Find critical region

A question is this type if and only if it asks to determine the critical region for a hypothesis test at a given significance level, without carrying out the test.

3 questions · Standard +0.3

2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail5.02i Poisson distribution: random events model
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Edexcel S2 2022 January Q3
9 marks Standard +0.3
3 A photocopier in a school is known to break down at random at a mean rate of 8 times per week.
  1. Give a reason why a Poisson distribution could be used to model the number of breakdowns. The headteacher of the school replaces the photocopier with a refurbished one and wants to find out if the rate of breakdowns has increased or decreased.
  2. Write down suitable null and alternative hypotheses that the headteacher should use. The refurbished photocopier was monitored for the first week after it was installed.
  3. Using a \(5 \%\) level of significance, find the critical region to test whether the rate of breakdowns has now changed.
  4. Find the actual significance level of a test based on the critical region from part (c). During the first week after it was installed there were 4 breakdowns.
  5. Comment on this finding in the light of the critical region found in part (c).
CAIE S2 2016 June Q4
7 marks Standard +0.3
The number of sightings of a golden eagle at a certain location has a Poisson distribution with mean 2.5 per week. Drilling for oil is started nearby. A naturalist wishes to test at the 5\% significance level whether there are fewer sightings since the drilling began. He notes that during the following 3 weeks there are 2 sightings.
  1. Find the critical region for the test and carry out the test. [5]
  2. State the probability of a Type I error. [1]
  3. State why the naturalist could not have made a Type II error. [1]
CAIE S2 2002 November Q4
7 marks Standard +0.3
The number of accidents per month at a certain road junction has a Poisson distribution with mean 4.8. A new road sign is introduced warning drivers of the danger ahead, and in a subsequent month 2 accidents occurred.
  1. A hypothesis test at the 10% level is used to determine whether there were fewer accidents after the new road sign was introduced. Find the critical region for this test and carry out the test. [5]
  2. Find the probability of a Type I error. [2]