Hypothesis Tests

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Cambridge International AS & A Level Mathematics 9709 — Probability & Statistics 2

Hypothesis Tests

The big idea: Assume nothing has changed (H₀), work out how unlikely the evidence would be under that assumption, and reject H₀ only if it is less likely than the significance level.

Summary — What This Chapter Covers

H₀ states the parameter value; H₁ states how it might differ (one-tailed: < or >; two-tailed: ≠).
Compare the probability of the observed value or more extreme with the significance level (halve it for each tail of a two-tailed test), or find the critical region.
Binomial and Poisson tests use exact probabilities (or a normal approximation for large samples).
Test for a mean: z = (x̄ − μ₀)/(σ/√n).
Type I error: rejecting H₀ when it is true. Type II error: accepting H₀ when it is false.
Always conclude in context, without claiming certainty.

1. Tests for Binomial and Poisson Parameters

Discrete tests

Probability of the observed value or more extreme

Define the test statistic (X = number of heads, X = number of accidents) and its distribution under H₀. Find P(X ≥ observed) for an upper-tail test or P(X ≤ observed) for a lower-tail test — not P(X = observed). If this probability is less than the significance level, reject H₀.

Alternatively find the critical (rejection) region: the set of values whose total tail probability is as large as possible without exceeding the significance level. For a two-tailed test, compare each tail with half the significance level. For large samples use the normal approximation with a continuity correction.

Worked Example 1

A coin is suspected of being biased towards heads. In 15 tosses it shows 12 heads. Test at the 5% significance level.

Worked Example 2

A road averaged 4 accidents per week. After new speed limits there was 1 accident in a week. Test at the 5% level whether the rate has decreased.

Practice Question 1

X ~ B(20, p). Find the critical region for testing H₀: p = 0.3 against H₁: p < 0.3 at the 5% level.

2. Tests for a Population Mean

Mean tests

The z-test

Use when the population is normal with known variance, or when the sample is large (then σ may be estimated by s, and the CLT justifies normality). Under H₀, X̄ ~ N(μ₀, σ²/n). Calculate z and compare with the critical value (one-tail 5%: 1.645; two-tail 5%: 1.96; one-tail 1%: 2.326; two-tail 1%: 2.576), or compare the tail probability with the significance level.

Test statistic z = (x̄ − μ₀) / (σ/√n)

Worked Example

Bags are labelled 500 g; masses are normal with σ = 12 g. A sample of 36 bags has mean 496 g. Test at the 5% level whether the mean mass is less than 500 g. Would the conclusion be the same for a two-tailed test at 1%?

3. Type I and Type II Errors

Errors

Two ways to be wrong

A Type I error is rejecting H₀ when it is true; its probability is the probability of the critical region assuming H₀. For a normal test this equals the significance level; for a discrete test it is the actual tail probability of the critical region (at most the significance level).

Memorise

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