---
title: "Hypothesis testing; difference of two proportions"
subject: "Probability and Statistics"
chapter: 7
key: "semester_5/probability_and_statistics:hypothesis-procedure-diff-two-proportions"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/hypothesis-procedure-diff-two-proportions
site: BCT Engineering
---

# Hypothesis testing; difference of two proportions

Probability and Statistics · Chapter 7: Inference concerning proportion

- Asked in: 2075 Chaitra, 5 marks; 2076 Asoj, 5 marks (asked 2 times)
- Video: [Hypothesis Testing With Two Proportions](https://www.youtube.com/watch?v=pCbNUnZ98oE) — The Organic Chemistry Tutor
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/hypothesis-procedure-diff-two-proportions
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-7

## Question

What is testing of hypothesis? Explain the procedure followed in testing of significance difference between two population proportion for large sample.

## Short answer

Hypothesis testing decides whether sample data support rejecting a claim about a population parameter. For two proportions, use pooled p̂=(x₁+x₂)/(n₁+n₂), then z=(p̂₁−p̂₂)/√[p̂(1−p̂)(1/n₁+1/n₂)]; reject H0 if |z| exceeds the normal critical value for the chosen α.

## Answer

**Testing of hypothesis** is a statistical procedure for deciding, from sample data, whether a claim (hypothesis) about a population parameter should be rejected or not. The claim to be tested is the null hypothesis H0; it is compared with an alternative H1 using a test statistic and a level of significance α.

#### Test for the difference of two population proportions (large samples)

Used to test whether proportions p₁ and p₂ in two populations are equal, from samples of sizes n₁, n₂ with x₁, x₂ successes. Samples must be large (n·p ≥ 5 and n·q ≥ 5 in each).

1. **Hypotheses.** H0: p₁ = p₂. H1: p₁ ≠ p₂ (two-tailed), or p₁ > p₂ / p₁ < p₂ (one-tailed).
2. **Level of significance** α (5% or 1%).
3. **Sample proportions and pooled proportion.**

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

4. **Test statistic.**

```text
z = (p̂₁ − p̂₂) / √( p̂ q̂ (1/n₁ + 1/n₂) )
```

5. **Critical value** from the normal table: ±1.96 (two-tail, 5%), 1.645 (one-tail, 5%), ±2.576 (two-tail, 1%), 2.326 (one-tail, 1%).
6. **Decision.** If the calculated z falls in the rejection region (|z| > critical value for two-tail), reject H0; otherwise do not reject H0.
7. **Conclusion** in words: the two population proportions differ significantly, or they do not.

## Check yourself

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

1. What is testing of hypothesis?
   A. Proving H0 true
   B. Estimating a parameter with an interval
   C. Collecting a sample
   D. Deciding from sample data whether a claim about a population parameter should be rejected
   Answer: D. It compares H0 with H1 using a test statistic and level of significance to decide whether to reject the claim. (Key point: Definition)

2. Why is a pooled proportion p̂ = (x1 + x2)/(n1 + n2) used in the standard error?
   A. H0 says p1 = p2, so one common proportion is estimated
   B. It reduces alpha
   C. It makes the sample larger
   D. It converts z to t
   Answer: A. Under H0 the two proportions are equal, so the best estimate of the common p combines both samples. (Key point: Pooled proportion)

3. Which is the test statistic?
   A. (p̂1 + p̂2)/sqrt(p̂q̂)
   B. (p̂1 − p̂2)/sqrt(p̂q̂(1/n1 + 1/n2))
   C. (p̂1 − p̂2)/(n1 + n2)
   D. (x1 − x2)/p̂
   Answer: B. z = (p̂1 − p̂2)/sqrt(p̂ q̂ (1/n1 + 1/n2)), approximately N(0,1) for large samples. (Key point: Test statistic)

4. What condition makes the large-sample z-test valid?
   A. n < 30
   B. p1 = 0.5
   C. n·p ≥ 5 and n·q ≥ 5 in each sample
   D. Unknown sigma
   Answer: C. These conditions make the normal approximation to the binomial adequate. (Key point: Large-sample condition)

## Related in Probability and Statistics

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