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CCEA GCE Mathematics (2210) · Hypothesis testing
Mini-Lesson

Hypothesis testing

This mini-lesson covers the logic of a hypothesis test: H₀ and H₁, the significance level, one- and two-tailed tests, critical regions and p-values for a binomial test, and testing the mean of a normal distribution — plus how to word a conclusion that earns the mark.

Where this sits: Unit AS 2: Applied Mathematics — hypothesis testing with the binomial distribution. Unit A2 2 — hypothesis testing for the mean of a normal distribution.

Work through each screen, answer the questions as you go (some are reasoning, some are calculations) and collect ⭐ stars. Press Start when you are ready.

The logic

Null and alternative hypotheses

A hypothesis test asks: is the evidence unusual enough, if the null hypothesis were true, to reject it?

  • H₀ (null): the "no change" assumption, always an equality, e.g. p = 0.25 or μ = 50.
  • H₁ (alternative): what you suspect, e.g. p > 0.25 (one-tailed) or p ≠ 0.25 (two-tailed).
  • Significance level α (5%, 1%): the probability of rejecting a TRUE H₀ — the risk you accept of a false alarm.

You NEVER prove H₀. The two permitted conclusions are "reject H₀" or "do not reject H₀" (insufficient evidence). Writing "H₀ is true" or "this proves H₁" loses the mark.

Binomial test

p-values and critical regions

Method 1 — p-value: assume H₀, compute the probability of a result at least as extreme as the one observed. If p-value ≤ α, reject H₀.

Method 2 — critical region: find the set of extreme values with total probability ≤ α under H₀. If the observation lands in it, reject H₀.

Worked example — X ~ B(20, 0.25), testing H₁: p > 0.25 at 5%

P(X ≥ 9) = 1 − P(X ≤ 8) = 1 − 0.9591 = 0.0409 ≤ 0.05 ✓

P(X ≥ 8) = 1 − P(X ≤ 7) = 1 − 0.8982 = 0.1018 > 0.05 ✗

So the critical region is X ≥ 9, and its actual significance is 4.09% (not exactly 5% — the distribution is discrete).

Calculate

Binomial p-value

1Under H₀, X ~ B(20, 0.25). Find P(X ≥ 9) to 3 decimal places.
Hint: P(X ≥ 9) = 1 − P(X ≤ 8). From the binomial tables (or your calculator), P(X ≤ 8) = 0.9591, so the answer is 0.0409.
Calculate

Critical value

2For X ~ B(20, 0.25) testing H₁: p > 0.25 at the 5% level, find the smallest value c such that the critical region is X ≥ c.
Hint: You need P(X ≥ c) ≤ 0.05. P(X ≥ 9) = 0.0409 ✓ but P(X ≥ 8) = 0.1018 ✗, so c = 9.
Quick check

Which tail?

?A coin is suspected of being biased — but you do not know in which direction. Which alternative hypothesis is right?
Two-tailed tests

Splitting the significance level

For a two-tailed test at the 5% level you split α: 2.5% in each tail. Comparing a p-value from one tail to the full 5% is a classic error.

Equivalently, double the one-tail p-value before comparing it with 5%.

Standard critical z-values (normal tests):
  • one-tailed 5%: z = 1.645
  • one-tailed 1%: z = 2.326
  • two-tailed 5%: z = ±1.96
  • two-tailed 1%: z = ±2.576
A2 test

Testing the mean of a normal distribution

If X ~ N(μ, σ²) with σ known, then the sample mean of n observations satisfies:

X̄ ~ N(μ, σ²/n) ⇒ z = (x̄ − μ) / (σ/√n)the standard error σ/√n shrinks as the sample grows — bigger samples detect smaller shifts
Worked example

H₀: μ = 50, H₁: μ > 50. σ = 6, n = 36, sample mean x̄ = 52.

Standard error = 6/√36 = 1, so z = (52 − 50)/1 = 2

p-value = P(Z > 2) = 1 − 0.9772 = 0.0228 < 0.05 ⇒ reject H₀

Conclusion: there is significant evidence at the 5% level that the mean has increased above 50.

Calculate

Test statistic

3H₀: μ = 50 against H₁: μ > 50, with σ = 6, n = 36 and sample mean 52. Find the test statistic z.
Hint: z = (x̄ − μ) ÷ (σ/√n) = (52 − 50) ÷ (6/√36) = 2 ÷ 1.
Calculate

p-value from z

4For that one-tailed test, find the p-value P(Z > 2) to 4 decimal places.
Hint: P(Z > 2) = 1 − P(Z < 2) = 1 − 0.9772 = 0.0228.
Calculate

Two-tailed critical value

5For a two-tailed z-test at the 5% level, state the positive critical value of z to 2 decimal places.
Hint: Split 5% into 2.5% in each tail. The z-value with 0.975 of the area below it is 1.96.
Sort it

What is the conclusion?

Tap the evidence, then tap the correct verdict.

❌ Reject H₀

✅ Do not reject H₀

🚨 Not a valid conclusion

Quick check

Wording the conclusion

?A binomial test gives a p-value of 0.12 at the 5% level. Which conclusion is correctly worded?
Match it

Match the term

Tap the term on the left, then its meaning on the right.

Statement
Answer
Quick check

Significance level meaning

?What does testing at the 1% level rather than the 5% level do?
Quick check

Discrete critical regions

?For a binomial test, why is the actual significance level usually not exactly 5%?
Examiner traps

Hypothesis testing pitfalls

  • Never "accept H₀" or "prove H₁". Write "reject H₀" or "insufficient evidence to reject H₀".
  • Two-tailed means α/2 in each tail — compare with 2.5%, not 5%.
  • The tail must match H₁: for p > p₀ compute P(X ≥ observed), for p < p₀ compute P(X ≤ observed).
  • Include the observed value in the tail probability — it is "at least as extreme".
  • Conclude in context, quoting the significance level.
  • Actual significance ≠ nominal for a discrete binomial critical region.
Recap

The big ideas to know

H₀ is an equality (p = …, μ = …); H₁ is the suspicion (>, <, or ≠)

p-value ≤ α ⇒ reject H₀; otherwise do not reject (never 'accept' or 'prove')

Two-tailed at 5% means 2.5% per tail — critical z = ±1.96

Binomial test: find the critical region from the tails; actual significance ≠ nominal, because X is discrete

Normal mean test: z = (x̄ − μ)/(σ/√n) — the standard error uses √n

Conclusion must be in context, at the stated significance level

Hypothesis testing turns the distributions of the previous topic into evidence-based decisions. Press Finish to see your score.

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