A-level · Hypothesis testing

One-tailed tests (the probability method)Exam-style questions

Carrying out a one-tailed binomial hypothesis test by the probability (p-value) method: assume H0, model X ~ B(n, p0), find the probability of a result at least as extreme as the one observed in the direction of H1 — P(X ≥ x) for an upper test or P(X ≤ x) for a lower test — compare it with the significance level, and write a full conclusion in context.

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11 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
Historically $15\%$ of seeds fail to germinate. A new batch of $40$ seeds has $11$ failures. Test, at the $5\%$ level, whether the failure rate has increased ( $X \sim B(40, 0.15)$ ).
Level 5 · 5 marksTry it
2
At a cinema, $35\%$ of customers buy popcorn. The price of popcorn is increased, and the manager suspects that the proportion of customers who buy popcorn has decreased. He chooses a random sample of $20$ customers and finds that $3$ of them buy popcorn.
(a) Use the binomial distribution to test, at the $5\%$ significance level, whether the manager's suspicion is justified.

(5 marks)

A month later the manager carries out another test of the same hypotheses, again using a random sample of $20$ customers, but this time at the $10\%$ significance level.
(b) Calculate the probability of a Type I error.

(2 marks)

The largest screen at the cinema has $250$ seats. On an evening when all $250$ seats are sold, the number, $Y$ , of customers who leave an item behind is modelled by the distribution $\mathrm{B}(250, 0.016)$ .
(c) Use a suitable approximating distribution to find $\mathrm{P}(Y = 5)$ .

(3 marks)
(d) Justify your approximating distribution in this context.

(1 mark)
Level 5 · 11 marksTry it