OxfordAQA International AS and A-level Mathematics: Hypothesis tests for means: z-tests and t-tests (9660) – Revision Notes

These notes condense S2.6 (tests for a mean) from the OxfordAQA International AS and A-level Mathematics (9660) specification, Version 5.2. The content belongs to International A-level Unit S2 (Statistics) and is not examined at International AS. For full explanations and longer worked examples, read the study guide. Then test yourself with the practice questions. See also the course hub, the printable checklist and the free 10-minute diagnostics.

Scope in one table

Spec outcome	Test statistic	Reference distribution
Normal population, variance known	z = (x̄ − μ₀)/(σ/√n)	N(0, 1)
Any population, large n, variance known	z = (x̄ − μ₀)/(σ/√n)	N(0, 1), approximate
Any population, large n, variance unknown	z = (x̄ − μ₀)/(s/√n)	N(0, 1), approximate
Normal population, variance unknown	t = (x̄ − μ₀)/(s/√n)	t with n − 1 degrees of freedom

The significance level is given in the question. You must understand Type I and Type II errors, but you will not be asked to calculate the risk of a Type II error.

Definitions

Term	Meaning
Null hypothesis H₀	μ takes one stated value, μ = μ₀
Alternative hypothesis H₁	μ > μ₀, μ < μ₀ (one-tailed) or μ ≠ μ₀ (two-tailed)
Test statistic	The z or t value worked out from the sample
Significance level α	The probability of rejecting H₀ that the test is designed to allow when H₀ is true
Critical value	The boundary of the critical (rejection) region
Critical region	Values of the statistic (or of x̄) that lead to rejecting H₀
Acceptance region	Values that lead to not rejecting H₀
p-value	The probability, if H₀ is true, of a result at least as extreme as the one observed
Type I error	Rejecting H₀ when H₀ is true
Type II error	Accepting H₀ when H₀ is false

Key facts to recall

• Unbiased estimators: x̄ for μ, and s² = [Σx² − (Σx)²/n]/(n − 1) for σ².
• Standard error of the mean: σ/√n, estimated by s/√n.
• Under H₀ with a normal population, X̄ ~ N(μ₀, σ²/n).
• For a fixed critical region, P(Type I error) equals the significance level.
• The t-distribution has heavier tails than N(0, 1), so its critical values are larger. It approaches N(0, 1) as the degrees of freedom increase.
• Useful z critical values: one-tailed 5% is 1.645, one-tailed 1% is 2.326, two-tailed 5% is ±1.960, two-tailed 10% is ±1.645.

Method in steps

1. Define μ and write H₀ and H₁ from the wording.
2. State the distribution of X̄ under H₀ (or say t with n − 1 degrees of freedom).
3. Find x̄, and s from summary data if σ is not known.
4. Calculate the test statistic using the standard error.
5. Compare with the critical value, or find the p-value and compare with α (or α/2 for one tail of a two-tailed test).
6. Write: “Reject H₀” or “Do not reject H₀”, then a conclusion in context, without claiming proof.

Critical region for the sample mean

Work backwards from the statistic. For H₀: μ = μ₀ with σ known and a two-tailed test at 5%:

x̄ ≤ μ₀ − 1.960 × σ/√n      or      x̄ ≥ μ₀ + 1.960 × σ/√n

For a one-tailed test use 1.645 (5%) or 2.326 (1%) and put the region in the tail given by H₁.

Small worked reminders

z-test, known σ. H₀: μ = 14.0, H₁: μ > 14.0, σ = 2.1, n = 20, x̄ = 15.1, 1% level. Here z = 1.1/(2.1/√20) = 1.1/0.4696 = 2.343. This is above 2.326 (p-value 0.0096), so reject H₀.

Large sample, σ unknown. n = 120, Σx = 4032, Σx² = 136 550. Then x̄ = 33.6 and s² = [136 550 − 4032²/120]/119 = 9.03, so s = 3.005. The standard error estimate is 3.005/√120 = 0.2743. For H₀: μ = 33, z = 0.6/0.2743 = 2.19.

t-test, small sample. n = 20, x̄ = 18.4, s = 2.5, H₀: μ = 17.5, H₁: μ > 17.5, 5% level, critical value 1.729 for 19 degrees of freedom. Here t = 0.9/(2.5/√20) = 0.9/0.559 = 1.610, which is below 1.729. Do not reject H₀.

