AQA A-Level Mathematics: O: Statistical hypothesis testing (7357) – Revision Notes

These notes condense Section O: Statistical hypothesis testing (O1 to O3) of the AQA A-level Mathematics (7357) specification, version 1.3, for A-level exams from June 2018 onwards. Section O is listed under Paper 3, and a calculator is required in every 7357 paper. For full explanations and six worked examples, use the hypothesis testing study guide; for exam-style questions, use the practice set.

Course links: 7357 hub | printable checklist | free 10-minute diagnostics | statistical sampling notes | data presentation notes

O1: Vocabulary you must use correctly

Term	Meaning
Null hypothesis H₀	The starting belief; one value of the parameter, e.g. p = 0.3, μ = 50, ρ = 0
Alternative hypothesis H₁	Accepted if evidence is strong enough; uses <, > or ≠
Test statistic	Value from the sample used to decide, e.g. X (count) or X̄ (sample mean)
1-tail test	H₁ is one-directional (< or >)
2-tail test	H₁ uses ≠; significance level split equally between the tails
Significance level	Threshold probability for rejecting H₀; the probability of incorrectly rejecting H₀
Critical region	Values of the test statistic that lead to rejecting H₀
Critical value	The boundary value of the critical region (first value inside it)
Acceptance region	Values not in the critical region; H₀ is not rejected
p-value	P(result at least as extreme as observed, assuming H₀ is true)

Decision rules:

• p-value < significance level → reject H₀.
• Test statistic in critical region → reject H₀.
• Otherwise → do not reject H₀ (never “accept H₁” or “prove H₀”).

A test uses a sample to make an inference about the population. Hypotheses always refer to the population parameter, not to the sample result.

Key distributions and formulas

Test	Under H₀	Notes
Binomial proportion	X ~ B(n, p₀)	X = number of successes in the sample
Normal mean	X̄ ~ N(μ₀, σ²/n)	σ known, given or assumed unchanged
Standardised mean	z = (x̄ − μ₀)/(σ/√n)	compare with critical z
Correlation	H₀: ρ = 0	r is given; compare with a given critical value or p-value

Common critical z values (to 3 d.p.):

Level	1-tail	2-tail
10%	1.282	±1.645
5%	1.645	±1.960
1%	2.326	±2.576

O2: Binomial test, method in steps

1. "Let p = proportion of ... in the population."
2. H₀: p = p₀    H₁: p > p₀  (or < or ≠)
3. Under H₀, X ~ B(n, p₀)
4. p-value: P(X ≥ x) for H₁: p > p₀;  P(X ≤ x) for H₁: p < p₀
   2-tail: find the tail the observed x lies in, compare with half the level
5. Compare, decide, conclude in context with "suggest"

Remember: P(X ≥ k) = 1 − P(X ≤ k − 1).

Worked reminder: lower-tail critical region

Under H₀, X ~ B(20, 0.35). Find the critical region for H₁: p < 0.35 at 5%.

P(X ≤ 3) = 0.0444  ≤ 0.05
P(X ≤ 4) = 0.1182  > 0.05
Critical region: X ≤ 3
Actual significance level: 0.0444 (4.44%)

Always quote both probabilities either side of the boundary.

Worked reminder: 2-tail p-value

Under H₀, X ~ B(16, 0.5), with H₁: p ≠ 0.5 at 5%. You observe x = 13, which is well above half of 16, so it lies in the upper tail.

P(X ≥ 13) = 0.0106
Compare with 0.025 (half of 5%): 0.0106 < 0.025
Reject H₀: evidence that p is not 0.5

Had you observed 12, P(X ≥ 12) = 0.0384 > 0.025, so you would not reject H₀, even though 0.0384 < 0.05.

Actual significance level

For a binomial test, the actual significance level is P(test statistic in critical region | H₀ true). It is the true probability of incorrectly rejecting H₀ and is usually less than the stated level, because X is discrete. In a 2-tail test, add the two tail probabilities.

O3: Normal mean test, method in steps

1. "Let μ = mean ... of the population."
2. H₀: μ = μ₀    H₁: μ > μ₀  (or < or ≠)
3. Under H₀, X̄ ~ N(μ₀, σ²/n)
4. Either: z = (x̄ − μ₀)/(σ/√n) and compare with critical z
   Or:     p-value = P(X̄ ≥ x̄) or P(X̄ ≤ x̄) from the calculator
   Or:     critical value of X̄ = μ₀ ± z × σ/√n
5. Decide and conclude in context

For a continuous test statistic, the probability of incorrectly rejecting H₀ equals the stated significance level exactly.

