Two-tailed test setup or execution

A question is this type if and only if it involves a two-tailed hypothesis test (H₁: λ ≠ λ₀) requiring critical regions in both tails.

8 questions · Standard +0.8

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OCR S2 2008 January Q3
8 marks Standard +0.8
3 The random variable \(G\) has the distribution \(\operatorname { Po } ( \lambda )\). A test is carried out of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 4.5\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \lambda \neq 4.5\), based on a single observation of \(G\). The critical region for the test is \(G \leqslant 1\) and \(G \geqslant 9\).
  1. Find the significance level of the test.
  2. Given that \(\lambda = 5.5\), calculate the probability that the test results in a Type II error.
Edexcel S2 2015 June Q4
5 marks Challenging +1.2
  1. A single observation \(x\) is to be taken from a Poisson distribution with parameter \(\lambda\) This observation is to be used to test, at a \(5 \%\) level of significance,
$$\mathrm { H } _ { 0 } : \lambda = k \quad \mathrm { H } _ { 1 } : \lambda \neq k$$ where \(k\) is a positive integer.
Given that the critical region for this test is \(( X = 0 ) \cup ( X \geqslant 9 )\)
  1. find the value of \(k\), justifying your answer.
  2. Find the actual significance level of this test.
Edexcel S2 2003 January Q2
8 marks Standard +0.8
2. A single observation \(x\) is to be taken from a Poisson distribution with parameter \(\lambda\). This observation is to be used to test \(\mathrm { H } _ { 0 } : \lambda = 7\) against \(\mathrm { H } _ { 1 } : \lambda \neq 7\).
  1. Using a \(5 \%\) significance level, find the critical region for this test assuming that the probability of rejection in either tail is as close as possible to \(2.5 \%\).
  2. Write down the significance level of this test. The actual value of \(x\) obtained was 5 .
  3. State a conclusion that can be drawn based on this value.
Edexcel S2 2008 January Q7
14 marks Standard +0.3
  1. (a) Explain what you understand by
    1. a hypothesis test,
    2. a critical region.
    During term time, incoming calls to a school are thought to occur at a rate of 0.45 per minute. To test this, the number of calls during a random 20 minute interval, is recorded.
    (b) Find the critical region for a two-tailed test of the hypothesis that the number of incoming calls occurs at a rate of 0.45 per 1 minute interval. The probability in each tail should be as close to \(2.5 \%\) as possible.
    (c) Write down the actual significance level of the above test. In the school holidays, 1 call occurs in a 10 minute interval.
    (d) Test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of incoming calls is less during the school holidays than in term time.
Edexcel S2 2014 June Q3
13 marks Standard +0.3
  1. A company claims that it receives emails at a mean rate of 2 every 5 minutes.
    1. Give two reasons why a Poisson distribution could be a suitable model for the number of emails received.
    2. Using a \(5 \%\) level of significance, find the critical region for a two-tailed test of the hypothesis that the mean number of emails received in a 10 minute period is 4 . The probability of rejection in each tail should be as close as possible to 0.025
    3. Find the actual level of significance of this test.
    To test this claim, the number of emails received in a random 10 minute period was recorded. During this period 8 emails were received.
  2. Comment on the company's claim in the light of this value. Justify your answer. During a randomly selected 15 minutes of play in the Wimbledon Men's Tennis Tournament final, 2 emails were received by the company.
  3. Test, at the \(10 \%\) level of significance, whether or not the mean rate of emails received by the company during the Wimbledon Men’s Tennis Tournament final is lower than the mean rate received at other times. State your hypotheses clearly.
Edexcel S2 2015 June Q5
12 marks Standard +0.3
  1. Liftsforall claims that the lift they maintain in a block of flats breaks down at random at a mean rate of 4 times per month. To test this, the number of times the lift breaks down in a month is recorded.
    1. Using a 5\% level of significance, find the critical region for a two-tailed test of the null hypothesis that 'the mean rate at which the lift breaks down is 4 times per month'. The probability of rejection in each of the tails should be as close to \(2.5 \%\) as possible.
    Over a randomly selected 1 month period the lift broke down 3 times.
  2. Test, at the \(5 \%\) level of significance, whether Liftsforall's claim is correct. State your hypotheses clearly.
  3. State the actual significance level of this test.
    ! The residents in the block of flats have a maintenance contract with Liftsforall. The residents pay Liftsforall \(\pounds 500\) for every quarter ( 3 months) in which there are at most 3 breakdowns. If there are 4 or more breakdowns in a quarter then the residents do not pay for that quarter. Liftsforall installs a new lift in the block of flats.
    Given that the new lift breaks down at a mean rate of 2 times per month,
  4. find the probability that the residents do not pay more than \(\pounds 500\) to Liftsforall in the next year.
Edexcel S4 2004 June Q3
9 marks Challenging +1.2
3. It is suggested that a Poisson distribution with parameter \(\lambda\) can model the number of currants in a currant bun. A random bun is selected in order to test the hypotheses \(\mathrm { H } _ { 0 } : \lambda = 8\) against \(\mathrm { H } _ { 1 } : \lambda \neq 8\), using a \(10 \%\) level of significance.
  1. Find the critical region for this test, such that the probability in each tail is as close as possible to \(5 \%\).
  2. Given that \(\lambda = 10\), find
    1. the probability of a type II error,
    2. the power of the test.
      (4)
Edexcel FS1 2019 June Q5
12 marks Challenging +1.2
  1. Information was collected about accidents on the Seapron bypass. It was found that the number of accidents per month could be modelled by a Poisson distribution with mean 2.5 Following some work on the bypass, the numbers of accidents during a series of 3-month periods were recorded. The data were used to test whether or not there was a change in the mean number of accidents per month.
    1. Stating your hypotheses clearly and using a \(5 \%\) level of significance, find the critical region for this test. You should state the probability in each tail.
    2. State P(Type I error) using this test.
    Data from the series of 3-month periods are recorded for 2 years.
  2. Find the probability that at least 2 of these 3-month periods give a significant result. Given that the number of accidents per month on the bypass, after the work is completed, is actually 2.1 per month,
  3. find P (Type II error) for the test in part (a)