A-level · Hypothesis testing

Hypothesis testing for a normal meanExam-style questions

Testing a claim about the mean μ of a normal population from a sample, using that the sample mean is N(μ, σ²/n) — forming the test statistic z = (x̄ − μ₀)/(σ/√n) and comparing it to the critical z-value.

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

DifficultyAll Level 3 Level 4 Level 5

1
A machine fills bottles with a mean it claims is $500$ ml, with $\sigma = 15$ ml. A sample of $30$ bottles has mean $\bar{x} = 502$ ml. Test at the $5\%$ level whether the mean differs from $500$ ml.
Level 5 · 5 marksTry it
2
A normal population has $\sigma = 4$ . A sample of $16$ has mean $\bar{x} = 31$ . Test $H_0: \mu = 30$ against $H_1: \mu > 30$ at the $5\%$ level, using the $p$ -value.
Level 5 · 4 marksTry it
3
A machine is supposed to fill bags with $750$ g of flour, with $\sigma = 9$ g. An inspector suspects the machine is UNDER-filling. A random sample of $36$ bags has mean $746.4$ g.

Test the inspector's suspicion at the $1\%$ significance level. (6 marks)
Level 5 · 6 marksTry it
4
A machine dispenses a liquid into bottles. The volume dispensed is normally distributed with mean $500$ ml and standard deviation $4$ ml.

After the machine is serviced, the manager suspects that the mean volume may have changed. She takes a random sample of $15$ bottles and finds that the sample mean volume is $502.4$ ml.

You may assume that the standard deviation is unchanged.
(a) Stating your hypotheses clearly, test at the $5\%$ level of significance whether the mean volume dispensed has changed. (6 marks)
(b) Find the critical region for this test, giving the boundaries to $2$ decimal places. (3 marks)
(c) Explain what is meant by the $5\%$ level of significance in this context. (1 mark)
(d) The machine is serviced again. A further random sample of $15$ bottles has a mean volume of $501.8$ ml.

Explain, without further calculation, whether this sample would lead to the same conclusion. (2 marks)
Level 5 · 12 marksTry it
5
A machine dispenses a liquid into bottles. The volume dispensed is normally distributed with mean $500$ ml and standard deviation $4$ ml.

After the machine is serviced, the manager suspects that the mean volume may have changed. She takes a random sample of $15$ bottles and finds that the sample mean volume is $502.4$ ml.

You may assume that the standard deviation is unchanged.
(a) Stating your hypotheses clearly, test at the $5\%$ level of significance whether the mean volume dispensed has changed. (6 marks)
(b) Find the critical region for this test, giving the boundaries to $2$ decimal places. (3 marks)
(c) The machine is serviced again. A further random sample of $15$ bottles has a mean volume of $501.8$ ml.

Explain, without further calculation, whether this sample would lead to the same conclusion. (2 marks)
Level 5 · 11 marksTry it

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