Two-tailed t-test, summary data. n = 15, Σx = 412.5, Σx² = 11 420, H₀: μ = 28, H₁: μ ≠ 28, 5% level, critical values ±2.145 for 14 degrees of freedom. Then x̄ = 27.5 and s² = [11 420 − 412.5²/15]/14 = 5.446, so s = 2.334. Then t = (27.5 − 28)/(2.334/√15) = −0.5/0.6026 = −0.830, which lies inside ±2.145. Do not reject H₀.

Assumptions to state

• z-test with known variance: the population is normal (or n is large), and the sample is random.
• Large-sample z-test: n is large, so the Central Limit Theorem makes X̄ approximately normal and s is a good estimate of σ. Say that the conclusion is approximate.
• t-test: the population is normal, σ is unknown, and the sample is random. State “assume the population is normal” in your answer unless the question already tells you.

Wording a conclusion

A good final sentence has three parts: the decision, the level, and the context. For example: “Reject H₀. There is evidence at the 5% level that the mean delivery time is more than 36 minutes.” For a result that is not significant: “Do not reject H₀. There is insufficient evidence at the 5% level that the mean has changed.” Avoid certainty words such as “proves” or “shows definitely”.

Must-know distinctions

• z or t. Normal population with small n and unknown σ needs t. Large n allows z with s in place of σ. Known σ with a normal population allows z at any n.
• σ or s. σ is the population value, given. s is calculated from the sample, with divisor n − 1.
• Standard deviation or standard error. The standard deviation describes single values. The standard error σ/√n describes the sample mean.
• One-tailed or two-tailed. The words “greater”, “less” or “fallen” give one tail. The words “changed”, “different” or “differs” give two tails.
• Significance level or p-value. The level is chosen in advance. The p-value comes from the data.
• “Not rejected” or “accepted”. A test does not prove H₀. Write “insufficient evidence to reject”.

Quick self-test

1. A population has σ = 6. Find the standard error of the mean for a sample of 36.
2. Find z when x̄ = 52.4, μ₀ = 50, σ = 8 and n = 64.
3. How many degrees of freedom does a t-test have for a sample of 14?
4. A sample of 10 has Σx = 50 and Σx² = 290. Find s².
5. State the one-tailed 1% critical value of z.
6. A test is carried out at the 5% level using a fixed critical region. What is the probability of a Type I error?
7. A normal population has unknown variance and you have 8 observations. Which statistic do you use, and with how many degrees of freedom?
8. A two-tailed z-test at the 5% level gives a two-tailed p-value of 0.03. State the decision.
9. H₀: μ = 100, σ = 10, n = 25, two-tailed 5% test. Give the critical region for x̄.
10. A lower-tail t-test with 15 degrees of freedom has critical value −1.753 at the 5% level, and t = −1.9. State the decision.
11. Why is it wrong to use N(0, 1) for a sample of 5 from a normal population with unknown variance?

Answers

1. 6/√36 = 1.
2. z = 2.4/(8/8) = 2.4.
3. 14 − 1 = 13.
4. s² = (290 − 50²/10)/9 = 40/9 = 4.44.
5. 2.326.
6. 0.05.
7. t = (x̄ − μ₀)/(s/√n), with 7 degrees of freedom.
8. 0.03 < 0.05, so reject H₀.
9. Standard error = 2, and 1.960 × 2 = 3.92. Critical region: x̄ ≤ 96.08 or x̄ ≥ 103.92.
10. −1.9 < −1.753, so reject H₀.
11. s from 5 values is an unreliable estimate of σ, so the statistic does not follow N(0, 1). It follows a t-distribution with 4 degrees of freedom, which has heavier tails.

Where marks are usually lost

• Forgetting the √n and dividing by σ instead of σ/√n.
• Using divisor n instead of n − 1 when calculating s² from Σx and Σx².
• Using n instead of n − 1 as the degrees of freedom.
• Using a z critical value for a small sample with unknown variance.
• Writing hypotheses in terms of x̄ instead of the population mean μ.
• Mixing up tails, for example comparing with a positive critical value in a lower-tail test.
• Comparing a two-tailed p-value with α/2, or a one-tail probability with α in a two-tailed test.
• Stopping at “reject H₀” with no sentence in context, or writing “the mean is proved to be higher”.
• Not stating the assumption that the population is normal when using t.

Official syllabus

OxfordAQA International AS and A-level Mathematics (9660), Version 5.2 (International AS exams from May/June 2018, A-level from May/June 2019), published by OxfordAQA. Content: S2.6 Hypothesis testing, tests for the mean (z-statistic and t-statistic), within Unit S2 (Statistics).