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

This guide teaches Section O: Statistical hypothesis testing (content references 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 in the specification, with the rest of the statistics content (Sections K to O). A calculator is required in every 7357 paper, and for Sections K to O the specification says you must be able to use calculator technology to access probabilities from standard statistical distributions.

Use it with the hypothesis testing revision notes and the hypothesis testing practice questions. The 7357 course hub lists every topic, the printable checklist lets you tick off outcomes, and the free 10-minute diagnostics show where to start. Testing builds on sampling, so read the statistical sampling guide first if “population” and “sample” are not secure. Scatter diagrams and describing correlation informally are in the data presentation and interpretation guide.

What Section O covers

Ref	What you must be able to do
O1	Understand and apply the language of hypothesis testing, developed through a binomial model: null hypothesis, alternative hypothesis, significance level, test statistic, 1-tail test, 2-tail test, critical value, critical region, acceptance region, p-value. Extend to correlation coefficients as measures of how close data points lie to a straight line, and interpret a given correlation coefficient using a given p-value or critical value (calculating correlation coefficients is excluded).
O2	Conduct a hypothesis test for the proportion in the binomial distribution and interpret the result in context. Understand that a sample is used to make an inference about the population, and that the significance level is the probability of incorrectly rejecting the null hypothesis.
O3	Conduct a hypothesis test for the mean of a Normal distribution with known, given or assumed variance and interpret the result in context.

The specification’s notation is H₀ for the null hypothesis, H₁ for the alternative hypothesis, B(n, p) for the binomial distribution, N(μ, σ²) for the Normal distribution, r for the sample product moment correlation coefficient and ρ for the population one.

O1: The language of hypothesis testing

A hypothesis test uses a sample to decide whether there is enough evidence to change a belief about a population parameter, such as a proportion p, a mean μ or a correlation coefficient ρ.

• Null hypothesis, H₀: the belief you start from. It always states one value of the parameter, for example H₀: p = 0.2.
• Alternative hypothesis, H₁: what you accept if the evidence is strong enough. It is written with <, > or ≠, for example H₁: p > 0.2.
• Test statistic: the quantity calculated from the sample that you use to decide, for example X, the number of successes in the sample, or X̄, the sample mean.
• 1-tail test: H₁ looks in one direction only (p > 0.2 or p < 0.2).
• 2-tail test: H₁ is p ≠ 0.2. The significance level is split equally between the two tails.
• Significance level: the threshold probability for rejecting H₀, such as 5% or 1%. It is the probability of incorrectly rejecting H₀ when H₀ is true.
• Critical region: the set of values of the test statistic that lead you to reject H₀.
• Critical value: the first value inside the critical region (its boundary).
• Acceptance region: all values not in the critical region. If the test statistic lands here, you do not reject H₀.
• p-value: the probability, assuming H₀ is true, of a result at least as extreme as the one observed. If the p-value is less than the significance level, reject H₀.

Two points about wording. First, you never “prove” H₀ or H₁; the sample gives evidence. Write “there is sufficient evidence to suggest…” or “there is insufficient evidence to suggest…”. Second, the hypotheses are about the population parameter, never the sample value.

O2: Testing a binomial proportion

Method

1. Define the parameter: “let p be the proportion of … in the population”.
2. Write H₀ and H₁ in terms of p.
3. Define the test statistic and its distribution under H₀: X ~ B(n, p₀).
4. Either find the p-value and compare it with the significance level, or find the critical region and see whether the observed value lies in it.
5. Conclude in context, with non-definite language.

Worked example 1: one-tail test using a p-value

Historically, 20% of a shop’s customers buy an extended warranty. After a new display is installed, 9 of a random sample of 25 customers buy one. Test at the 5% significance level whether the proportion has increased.

Let p = proportion of all customers who buy a warranty.
H₀: p = 0.2     H₁: p > 0.2
Under H₀, X ~ B(25, 0.2), where X = number in the sample who buy.
p-value = P(X ≥ 9) = 1 − P(X ≤ 8) = 0.0468 (3 s.f.)
0.0468 < 0.05, so reject H₀.

There is sufficient evidence at the 5% level to suggest that the proportion of customers buying an extended warranty has increased.

For H₁: p > 0.2 you find P(X ≥ 9), not P(X = 9).

Worked example 2: two-tail critical region

A spinner should land on red with probability 0.25. It is spun 40 times to test whether this is true, at the 10% significance level. Find the critical region and the actual significance level.

For a 2-tail test at 10%, each tail gets 5%. Find the largest lower value and the smallest upper value whose tail probabilities do not exceed 0.05. Under H₀, X ~ B(40, 0.25).

Lower tail: P(X ≤ 5) = 0.0433   P(X ≤ 6) = 0.0962  → X ≤ 5
Upper tail: P(X ≥ 16) = 0.0262  P(X ≥ 15) = 0.0544 → X ≥ 16
Critical region: X ≤ 5 or X ≥ 16
Actual significance level = 0.0433 + 0.0262 = 0.0695 (3 s.f.), about 6.95%

Showing the probability either side of each boundary (0.0433 and 0.0962; 0.0262 and 0.0544) is what proves your boundary is correct.

