---
title: "t-test assumptions; two means small sample"
subject: "Probability and Statistics"
chapter: 6
key: "semester_5/probability_and_statistics:t-test-assumptions-small-sample-procedure"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/t-test-assumptions-small-sample-procedure
site: BCT Engineering
---

# t-test assumptions; two means small sample

Probability and Statistics · Chapter 6: Inference concerning mean

- Asked in: 2076 Asoj, 5 marks
- Video: [Two-sample t test for difference of means | AP Statistics | Khan Academy](https://www.youtube.com/watch?v=NkGvw18zlGQ) — Khan Academy
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/t-test-assumptions-small-sample-procedure
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-6

## Question

What are the assumptions for the t-test? Describe the procedure of test of significance between two means for small sample.

## Short answer

Populations must be normal, small random samples, σ unknown. For equal variances, test H₀: μ₁=μ₂ with t = (x̄₁−x̄₂)/[Sₚ√(1/n₁+1/n₂)], df=n₁+n₂−2, and reject if |t| > critical t.

## Answer

#### Assumptions for the t-test

1. The population(s) from which the sample is drawn is (are) normally distributed.
2. The sample is small (n < 30) and is a random sample; observations are independent.
3. The population standard deviation σ is unknown (estimated by the sample standard deviation s).
4. For two-sample t-test: the two samples are independent and the population variances are equal (σ₁² = σ₂²).
5. The data are measured on an interval or ratio scale.

#### Test of significance for the difference of two means (small samples)

Let samples of sizes n₁ and n₂ (each small) come from normal populations with means μ₁, μ₂ and equal unknown variance.

1. **Hypotheses:** H₀: μ₁ = μ₂. H₁: μ₁ ≠ μ₂ (two-tailed) or μ₁ > μ₂ / μ₁ < μ₂ (one-tailed).
2. **Level of significance:** α (usually 0.05).
3. **Test statistic:**
   t = (x̄₁ − x̄₂) / [S_p √(1/n₁ + 1/n₂)]
   where the pooled standard deviation is
   S_p² = [(n₁ − 1)s₁² + (n₂ − 1)s₂²] / (n₁ + n₂ − 2)
4. **Degrees of freedom:** ν = n₁ + n₂ − 2.
5. **Critical value:** t_α/2(ν) for a two-tailed test (t_α(ν) for a one-tailed test) from the t-table.
6. **Decision:** reject H₀ if |t| > critical value; otherwise do not reject H₀.
7. **Conclusion:** state whether the two population means differ significantly.

If the variances are not equal, Welch's test with t = (x̄₁ − x̄₂)/√(s₁²/n₁ + s₂²/n₂) and adjusted degrees of freedom is used. For paired (dependent) samples, use t = d̄/(s_d/√n) with n − 1 degrees of freedom.

## Check yourself

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

1. Which assumption belongs to the t-test?
   A. Population normal, σ unknown, small sample
   B. Population non-normal
   C. σ known
   D. Large dependent samples
   Answer: A. The t-test assumes normality and unknown σ with a small sample. (Key point: Assumptions)

2. Two-sample pooled t-test additionally needs:
   A. Unequal variances
   B. Equal population variances and independent samples
   C. Paired data
   D. n ≥ 100
   Answer: B. σ1² = σ2² and independence. (Key point: Two-sample assumption)

3. Degrees of freedom of the pooled test?
   A. n1 + n2 − 2
   B. n1 + n2
   C. n1 − 1
   D. n1·n2
   Answer: A. Two means are estimated, losing 2 df. (Key point: Degrees of freedom)

4. Sp² is:
   A. [(n1−1)s1² + (n2−1)s2²]/(n1+n2−2)
   B. (s1² + s2²)
   C. (s1−s2)²
   D. s1²/s2²
   Answer: A. It is a weighted average of the two sample variances. (Key point: Pooled variance)

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