---
title: "Null, alternative hypothesis; errors"
subject: "Probability and Statistics"
chapter: 7
key: "semester_5/probability_and_statistics:null-alternative-hypothesis-errors"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/null-alternative-hypothesis-errors
site: BCT Engineering
---

# Null, alternative hypothesis; errors

Probability and Statistics · Chapter 7: Inference concerning proportion

- Asked in: 2068 Jestha (old course), 6 marks
- Video: [Hypothesis Testing - Null and Alternative Hypotheses](https://www.youtube.com/watch?v=wiYJWyfdGg4) — 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/null-alternative-hypothesis-errors
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-7

## Question

Explain Null hypothesis, alternative hypothesis and the errors arising in the test of hypothesis.

## Short answer

Null hypothesis states no difference or effect and uses equality; alternative states a difference or direction. Type I error (α) is rejecting a true H0; Type II error (β) is accepting a false H0. Decreasing α increases β unless sample size grows.

## Answer

#### Null hypothesis (H0)

The statement of no difference or no effect that is tested, and is assumed true until the sample evidence rejects it. It always contains an equality sign. Example: H0: μ = 50 (mean life is 50 hours).

#### Alternative hypothesis (H1 or Ha)

The statement that is accepted if H0 is rejected. It says there is a difference or effect. It decides the type of test:

- H1: μ ≠ 50: two-tailed test
- H1: μ > 50: right-tailed test
- H1: μ < 50: left-tailed test

#### Errors in testing

The decision rests on sample data, so it can be wrong.

| Decision | H0 true | H0 false |
|---|---|---|
| Reject H0 | **Type I error**, probability α | Correct decision (power 1 − β) |
| Accept H0 | Correct decision (1 − α) | **Type II error**, probability β |

- **Type I error:** rejecting H0 when it is actually true. Its probability is the level of significance α (e.g. 0.05). Example: rejecting a good lot.
- **Type II error:** accepting H0 when it is actually false. Its probability is β. Example: accepting a bad lot.
- For a fixed sample size, decreasing α increases β. Both can be reduced only by increasing the sample size.

## Check yourself

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

1. What best describes a null hypothesis?
   A. A statement of no difference or no effect, assumed true until evidence rejects it
   B. The statement we hope to prove
   C. A statement that never contains an equality sign
   D. The hypothesis accepted if H0 is rejected
   Answer: A. H0 states no effect, always contains equality, and is assumed true until the sample rejects it. (Key point: Null hypothesis)

2. If H1 is mu > 50, what type of test is it?
   A. Two-tailed
   B. Left-tailed
   C. Right-tailed
   D. No tail
   Answer: C. The direction of H1 decides the test: > is right-tailed, < is left-tailed, not equal is two-tailed. (Key point: Alternative hypothesis and tails)

3. What is a Type I error and its probability?
   A. Accepting a false H0, probability beta
   B. Rejecting a true H0, probability alpha
   C. Rejecting a false H0, probability 1 - beta
   D. Accepting a true H0, probability 1 - alpha
   Answer: B. Type I error is rejecting H0 when it is true; its probability is the level of significance alpha. (Key point: Type I error)

4. For a fixed sample size, what happens to beta if alpha is decreased?
   A. Beta decreases
   B. Beta increases
   C. Beta stays the same
   D. Beta becomes zero
   Answer: B. Reducing alpha makes rejection harder, so Type II errors become more likely; only a larger sample can reduce both. (Key point: Alpha and beta trade-off)

## Related in Probability and Statistics

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- [Test: soft drink filling machines](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Asoft-drink-machines-same-volume) — Probability and Statistics, chapter 6
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## Related in other subjects

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