Significance level and wrong decisions

Because X is discrete, the critical region rarely has a probability of exactly 10%. The actual significance level is P(X in critical region | H₀ true), here 0.0695. This is the probability of incorrectly rejecting H₀. With a discrete test statistic it is usually below the stated level.

A different random sample of 40 spins could give a different count, so a conclusion is an inference about the population, with a known chance of being wrong.

O3: Testing the mean of a Normal distribution

If X ~ N(μ, σ²) and you take a random sample of size n, the sample mean has the distribution

X̄ ~ N(μ, σ²/n)

so the standard deviation of X̄ is σ/√n. The variance σ² is known, given in the question, or assumed to be unchanged from a previous value. You can test using the standardised value

z = (x̄ − μ₀) / (σ/√n)

and compare it with a critical z value, or find the p-value directly from the N(μ₀, σ²/n) distribution on your calculator.

Worked example 3: 2-tail test

A machine fills bags of flour. The masses are Normally distributed with mean 500 g and standard deviation 4 g. After a repair, a random sample of 16 bags has mean 497.8 g. Assuming the standard deviation is unchanged, test at the 5% level whether the mean mass has changed.

Let μ = mean mass of all bags filled after the repair.
H₀: μ = 500     H₁: μ ≠ 500
Under H₀, X̄ ~ N(500, 4²/16) = N(500, 1²)
z = (497.8 − 500)/1 = −2.2
2-tail at 5%: critical values z = ±1.960
−2.2 < −1.960, so reject H₀.
(Equivalently, P(X̄ ≤ 497.8) = 0.0139 < 0.025.)

There is sufficient evidence at the 5% level to suggest that the mean mass has changed. When you use a p-value in a 2-tail test, compare the one-tail probability with half the significance level, as in the bracketed line.

Worked example 4: critical region for the sample mean

Reaction times in a population are N(72, 9²) in milliseconds. A coach claims a training programme increases mean reaction time. A random sample of 36 trained people is taken. Find the critical region for X̄ at the 1% level, then test a sample mean of 75.2.

H₀: μ = 72     H₁: μ > 72
Under H₀, X̄ ~ N(72, 9²/36), so σ/√n = 1.5
Upper 1% point: z = 2.326
Critical value = 72 + 2.326 × 1.5 = 75.49 (2 d.p.)
Critical region: X̄ > 75.49
75.2 is not in the critical region, so do not reject H₀.

There is insufficient evidence at the 1% level to suggest the mean reaction time has increased. For a continuous test statistic, the probability of incorrectly rejecting H₀ is exactly the significance level, 1%.

O1 extended: testing a correlation coefficient

The product moment correlation coefficient measures how close data points lie to a straight line. It takes values from −1 to 1. Values near ±1 mean points lie close to a line; values near 0 mean there is little linear relationship. You are not asked to calculate r in this section: it is given, together with either a critical value or a p-value.

Hypotheses are about the population coefficient ρ:

• H₀: ρ = 0 (no linear correlation in the population)
• H₁: ρ > 0, ρ < 0 (1-tail) or ρ ≠ 0 (2-tail)

Worked example 5: using a critical value

For a random sample of 20 students, the correlation between hours of revision and test score is r = 0.412. Test at the 5% level whether there is positive correlation in the population. The critical value for a 1-tail test at 5% with a sample of 20 is given as 0.3783.

H₀: ρ = 0     H₁: ρ > 0
r = 0.412 > 0.3783, so reject H₀.

There is sufficient evidence at the 5% level to suggest positive correlation between revision time and test score for students in the population. For H₁: ρ < 0, compare a negative r with the negative critical value: reject H₀ if r < −0.3783.

Worked example 6: using a p-value

For 30 towns, r = 0.31 between population density and bus use, and software gives a 2-tail p-value of 0.095 for H₁: ρ ≠ 0. At 5%: 0.095 > 0.05, so do not reject H₀; there is insufficient evidence of correlation. At 10%: 0.095 < 0.10, so you would reject H₀. The same data can lead to different conclusions at different significance levels, which is why the level is chosen before the test.

Using your calculator

The specification requires a calculator in every paper, with the ability to access probabilities from standard statistical distributions and an inverse Normal function.

• Binomial cumulative: P(X ≤ k) directly. For P(X ≥ k), use 1 − P(X ≤ k − 1).
• Critical regions: tabulate cumulative probabilities near the boundary and write the two values either side down.
• Normal: enter the mean μ₀ and the standard deviation σ/√n (not σ) to get P(X̄ ≤ x̄).
• Inverse Normal: gives critical z values (1.645, 1.960, 2.326, 2.576) or critical sample means.

Always write down the distribution and the probability you found.

Common errors

• Writing hypotheses with the sample value: “H₀: p = 9/25” is wrong.
• Using P(X = k) instead of P(X ≥ k) or P(X ≤ k) as the p-value.
• Using P(X ≥ k) = 1 − P(X ≤ k), which drops the value k itself.
• In a 2-tail test, comparing each tail with the full significance level.
• Using σ instead of σ/√n for the sample mean.
• Stating a definite conclusion (“the mean has changed”) or missing the context.
• Testing r against ρ = r, or claiming the test proves causation.

Next steps

Go through the revision notes for a condensed version and a quick self-test, then try the practice questions. The exam preparation guide covers the three papers.

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.