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.

6 questions

CAIE S2 2016 June Q2
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
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
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
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.
Edexcel S4 Q3
3. A train company claims that the probability \(p\) of one of its trains arriving late is \(10 \%\). A regular traveller on the company's trains believes that the probability is greater than \(10 \%\) and decides to test this by randomly selecting 12 trains and recording the number \(X\) of trains that were late. The traveller sets up the hypotheses \(\mathrm { H } _ { 0 } : p = 0.1\) and \(\mathrm { H } _ { 1 } : p > 0.1\) and accepts the null hypothesis if \(x \leq 2\).
  1. Find the size of the test.
  2. Show that the power function of the test is $$1 - ( 1 - p ) ^ { 10 } \left( 1 + 10 p + 55 p ^ { 2 } \right)$$
  3. Calculate the power of the test when
    1. \(p = 0.2\),
    2. \(p = 0.6\).
  4. Comment on your results from part (c).
Edexcel S4 2002 June Q7
  1. A proportion \(p\) of the items produced by a factory is defective. A quality assurance manager selects a random sample of 5 items from each batch produced to check whether or not there is evidence that \(p\) is greater than 0.10 . The criterion that the manager uses for rejecting the hypothesis that \(p\) is 0.10 is that there are more than 2 defective items in the sample.
    1. Find the size of the test.
      (2)
    Table 1 gives some values, to 2 decimal places, of the power function of this test. \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Table 1}
    \(p\)0.150.200.250.300.350.40
    Power0.03\(r\)0.100.160.240.32
    \end{table}
  2. Find the value of \(r\). One day the manager is away and an assistant checks the production by random sample of 10 items from each batch produced. The hypothesis that \(p = 0.10\) is rejected if more than 4 defectives are found in the sample.
  3. Find P (Type I error) using the assistant's test. Table 2 gives some values, to 2 decimal places, of the power function for this test. \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Table 2}
    \(p\)0.150.200.250.300.350.40
    Power0.010.030.080.150.25\(s\)
    \end{table}
  4. Find the value of \(s\).
  5. Using the same axes, draw the graphs of the power functions of these two tests.
    1. State the value of \(p\) where these graphs cross.
    2. Explain the significance if \(p\) is greater than this value. The manager studies the graphs in part ( \(e\) ) but decides to carry on using the test based on a sample of size 5 .
  7. Suggest 2 reasons why the manager might have made this decision.