Hypothesis tests in A Level Math Paper 6 (Statistics 2): common mistakes and examiner advice

Hypothesis test questions follow a pattern: hypotheses, a test (using a binomial, Poisson or Normal distribution), a comparison, a conclusion, and often a Type I or Type II error. The reports show the same weaknesses in every session: the parameter in the hypotheses, the comparison and the wording of the conclusion. This guide collects what Cambridge examiners say students get wrong, based on recent reports.

Summarised in our own words from a reading of the examiner reports.

Mistakes that cost marks

• The hypotheses. Omitting them, or giving them incorrectly. Using x, X̄ or μ for a proportion where p is needed, writing "mean" and not "population mean", and omitting the parameter altogether (writing H₀ = 0.15). Some did a one-tail test when a two-tail test was needed.
• The comparison. Not showing one, or showing an invalid one, such as a z-value with an area. For a two-tail test, comparing with 0.05 rather than 0.025 (or the reverse), or using the one-tail z-value even when a two-tail test had been stated. Reports mention this in many sessions.
• The conclusion. Not in context, or definite in language. The reports say it should include a level of uncertainty, for example "there is evidence to suggest ...".
• The Poisson test. Using an incorrect λ, including an extra term, omitting a term, and calculating P(0, 1, 2) when 1 − P(0, 1, 2) was needed (and the other way round).
• The binomial test. Using the wrong inequality (≥ or >, ≤ or <) when carrying out the test. In one question, errors in 1 − P(X ≤ 3) included omitting "1 −", using the wrong value of p, and omitting a term.
• The Normal approximation. Using an incorrect Normal distribution (for B(100, 0.25), the approximation should be N(25, 18.75)), and a wrong, or no, continuity correction. Some continued to use a Normal distribution where a Poisson was needed.
• Standardising. Omitting a √n, confusing variance with standard deviation, using a wrong sign for z, using 0.95 or 0.05 in place of a z value, and finding the wrong probability area (greater than 0.5 when less than 0.5 was needed).
• A single value rather than an interval. Considering one value (such as = 3.25) instead of the whole range of values the question required.
• Critical value. Many did not find the critical value, which the question required.
• Significance level and Type I error. Many candidates assumed the significance level had to be a familiar value such as 1% or 2%, many did not see that the required answer was the probability already calculated in an earlier part, and some thought the probability of a Type I error was 0.02 from the significance level. Few candidates fully justified a value such as 0.0239 for the probability of a Type I error.
• Type II error. Using an incorrect definition, defining it without applying it to the context of the question, not saying that the dice was not actually fair, not commenting on the claim, and confusion over which region relates to a Type II error. The reports say generic definitions are not acceptable.
• Accuracy and form. Giving two significant figures rather than three, omitting a percentage sign (3.55 for 3.55%), and recalculating a test statistic with incorrect figures from an earlier part.

What examiners advise

• Write the hypotheses in terms of the right parameter: p for a population proportion, and μ (or the words "population mean") for a population mean.
• Show a clear and valid comparison: the test statistic with the critical value, or the probability with the significance level (halved for a two-tail test where necessary).
• State the conclusion in context, with a level of uncertainty in the language.
• For Type I and Type II errors, write the definition in the context of the question, and say what the real situation is when the error has been made.
• Give probabilities to 3 significant figures, and percentages with the % sign.

Habits that help

• Before you calculate, say whether the test is one-tail or two-tail and write the significance level to compare with.
• Write your rejection region as an inequality, such as X ≥ 9, and check it using the exact probabilities.
• Finish with a one-sentence conclusion that mentions the situation by name.

Questions students ask

How do I write the hypotheses?

In terms of the right parameter: p for a population proportion, μ or "population mean" for a mean. Do not use x or X̄, and do not omit the parameter, as in writing H₀ = 41.5.

How do I make a valid comparison?

Compare the test statistic with the critical value, or the probability with the significance level, halved for a two-tail test. Comparing a z-value with an area is not valid, and a two-tail test needs 0.025, not 0.05.

How should I word the conclusion?

In context and with a level of uncertainty, for example "there is evidence to suggest …". Definite language such as "proves" loses marks. Mention the situation by name.

How do I define Type I and Type II errors?

Write the definition in the context of the question and say what the real situation is when the error has been made. Many candidates assumed the significance level had to be a familiar value such as 1% or 2%.

What goes wrong in a Normal approximation to a binomial test?

Using the wrong distribution (for B(100, 0.25), the approximation should be N(25, 18.75)), a wrong or missing continuity correction, and confusing variance with standard deviation.