Hypothesis test for a normal mean (known variance)

The steps

1. Hypotheses in terms of μ: H0: μ = μ0 and H1: μ < μ0, μ > μ0 or μ ≠ μ0 (use μ, never x̄).

2. Test statistic: z = (x̄ - μ0)/(σ/sqrt(n)).

3. Critical value from the percentage points table: 1.6449 one-tailed 5%, 1.96 two-tailed 5%, 2.3263 one-tailed 1%, 2.5758 two-tailed 1% (examiners deduct marks for 1.645).

4. Compare and decide: reject H0 or do not reject H0.

5. Conclusion in context, with the direction: 'there is evidence that the mean volume is less than 500 ml'.

Example

Bottles should hold 500 ml; volumes are N(μ, 4²). A sample of 20 has x̄ = 498.1. Test at 5% whether the mean is less than 500.

H0: μ = 500, H1: μ < 500. z = (498.1 - 500)/(4/sqrt(20)) = -2.1243. Critical value -1.6449. -2.124 < -1.6449, so reject H0: there is evidence that the mean volume is less than 500 ml.

Critical regions and p-values

Equivalent methods:

• Critical region for x̄: reject if x̄ < 500 - 1.6449 × 4/sqrt(20) = 498.529.
• p-value: P(Z < -2.12) = 0.0170, which is less than 0.05, so reject H0.

State the conclusion without contradicting it: 'reject H0' with 'no evidence' scores nothing.

Worked example

Scores are N(μ, 6²). A sample of 30 has x̄ = 73.9. Test at the 5% level whether μ differs from 72.

1. H0: μ = 72, H1: μ ≠ 72 (two-tailed).
2. z = 1.9/(6/sqrt(30)) = 1.7345
3. Critical values ±1.96; 1.734 < 1.96
4. Not significant: do not reject H0.

Answer: There is insufficient evidence that the mean score differs from 72.

Check yourself

1. Volumes are N(μ, 4²). A sample of 20 bottles has mean 498.1 ml. Calculate the test statistic for H0: μ = 500. Give your answer to 3 significant figures.
   Show answer

   -2.12. (498.1 - 500)/(4/sqrt(20)) = -2.1243.

2. Find the critical value of Z for a one-tailed test at the 1% level with H1: μ > μ0. Give your answer to 4 decimal places.
   Show answer

   2.3263. P(Z > z) = 0.01 gives z = 2.3263.

3. Volumes are N(μ, 4²). For a sample of 20, H0: μ = 500 is tested against H1: μ < 500 at the 5% level. Find the critical value of x̄. Give your answer to 2 decimal places.
   Show answer

   498.53. 500 - 1.6449 × 4/sqrt(20) = 498.529.

4. A test of H0: μ = 500 against H1: μ < 500 gives z = -2.124. Find the p-value. Give your answer to 4 decimal places.
   Show answer

   0.0168. P(Z < -2.12) = 0.0170 from tables.

5. H0: μ = 500, H1: μ < 500 at the 5% level gives z = -2.12. Which conclusion earns the mark?
   • Accept H0: the mean volume is 500 ml
   • Reject H0: the mean volume is exactly 498.1 ml
   • Do not reject H0: there is no evidence the volume has changed
   • Reject H0: there is evidence that the mean volume of the bottles is less than 500 ml
   Show answer

   Reject H0: there is evidence that the mean volume of the bottles is less than 500 ml. -2.12 < -1.6449, so the result is significant; the conclusion must be in context and include the direction 'less than'.

6. Scores are N(μ, 6²). A sample of 30 has mean 73.9, giving z = 1.734 for H0: μ = 72 against H1: μ ≠ 72 at the 5% level. What is the outcome?
   • Reject H0, because the statistic exceeds 1.6449
   • Do not reject H0, because the statistic is less than 1.96
   • Reject H0, because 73.9 is not 72
   • Do not reject H0, because the statistic is positive
   Show answer

   Do not reject H0, because the statistic is less than 1.96. A two-tailed 5% test uses ±1.96, and 1.73 is inside these.

7. A test is to check a claim that the mean time is more than 20 minutes. Which hypotheses are correct?
   • H0: x̄ = 20, H1: x̄ > 20
   • H0: μ > 20, H1: μ = 20
   • H0: μ = 20, H1: μ ≠ 20
   • H0: μ = 20, H1: μ > 20
   Show answer

   H0: μ = 20, H1: μ > 20. Hypotheses are about the population mean μ, with the claim in H1.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S3
• Pearson Edexcel IAL Statistics S3 (WST03) examiners' report, June 2025
• Pearson Edexcel IAL Statistics S3 (WST03) mark scheme, June 2024
• Pearson Edexcel Mathematical Formulae and Statistical Tables

Last updated 8 November 2026. Spotted a mistake? Tell us.