S7.1StatisticsStretch

Hypothesis Testing

Work through a binomial hypothesis test from assumptions and hypotheses to a tail probability and a conclusion in context. Understand significance levels, one-tailed and two-tailed alternatives, and why a test does not prove a claim.

30 min Hypothesis Testing

Edexcel AS Level Maths: 7.1 Hypothesis TestingCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

State null and alternative hypotheses
Understand the significance level
Define the test statistic
Outline the logic of a hypothesis test

Null and alternative hypotheses

The $\textbf{null hypothesis}$ $H_0$ states the assumed value (e.g. $p = 0.5$ ); the $\textbf{alternative hypothesis}$ $H_1$ states what you suspect instead (e.g. $p > 0.5$ ). You test against $H_0$ .

H_0: p = p_0 \qquad H_1: p \ne p_0 \ \text{(or } >, <)

Always state both hypotheses in terms of the parameter.

Significance level

The $\textbf{significance level}$ (e.g. 5%) is the chosen upper bound on the probability of rejecting $H_0$ when it is true. A smaller level demands stronger evidence. For a discrete distribution, the actual probability of a false rejection can be smaller than the stated level because whole outcomes must be included in the critical region.

Tip — The significance level is chosen before seeing the data — typically 5% or 1%.

The logic

Assume $H_0$ is true, then find the probability of the observed result or one more extreme in the direction of $H_1$ . Compare this tail probability with the significance level; reject $H_0$ if it is at most the level. Otherwise do not reject $H_0$ . Conclusions are stated in context, never as absolute proof.

Worked binomial test: is a coin biased towards heads?

A coin is tossed 10 times and produces 9 heads. Test whether it is biased towards heads at the 5% significance level. Assume tosses are independent and the probability of a head is constant. Define p as the population probability of a head and X as the number of heads in 10 tosses.

Example 1Carry out the test and state a conclusion in context.

H_0:p=0.5

and

H_1:p>0.5

. The direction is chosen from the claim, before looking at the sample.

2Under

H_0

X\sim\operatorname{B}(10,0.5)

. This model needs a fixed number of independent trials with two outcomes and constant success probability.

3An outcome at least as extreme as 9 heads is 9 or 10 heads:

P(X\geq9)=\binom{10}{9}(0.5)^{10}+\binom{10}{10}(0.5)^{10}=\tfrac{11}{1024}\approx0.0107

0.0107<0.05

, so reject

H_0

at the 5% level.

AnswerThere is sufficient evidence at the 5% significance level that the coin is biased towards heads.

Example 2What if the same experiment had produced only 7 heads?

P(X\geq7)=\tfrac{120+45+10+1}{1024}=\tfrac{176}{1024}=0.171875

0.171875>0.05

, so do not reject

H_0

AnswerThere is insufficient evidence that the coin is biased towards heads. This does not prove that it is fair.

Tip — Use the whole relevant tail, not just P(X=9). The tail probability is calculated assuming H0; it is not the probability that H0 is true.

One-tailed and two-tailed alternatives

A claim that p has increased gives $H_1:p>p_0$ and an upper-tail test. A decrease gives $H_1:p<p_0$ and a lower-tail test. A claim that p has changed in either direction gives $H_1:p\ne p_0$ .

For the usual equal-tail A-Level binomial test at 5%, allow at most 2.5% in each tail when choosing critical regions. In a discrete distribution the actual tail probabilities may be smaller. State the alternative first; do not choose a favourable direction after seeing the data.

Formula recap

H_0: p = p_0

Null hypothesis (assumed value).

H_1: p > p_0,\ <,\ \text{or} \ne

Alternative hypothesis.

\text{reject } H_0 \text{ if p-value} \leq \alpha

Compare the appropriate tail probability with the chosen significance level.

Common mistakes to avoid

Writing hypotheses about the sample instead of the population parameter.

H₀ and H₁ are statements about the population parameter p.

Saying a test “proves” H₁.

A test gives evidence to reject (or not) H₀ — never absolute proof.

Key takeaways

H₀ states the assumed parameter value; H₁ states the suspected change.
The significance level sets how strong the evidence must be.
Assume H₀, judge how surprising the data is, then reject or not — in context.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Hypothesis Testing. Use the feedback to decide what needs another look.

2 past-paper questions · 2 marks

Practise this topic

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