Statistical hypothesis testing — Edexcel A-Level Mathematics
Test yourself on Statistical hypothesis testing with PEARSON EDEXCEL A-Level practice questions.
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Statistical hypothesis testing explained
A hypothesis test starts with a null hypothesis H₀, usually a claim about a population parameter, and an alternative hypothesis H₁ stating the direction or difference being tested.
Read the full explanation
The significance level α is the probability of rejecting H₀ when it is true. The test statistic is calculated from the sample; for a binomial model it is the number of successes X ~ B(n, p). A 1-tail test uses one extreme region, a 2-tail test splits α between both tails. The critical region is the set of test statistic values leading to rejection of H₀; its boundary is the critical value, and the acceptance region is the complement. The p-value is the probability, assuming H₀, of a result at least as extreme as observed. Correlation coefficients measure how closely data lie to a straight line; interpret a given coefficient against a given p-value or critical value, without calculating it.
5.2 Conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in context. Understand that a sample is being used to make an inference about the population and appreciate that the significance level is the probability of incorrectly rejecting the null hypothesis.
To test a claim about a proportion p, take a random sample of size n and count successes X. Under H₀, X ~ B(n, p₀), where p₀ is the claimed proportion. Choose H₁ as p > p₀, p < p₀ or p ≠ p₀ to match the claim, and fix the significance level α. Find the critical region using cumulative binomial probabilities, or find the p-value as the probability of a result at least as extreme as the observed X. Compare the observed statistic with the critical region, or the p-value with α, then reject or do not reject H₀. The sample gives evidence about the unknown population proportion, so conclusions are inferences, not certainties. The significance level is the probability of incorrectly rejecting H₀ when it is true.
5.3 Conduct a statistical hypothesis test for the mean of a Normal distribution with known, given or assumed variance and interpret the results in context.
When a variable X is modelled as Normal with unknown mean μ but variance known, given or assumed, a sample mean x̄ from n independent observations is also Normal with mean μ and variance σ²/n, so its standard error is σ/√n. A hypothesis test sets H₀: μ = μ₀ against a one- or two-tailed H₁, then standardises: z = (x̄ − μ₀)/(σ/√n). For a two-tailed test at significance level α, reject H₀ if |z| exceeds the critical value z_(α/2); for a one-tailed test compare z with z_α in the correct direction. Equivalently compare x̄ with a critical region. The conclusion must refer to the original context: state whether there is sufficient evidence at the α level to reject H₀, and phrase the finding in terms of the real quantity measured, not merely 'accept H₀'.
Your focus
- Define null and alternative hypotheses and classify a test as 1-tail or 2-tail.
- Use critical values, critical regions or p-values to decide whether to reject H₀.
- Interpret a given correlation coefficient using a given p-value or critical value in context.
Show all 9 objectives
- Carry out a 1-tail or 2-tail binomial test for a population proportion.
- Use critical regions or p-values to reach and justify a decision about H₀.
- Interpret the outcome in context and explain the meaning of the significance level.
- Set up H₀ and H₁ for a stated context and identify whether the test is one- or two-tailed.
- Calculate the standard error and test statistic correctly from given values.
- Interpret the outcome as evidence about the population mean in the original context.
Statistical hypothesis testing exam tips
Marking Points
- Defines H₀ and H₁ in context, choosing a 1-tail or 2-tail alternative to match the wording of the claim.
- States the significance level and explains it as the probability of incorrectly rejecting a true H₀.
- Identifies the test statistic and its distribution, for example X ~ B(n, p) under H₀.
- Distinguishes the critical value, critical region and acceptance region, and uses them to reach a reject or do-not-reject decision.
- Interprets a p-value by comparing it with the significance level, and interprets a given correlation coefficient using a given p-value or critical value.
- Sets up H₀ and H₁ as statements about the population proportion p, matching the direction of the claim.
- States the binomial model X ~ B(n, p₀) under H₀ and identifies the observed test statistic from the sample.
- Finds the critical region or p-value using cumulative binomial probabilities, with the correct tail or tails for the chosen test.
- Compares the statistic with the critical region, or the p-value with α, and states the decision clearly.
- Writes a conclusion in context that links the decision back to the population proportion and acknowledges sampling uncertainty.
- State H₀ and H₁ clearly, choosing one- or two-tailed form from the wording of the problem and defining μ in context.
- Identify n, the sample mean x̄, the hypothesised mean μ₀ and the known, given or assumed standard deviation σ, then compute the standard error σ/√n.
- Calculate the test statistic z = (x̄ − μ₀)/(σ/√n), or find the critical region for x̄, showing the substitution accurately.
- Compare the test statistic with the appropriate critical value, or compare the sample mean with the critical region, and state the decision about H₀.
- Interpret the decision in the context of the question, referring to evidence about the population mean at the stated significance level.
Examiner Tips
- 💡Define H₀ and H₁ in words and symbols before calculating anything.
- 💡State whether the test is 1-tail or 2-tail and show how the critical region follows from α.
- 💡Finish with a conclusion in context that refers back to the original claim, not just to H₀.
- 💡Write down n, p₀, the observed value and α before calculating.
- 💡Show the cumulative probability calculation or calculator input that gives the critical region or p-value.
- 💡End with a sentence in context that answers the original question and mentions the evidence.
- 💡Write down H₀, H₁ and the significance level before any calculation so the tail direction is fixed.
- 💡Show the standard error and the substitution into z explicitly; method marks are available for correct structure even if arithmetic slips.
- 💡Finish with a sentence that names the context variable and states the strength of evidence at the given significance level.
Common Mistakes
- Writing H₁ as an equality; correct by using an inequality such as p > 0.5 or p ≠ 0.5.
- Comparing the test statistic directly with the significance level; correct by comparing the p-value with α, or the statistic with the critical value.
- Treating a large correlation coefficient as proof of causation; correct by interpreting it only as a measure of linear closeness, subject to the given p-value or critical value.
- Using the sample proportion in the binomial model under H₀; correct by using the claimed proportion p₀ in B(n, p₀).
- Forgetting to double the tail probability in a 2-tail test; correct by splitting α between both tails.
- Concluding that H₀ is proved true when it is not rejected; correct by saying there is insufficient evidence against H₀.
- Using the sample standard deviation in place of the known, given or assumed population standard deviation when computing the standard error; the correct divisor is σ/√n with the stated σ.
- Dividing by σ rather than σ/√n, which ignores sample size; the corrected standard error is σ/√n.
- Choosing a two-tailed critical value for a one-tailed test, or comparing against the wrong tail; the corrected approach selects the critical value matching the direction of H₁.