---
title: "Decision procedure by critical value approach"
subject: "Probability and Statistics"
chapter: 6
key: "semester_5/probability_and_statistics:decision-critical-value-approach"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/decision-critical-value-approach
site: BCT Engineering
---

# Decision procedure by critical value approach

Probability and Statistics · Chapter 6: Inference concerning mean

- Asked in: 2068 Jestha (old course), 6 marks
- Video: [Hypothesis Testing: Critical Value Approach versus P-Value Approach](https://www.youtube.com/watch?v=4mDemwLHzbo) — Maths and Stats
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/decision-critical-value-approach
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-6

## Question

Describe the decision making procedure using critical value approach.

## Short answer

Fix α and find the critical value(s) from the test statistic's distribution. Compute the statistic; if it exceeds the critical value (e.g., |z|>1.96 for two-tailed α=0.05), reject H0. Otherwise do not reject, and state conclusion in context.

## Answer

In the critical value (classical) approach, the sample test statistic is compared with a critical value fixed from the level of significance α. The critical value separates the acceptance region from the rejection (critical) region of the sampling distribution under H0.

#### Steps

1. **State H0 and H1.** The form of H1 (≠, >, <) fixes whether the test is two-tailed, right-tailed or left-tailed.
2. **Fix α**, the probability of a Type I error (commonly 0.05 or 0.01).
3. **Choose the test statistic** and its distribution under H0: z, t, χ² or F, depending on the problem.
4. **Find the critical value(s) from the table** for α and the degrees of freedom.

```text
Test          Rejection region
Two-tailed    |stat| > critical value (α/2 in each tail)
Right-tailed  stat > critical value
Left-tailed   stat < −critical value
```

5. **Compute the test statistic** from the sample data.
6. **Decision rule.**
   - If the calculated value lies in the rejection region, reject H0.
   - If it lies in the acceptance region, do not reject H0 (the evidence is not enough to reject it).
7. **Conclusion.** State the decision in the words of the problem.

#### Example

For a two-tailed z-test at α = 0.05 the critical values are ±1.96. If the calculated z = 2.4, then |z| > 1.96, so H0 is rejected. If z = 1.2, H0 is not rejected.

```text
      reject    |   accept H0    |    reject
  ------|-------|-------|-------|-------|-----
      −1.96             0            +1.96
```

The p-value approach gives the same decision: reject H0 when p-value < α.

## Check yourself

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

1. What does the critical value do in this approach?
   A. It gives the probability of H0
   B. It is the sample mean
   C. It separates the acceptance region from the rejection region
   D. It equals alpha
   Answer: C. It is read from the table for alpha and d.f. and divides the rejection region from the acceptance region. (Key point: Critical value role)

2. What determines whether the test is two-tailed, right-tailed or left-tailed?
   A. The sample size
   B. The test statistic used
   C. The value of alpha
   D. The form of H1 (≠, >, <)
   Answer: D. H1 with ≠ gives two tails; > gives right tail; < gives left tail. (Key point: Tail from H1)

3. For a left-tailed test, when is H0 rejected?
   A. stat < −critical value
   B. |stat| > critical value
   C. stat > critical value
   D. stat = 0
   Answer: A. A left-tailed rejection region lies below the negative critical value. (Key point: Decision rule)

4. For a two-tailed z-test at alpha = 0.05, what is the decision for z = 1.2?
   A. Reject H0 because z > 1
   B. Do not reject H0 because |z| < 1.96
   C. Reject H0 because z > 0
   D. Reject H0 because alpha = 0.05
   Answer: B. 1.2 is inside (−1.96, 1.96), the acceptance region, so H0 is not rejected. (Key point: Example)

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