State hypotheses only

Questions that ask only to state or write down the null and alternative hypotheses, with no further work required.

5 questions

Edexcel Paper 3 2022 June Q4
  1. A dentist knows from past records that \(10 \%\) of customers arrive late for their appointment.
A new manager believes that there has been a change in the proportion of customers who arrive late for their appointment. A random sample of 50 of the dentist's customers is taken.
  1. Write down
    • a null hypothesis corresponding to no change in the proportion of customers who arrive late
    • an alternative hypothesis corresponding to the manager's belief
    • Using a \(5 \%\) level of significance, find the critical region for a two-tailed test of the null hypothesis in (a) You should state the probability of rejection in each tail, which should be less than 0.025
    • Find the actual level of significance of the test based on your critical region from part (b)
    The manager observes that 15 of the 50 customers arrived late for their appointment.
  2. With reference to part (b), comment on the manager's belief.
Edexcel FS1 2024 June Q5
  1. Some of the components produced by a factory are defective. The management requires that no more than \(3 \%\) of the components produced are defective.
    Niluki monitors the production process and takes a random sample of \(n\) components.
    1. Write down the hypotheses Niluki should use in a test to assess whether or not the proportion of defective components is greater than 0.03
    Niluki defines the random variable \(D _ { n }\) to represent the number of defective components in a sample of size \(n\). She considers two tests \(\mathbf { A }\) and \(\mathbf { B }\) In test \(\mathbf { A }\), Niluki uses \(n = 100\) and if \(D _ { 100 } \geqslant 5\) she rejects \(H _ { 0 }\)
  2. Find the size of test \(\mathbf { A }\) In test B, Niluki uses \(n = 80\) and
    • if \(D _ { 80 } \geqslant 5\) she rejects \(\mathrm { H } _ { 0 }\)
    • if \(D _ { 80 } \leqslant 3\) she does not reject \(\mathrm { H } _ { 0 }\)
    • if \(D _ { 80 } = 4\) she takes a second random sample of size 80 and if \(D _ { 80 } \geqslant 1\) in this second sample then she rejects \(\mathrm { H } _ { 0 }\) otherwise she does not reject \(\mathrm { H } _ { 0 }\)
    • Find the size of test \(\mathbf { B }\)
    Given that the actual proportion of defective components is 0.06
    1. find the power of test \(\mathbf { A }\)
    2. find the expected number of components sampled using test \(\mathbf { B }\) Given also that, when the actual proportion of defective components is 0.06 , the power of test \(\mathbf { B }\) is 0.713
  4. suggest, giving your reasons, which test Niluki should use.
AQA AS Paper 2 2023 June Q18
18
  1. State the Null and Alternative hypotheses for this test. 18
  2. State, in context, the conclusion to this test. 18 It is believed that 25\% of the customers at a bakery buy a loaf of bread. is believed that \(25 \%\) of the customers at a bakery buy a loaf of bread.
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Edexcel AS Paper 2 Specimen Q5
5. (a) The discrete random variable \(X \sim \mathrm {~B} ( 40,0.27 )\) $$\text { Find } \quad \mathrm { P } ( X \geqslant 16 )$$ Past records suggest that \(30 \%\) of customers who buy baked beans from a large supermarket buy them in single tins. A new manager suspects that there has been a change in the proportion of customers who buy baked beans in single tins. A random sample of 20 customers who had bought baked beans was taken.
(b) Write down the hypotheses that should be used to test the manager's suspicion.
(c) Using a \(10 \%\) level of significance, find the critical region for a two-tailed test to answer the manager's suspicion. You should state the probability of rejection in each tail, which should be less than 0.05
(d) Find the actual significance level of a test based on your critical region from part (c). One afternoon the manager observes that 12 of the 20 customers who bought baked beans, bought their beans in single tins.
(e) Comment on the manager's suspicion in the light of this observation. Later it was discovered that the local scout group visited the supermarket that afternoon to buy food for their camping trip.
(f) Comment on the validity of the model used to obtain the answer to part (e), giving a reason for your answer.
WJEC Unit 2 Specimen Q1
  1. The events \(\mathrm { A } , B\) are such that \(P ( A ) = 0.2 , P ( B ) = 0.3\). Determine the value of \(P ( A \cup B )\) when
    1. \(A , B\) are mutually exclusive,
    2. \(A , B\) are independent,
    3. \(\quad A \subset B\).
    4. Dewi, a candidate in an election, believes that \(45 \%\) of the electorate intend to vote for him. His agent, however, believes that the support for him is less than this. Given that \(p\) denotes the proportion of the electorate intending to vote for Dewi,
    5. state hypotheses to be used to resolve this difference of opinion.
    They decide to question a random sample of 60 electors. They define the critical region to be \(X \leq 20\), where \(X\) denotes the number in the sample intending to vote for Dewi.
    1. Determine the significance level of this critical region.
    2. If in fact \(p\) is actually 0.35 , calculate the probability of a Type II error.
    3. Explain in context the meaning of a Type II error.
    4. Explain briefly why this test is unsatisfactory. How could it be improved while keeping approximately the same significance level?