A-level · Hypothesis testing

Two-tailed testsExam-style questions

Carrying out a two-tailed binomial test (H1 : p ≠ p0), where a change in either direction counts as evidence. Splitting the significance level α into two tails of α/2, finding both critical regions and the actual significance level (the sum of the two tail probabilities), the equivalent p-value method (compare the observed tail with α/2, or 2× the tail with α), and writing a full conclusion in context.

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

DifficultyAll Level 3 Level 4

1
The sheets of glass made by a factory should have a mean thickness of $6$ mm. A quality controller wishes to know whether the population mean thickness, $\mu$ mm, is different from $6$ . She measures the thickness, $x$ mm, of each sheet in a random sample of $120$ sheets. Her results are summarised below.

$\sum x = 723.6 \qquad \sum x^2 = 4370.8$

Test, at the $10\%$ significance level, whether $\mu$ is different from $6$ .

(7 marks)
Level 4 · 7 marksTry it
2
Records at a large gym show that $40\%$ $40%$ of its members go to at least one fitness class each week.

Dionne, the manager of the gym, thinks that this figure is different for the members who are aged over $60$ $60$

To investigate, she chooses $24$ $24$ members aged over $60$ $60$ at random and finds out how many of them go to at least one fitness class each week.
(a) Find the critical region for a suitable test to assess Dionne's belief.

You should
- state your hypotheses clearly
- use a $5\%$ level of significance
- state the probability of rejection in each tail
(4 marks)
(b) Write down the actual significance level of the test in part (a).

(1 mark)

Of the $24$ $24$ members aged over $60$ $60$ in Dionne's sample, $16$ $16$ go to at least one fitness class each week.
(c) With reference to your answer to part (a), comment on Dionne's belief.

(1 mark)
Level 4 · 6 marksTry it
3
Until last year, $30\%$ $30%$ of the households in a large town put out garden waste for recycling each fortnight.

The town council has since run a publicity campaign. Yusuf, who works for the council, thinks that the proportion of households in the town that put out garden waste for recycling is no longer $30\%$ $30%$

He will take a random sample of $40$ $40$ households in the town. The number of these households that put out garden waste for recycling is $X$ $X$
(a) Write down
- a null hypothesis corresponding to no change in the proportion of households that put out garden waste for recycling
- an alternative hypothesis corresponding to Yusuf's belief
(1 mark)
(b) Using a $5\%$ $5%$ level of significance, find the critical region for a two-tailed test of the null hypothesis in part (a).

You should state the probability of rejection in each tail, which should be less than $0.025$ $0.025$

(3 marks)
(c) Find the actual level of significance of the test based on your critical region from part (b).

(1 mark)

In Yusuf's sample, $17$ $17$ of the households put out garden waste for recycling.
(d) With reference to part (b), comment on Yusuf's belief.

(1 mark)
Level 4 · 6 marksTry it