Worked reminder

Under H₀, μ = 64, σ = 6, n = 9, so σ/√n = 2. For H₁: μ < 64 at 5%, the critical value of X̄ is 64 − 1.645 × 2 = 60.71 (2 d.p.). A sample mean of 61.0 is not in the critical region X̄ < 60.71, so do not reject H₀.

O1: Correlation test, method in steps

1. H₀: ρ = 0
   H₁: ρ > 0 (positive), ρ < 0 (negative) or ρ ≠ 0 (any correlation)
2. Use the critical value given for the right sample size, level and tails
3. Positive H₁: reject if r > critical value
   Negative H₁: reject if r < −critical value
   2-tail: reject if |r| > critical value
   Or: reject if the given p-value < significance level
4. Conclude about correlation in the population, in context

The correlation coefficient measures how close points lie to a straight line. Calculating r is excluded from this section. A significant correlation is not evidence of causation.

Must-know distinctions

• Sample vs population. r, x̄ and the observed count describe the sample. ρ, μ and p describe the population and appear in hypotheses.
• 1-tail vs 2-tail. Words like “increased”, “fewer” and “positive” give a 1-tail test. “Changed”, “different” and “any correlation” give a 2-tail test.
• Critical value vs critical region. The value is a single boundary; the region is a set, written as an inequality.
• Stated vs actual significance level. Stated: the level chosen before the test. Actual: the probability of the critical region under H₀ (only different for a discrete distribution).
• p-value vs significance level. The significance level is fixed before you collect data. The p-value comes from the data. Reject H₀ only when the p-value is the smaller of the two.
• σ vs σ/√n. The population standard deviation σ describes single values. Use σ/√n for the sample mean.

Quick self-test

1. Write H₀ and H₁ for a test of the claim that more than 10% of a school’s pupils are left-handed.
2. State what a 5% significance level means for a binomial test.
3. Under H₀, X ~ B(12, 0.5). H₁: p > 0.5 and 10 successes are observed. Test at 5%.
4. Under H₀, X ~ B(15, 0.4). H₁: p < 0.4 and 2 successes are observed. Test at 2.5%.
5. Under H₀, X ~ B(10, 0.3). Find the critical region for H₁: p > 0.3 at 5%, and the actual significance level.
6. A sample gives r = 0.52 with a 1-tail p-value of 0.03 for H₁: ρ > 0. Test at 1%.
7. X ~ N(μ, 5²). For n = 25, x̄ = 31.8. Test H₀: μ = 30 against H₁: μ > 30 at 5%.
8. State the critical z values for a 1-tail test at 1% and a 2-tail test at 5%.
9. X ~ N(μ, σ²). State the distribution of the mean X̄ of a random sample of size n.
10. For 12 pairs, r = −0.55. The 1-tail 5% critical value for a sample of 12 is 0.4973. Test H₁: ρ < 0.

Answers

1. H₀: p = 0.1, H₁: p > 0.1, where p is the proportion of all pupils who are left-handed.
2. The probability of rejecting H₀ when it is true is at most 0.05.
3. P(X ≥ 10) = 0.0193 < 0.05, so reject H₀: evidence that p > 0.5.
4. P(X ≤ 2) = 0.0271 > 0.025, so do not reject H₀.
5. P(X ≥ 6) = 0.0473 ≤ 0.05 and P(X ≥ 5) = 0.1503 > 0.05, so X ≥ 6; actual level 0.0473.
6. 0.03 > 0.01, so do not reject H₀: insufficient evidence of positive correlation at 1%.
7. σ/√n = 1, z = 1.8 > 1.645 (p-value 0.0359 < 0.05), so reject H₀.
8. 2.326 and ±1.960.
9. X̄ ~ N(μ, σ²/n).
10. −0.55 < −0.4973, so reject H₀: evidence of negative correlation in the population.

Where marks are usually lost

• Hypotheses written with the sample statistic (r, x̄ or a sample proportion) instead of ρ, μ or p.
• No definition of the parameter, so the hypotheses have no meaning.
• P(X = k) given as the p-value instead of a cumulative tail probability.
• P(X ≥ k) found as 1 − P(X ≤ k) rather than 1 − P(X ≤ k − 1).
• 2-tail test compared with the full level instead of half of it in each tail.
• Critical region given without the probabilities either side of the boundary.
• σ used instead of σ/√n for the distribution of X̄.
• Conclusions that are too definite (“this proves”) or not in context.
• Correlation conclusion that implies causation.
• Wrong critical value chosen because the number of tails was misread.

Official syllabus

AQA A-level Mathematics (7357) specification, version 1.3, 31 January 2018, for A-level exams June 2018 onwards, Section 3.16 O: Statistical hypothesis testing. Published by AQA.