---
title: "Procedure: test of single proportion"
subject: "Probability and Statistics"
chapter: 7
key: "semester_5/probability_and_statistics:single-proportion-procedure"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/single-proportion-procedure
site: BCT Engineering
---

# Procedure: test of single proportion

Probability and Statistics · Chapter 7: Inference concerning proportion

- Asked in: 2075 Asoj, 4 marks
- Video: [TEST FOR SINGLE PROPORTION - PROCEDURE](https://www.youtube.com/watch?v=4LsLBUuar40) — Statistics made simple
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/single-proportion-procedure
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-7

## Question

Write the procedure of testing of Hypothesis for single proportion.

## Short answer

Set H0:P=P0 and H1 two- or one-tailed; choose alpha; compute z=(p-hat-P0)/sqrt(P0Q0/n), p-hat=x/n; compare with normal critical values (±1.96 or 1.645 at 5%, ±2.576 or 2.326 at 1%); reject H0 if z lies in the rejection region and state the conclusion.

## Answer

The test of a single proportion checks whether the population proportion P equals a specified value P₀, using a large sample of size n with x items having the attribute.

#### Procedure

1. **Hypotheses.**
   H0: P = P₀
   H1: P ≠ P₀ (two-tailed), or P > P₀ (right-tailed), or P < P₀ (left-tailed)
2. **Level of significance** α (5% or 1%).
3. **Test statistic.** Sample proportion p̂ = x/n, Q₀ = 1 − P₀:

```text
z = (p̂ − P₀) / √(P₀Q₀/n)
```

   The normal approximation is valid when nP₀ ≥ 5 and nQ₀ ≥ 5.
4. **Critical value** from the standard normal table:

```text
α       two-tailed   one-tailed
0.05    ±1.96        ±1.645
0.01    ±2.576       ±2.326
```

5. **Decision.** Reject H0 if |z| > critical value (two-tailed), z > critical value (right-tailed) or z < −critical value (left-tailed). Otherwise do not reject H0.
6. **Conclusion.** Express the decision in the words of the problem, for example "the proportion of defectives is significantly higher than 5%".

## Check yourself

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

1. What is H0 in a single-proportion test?
   A. P ≠ P0
   B. P = P0
   C. P > P0
   D. p̂ = P0
   Answer: B. H0: P = P0, the specified population proportion. (Key point: Hypotheses)

2. When is the normal approximation valid?
   A. n < 30
   B. P0 = 0.5 only
   C. nP0 ≥ 5 and nQ0 ≥ 5
   D. sigma known
   Answer: C. Both nP0 and nQ0 must be at least 5. (Key point: Validity)

3. At alpha = 0.01, the one-tailed critical value is
   A. 1.645
   B. 2.576
   C. 1.96
   D. 2.326
   Answer: D. One-tailed 1% is 2.326; two-tailed 1% is ±2.576. (Key point: One-tailed critical value)

4. In a left-tailed test, H0 is rejected when
   A. z < −critical
   B. z > critical
   C. |z| < critical
   D. z = 0
   Answer: A. The left tail is the rejection region. (Key point: Decision rule)

## Close variants set in other exams

- [Types of error; single proportion test](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Asingle-proportion-types-of-error) — Probability and Statistics, chapter 7, asked 2×

## Related in Probability and Statistics

- [Hypothesis testing; difference of two proportions](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Ahypothesis-procedure-diff-two-proportions) — Probability and Statistics, chapter 7, asked 2×
- [Hypothesis testing steps; computer assemblies](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Asteps-hypothesis-testing-computers-readjust) — Probability and Statistics, chapter 7
- [Chi-square test of independence: 2×2 procedure](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Achi-square-independence-2x2-procedure) — Probability and Statistics, chapter 7
- [Decision criteria in hypothesis test with diagram](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Adecision-criteria-hypothesis-diagram) — Probability and Statistics, chapter 6
- [Test of proportion: pumps needing repairs](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apumps-repairs-30-percent-claim) — Probability and Statistics, chapter 7
- [Test of proportion: 60 Minutes TV share](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Atv-60-minutes-share-test) — Probability and Statistics, chapter 7

## Related in other subjects

- [Soundness test of cement and procedure](https://www.bctengineer.com/question.md?key=semester_3%2Fcivil_engineering_materials%3Acement-soundness-test-procedure) — Civil Engineering Materials, chapter 5
- [Testing of DC machines: retardation vs Swinburne's test](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Adc-testing-retardation-vs-swinburne) — Electrical Machines I, chapter 4
- [Marshall stability test procedure](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3Amarshall-stability-test) — Transportation Engineering I, chapter 6, asked 3×
- [Workability and slump test procedure](https://www.bctengineer.com/question.md?key=semester_5%2Fconcrete_technology_and_masonry_structure%3Aslump-test-procedure) — Concrete Technology and Masonry Structure, chapter 3