 = (0.0434, 0.1616)
```

**Answer: the 95% confidence interval for p₁ − p₂ is (0.043, 0.162).**

Since the interval does not contain zero, the proportion in favour of the higher speed limit is significantly greater in Maricopa County than in Pima County.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which standard error is used for the CI of p1 - p2?
   A. sqrt(p1q1/n1 + p2q2/n2)
   B. sqrt(p1q1/n1) - sqrt(p2q2/n2)
   C. sqrt(pq(1/n1 + 1/n2)) using a pooled p
   D. (p1q1 + p2q2)/(n1 + n2)
   Answer: A. For an interval the sample proportions are not pooled; the two variances p-hat q-hat / n are added. (Key point: CI formula for difference)

2. With p1 = 0.77, p2 = 0.6675 and margin 0.0591, what is the 95% interval for p1 - p2?
   A. (0.043, 0.162)
   B. (0.011, 0.194)
   C. (0.102, 0.162)
   D. (-0.016, 0.221)
   Answer: A. 0.1025 +/- 0.0591 gives (0.0434, 0.1616). (Key point: Result of the interval)

3. The interval (0.043, 0.162) does not contain zero. What does that imply?
   A. No difference between the counties
   B. Maricopa's proportion in favour is significantly higher than Pima's
   C. Pima's proportion is significantly higher
   D. The samples were too small
   Answer: B. Both limits are positive, so p1 - p2 > 0 with 95% confidence, meaning Maricopa is higher. (Key point: Interpreting the interval)

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