---
title: "Procedure: difference of two proportions, large sample"
subject: "Probability and Statistics"
chapter: 7
key: "semester_5/probability_and_statistics:difference-two-proportions-large-procedure"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/difference-two-proportions-large-procedure
site: BCT Engineering
---

# Procedure: difference of two proportions, large sample

Probability and Statistics · Chapter 7: Inference concerning proportion

- Asked in: 2073 Chaitra, 5 marks; 2071 Chaitra, 5 marks (asked 2 times)
- Video: [2 .Test for Two proportions.large sample tests.statistics ,4th semester](https://www.youtube.com/watch?v=dqeH_MDZWVI) — MS  EDUCATION 🖊📕
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/difference-two-proportions-large-procedure
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-7

## Question

Describe the procedure of the test of significance for difference of two proportions for large sample.

## Short answer

State null p1=p2 and alternative. Compute sample proportions p̂1, p̂2, pooled p̂=(x1+x2)/(n1+n2), SE=√(p̂q̂(1/n1+1/n2)), z=(p̂1−p̂2)/SE. Compare |z| with critical normal value; reject H0 if it exceeds the critical value, else do not reject.

## Answer

The test checks whether two population proportions p₁ and p₂ are equal, using large independent samples of sizes n₁ and n₂ with x₁ and x₂ items having the attribute.

#### Procedure

1. **Hypotheses.**
   H0: p₁ = p₂ (no difference)
   H1: p₁ ≠ p₂ (two-tailed); or p₁ > p₂ / p₁ < p₂ (one-tailed)
2. **Level of significance** α, for example 0.05.
3. **Sample proportions and the pooled estimate** (valid because H0 says p₁ = p₂):

```text
p̂₁ = x₁/n₁,  p̂₂ = x₂/n₂
p̂  = (x₁ + x₂)/(n₁ + n₂),  q̂ = 1 − p̂
```

4. **Standard error and test statistic:**

```text
SE = √( p̂ q̂ (1/n₁ + 1/n₂) )
z  = (p̂₁ − p̂₂)/SE          (follows N(0,1) for large n)
```

5. **Critical value.** Take it from the normal table for α: ±1.96 for a two-tailed test at 5%, 1.645 for a one-tailed test at 5%, ±2.576 and 2.326 at 1%.
6. **Decision.** Reject H0 when |z| exceeds the critical value (two-tailed) or z lies in the stated tail (one-tailed). Otherwise do not reject H0.
7. **Conclusion.** State in plain words whether the two proportions differ significantly.

#### Example layout

If 16 of 200 and 14 of 300 are defective: p̂₁ = 0.08, p̂₂ = 0.0467, p̂ = 0.06, SE = 0.0217, z = 1.54 < 1.96, so H0 is not rejected at 5%.

## Check yourself

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

1. What is H0 for the difference of two proportions?
   A. p1 = p2
   B. p̂1 = p̂2
   C. p1 ≠ p2
   D. p1 > p2
   Answer: A. H0 is about the population proportions: p1 = p2. (Key point: Null hypothesis)

2. What is SE of the test statistic?
   A. sqrt(p̂1q̂1/n1)
   B. sqrt(p̂q̂(1/n1 + 1/n2)) with pooled p̂
   C. (p̂1 − p̂2)/2
   D. p̂/sqrt(n1 + n2)
   Answer: B. The pooled estimate is used because H0 says p1 = p2. (Key point: Standard error)

3. In the example (16 of 200, 14 of 300) z = 1.54. What is the decision at 5%?
   A. Reject H0 since 1.54 > 1
   B. Reject H0 since z > 0
   C. Do not reject H0 since 1.54 < 1.96
   D. Use t instead
   Answer: C. |z| = 1.54 < 1.96, so the proportions do not differ significantly. (Key point: Example)

4. In the example, what is p̂ pooled?
   A. 0.0467
   B. 0.08
   C. 0.03
   D. 0.06
   Answer: D. p̂ = (16 + 14)/(200 + 300) = 30/500 = 0.06. (Key point: Pooled value)

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