---
title: "Test of difference of two means: large sample"
subject: "Probability and Statistics"
chapter: 6
key: "semester_5/probability_and_statistics:difference-two-means-large-procedure"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/difference-two-means-large-procedure
site: BCT Engineering
---

# Test of difference of two means: large sample

Probability and Statistics · Chapter 6: Inference concerning mean

- Asked in: 2076 Chaitra, 5 marks; 2072 Chaitra, 5 marks; 2072 Kartik, 5 marks; 2070 Asar, 4 marks (asked 4 times)
- Video: [8.1 Hypothesis Testing 2 Sample Means  (Large Samples)](https://www.youtube.com/watch?v=LGPrrB9aJ2Y) — Kaffenate Math
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/difference-two-means-large-procedure
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-6

## Question

Describe the procedure of the test of significance for difference of two population means for large samples.

## Short answer

For large independent samples, test H₀: μ₁ = μ₂ with Z = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂). Compare |Z| to the standard normal critical value Z_α/₂ (two‑tailed) or Z_α (one‑tailed); reject H₀ if the statistic exceeds the critical value.

## Answer

When two independent large samples (n₁, n₂ ≥ 30) are drawn from populations with means μ₁ and μ₂, we test whether the difference of the sample means x̄₁ − x̄₂ is significant, using the **Z-test** (by the central limit theorem, x̄₁ − x̄₂ is approximately normal).

#### Procedure

1. **Hypotheses:** H₀: μ₁ = μ₂ (or μ₁ − μ₂ = 0). H₁: μ₁ ≠ μ₂ (two-tailed), or μ₁ > μ₂ / μ₁ < μ₂ (one-tailed).
2. **Level of significance:** choose α (usually 0.05 or 0.01).
3. **Test statistic:**
   Z = (x̄₁ − x̄₂) / √(σ₁²/n₁ + σ₂²/n₂)
   If σ₁², σ₂² are unknown, replace them by the sample variances s₁², s₂². If σ₁² = σ₂² = σ², the standard error is σ√(1/n₁ + 1/n₂).
4. **Critical value:** from the normal table.

| α | Two-tailed | One-tailed |
|---|---|---|
| 0.05 | ±1.96 | 1.645 |
| 0.01 | ±2.576 | 2.326 |

5. **Decision:** reject H₀ if |Z| > Z_α/2 (two-tailed) [or Z > Z_α, Z < −Z_α for one-tailed]; otherwise accept (fail to reject) H₀.
6. **Conclusion:** state in words whether the two population means differ significantly.

#### Example

Sample A: n₁ = 100, x̄₁ = 1120, s₁ = 75; sample B: n₂ = 100, x̄₂ = 1062, s₂ = 82. Test H₀: μ₁ = μ₂ vs H₁: μ₁ ≠ μ₂ at α = 0.05.

Z = (1120 − 1062)/√(75²/100 + 82²/100) = 58/11.11 = 5.22 > 1.96, so H₀ is rejected: the means differ significantly.

## Check yourself

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

1. Which test is used for the difference of two means with large samples?
   A. t-test
   B. Z-test
   C. Chi-square test
   D. F-test
   Answer: B. By the CLT the difference of means is approximately normal, so Z is used. (Key point: Test type)

2. What is H0?
   A. μ1 ≠ μ2
   B. μ1 = μ2
   C. x̄1 = x̄2
   D. μ1 > μ2
   Answer: B. H0 states the population means are equal. (Key point: Hypotheses)

3. Which is the statistic when variances are unknown?
   A. Z = (x̄1 − x̄2)/√(s1²/n1 + s2²/n2)
   B. Z = (x̄1 + x̄2)/√(s1²/n1 + s2²/n2)
   C. Z = (x̄1 − x̄2)/(s1 + s2)
   D. Z = (x̄1 − x̄2)/√(s1²+s2²)
   Answer: A. Sample variances replace σ² and are divided by the sample sizes. (Key point: Test statistic)

4. At 5% two-tailed, when is H0 rejected?
   A. Z > 1.645
   B. |Z| > 1.96
   C. |Z| < 1.96
   D. |Z| > 2.576
   Answer: B. Two-tailed 5% critical value is ±1.96. (Key point: Decision)

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