Find or state significance level

A question is this type if and only if it asks the student to calculate or state the significance level of a test, given a specific critical region or test procedure.

4 questions · Standard +0.1

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CAIE S2 2016 June Q2
4 marks Moderate -0.3
2 Jacques is a chef. He claims that \(90 \%\) of his customers are satisfied with his cooking. Marie suspects that the true percentage is lower than \(90 \%\). She asks a random sample of 15 of Jacques' customers whether they are satisfied. She then performs a hypothesis test of the null hypothesis \(p = 0.9\) against the alternative hypothesis \(p < 0.9\), where \(p\) is the population proportion of customers who are satisfied. She decides to reject the null hypothesis if fewer than 12 customers are satisfied.
  1. In the context of the question, explain what is meant by a Type I error.
  2. Find the probability of a Type I error in Marie's test.
CAIE S2 2007 June Q4
7 marks Standard +0.3
4 At a certain airport 20\% of people take longer than an hour to check in. A new computer system is installed, and it is claimed that this will reduce the time to check in. It is decided to accept the claim if, from a random sample of 22 people, the number taking longer than an hour to check in is either 0 or 1 .
  1. Calculate the significance level of the test.
  2. State the probability that a Type I error occurs.
  3. Calculate the probability that a Type II error occurs if the probability that a person takes longer than an hour to check in is now 0.09 .
CAIE S2 2012 June Q5
8 marks Moderate -0.3
5
  1. Deng wishes to test whether a certain coin is biased so that it is more likely to show Heads than Tails. He throws it 12 times. If it shows Heads more than 9 times, he will conclude that the coin is biased. Calculate the significance level of the test.
  2. Deng throws another coin 100 times in order to test, at the \(5 \%\) significance level, whether it is biased towards Heads. Find the rejection region for this test.
OCR S2 2015 June Q8
7 marks Standard +0.8
8 The random variable \(S\) has the distribution \(\mathrm { B } ( 14 , p )\). A significance test is carried out of the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.3\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : p > 0.3\). The critical region for the test is \(S \geqslant 8\).
  1. Find the significance level of the test, correct to 3 significant figures.
  2. It is given that, on each occasion that the test is carried out, the true value of \(p\) is equally likely to be \(0.3,0.5\) or 0.7 , independently of any other test. Four independent tests are carried out. Find the probability that at least one of the tests results in a Type II error.