Type I/II errors and power

Calculate or explain Type I error, Type II error, significance level, power, or operating characteristic of a test.

3 questions · Standard +0.3

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CAIE S2 2023 November Q7
12 marks Standard +0.3
A biologist wishes to test whether the mean concentration \(\mu\), in suitable units, of a certain pollutant in a river is below the permitted level of 0.5. She measures the concentration, \(x\), of the pollutant at 50 randomly chosen locations in the river. The results are summarised below. \(n = 50 \quad \Sigma x = 23.0 \quad \Sigma x^2 = 13.02\)
  1. Carry out a test at the 5% significance level of the null hypothesis \(\mu = 0.5\) against the alternative hypothesis \(\mu < 0.5\). [7]
Later, a similar test is carried out at the 5% significance level using another sample of size 50 and the same hypotheses as before. You should assume that the standard deviation is unchanged.
  1. Given that, in fact, the value of \(\mu\) is 0.4, find the probability of a Type II error. [5]
CAIE S2 2011 November Q5
8 marks Standard +0.3
Records show that the distance driven by a bus driver in a week is normally distributed with mean 1150 km and standard deviation 105 km. New driving regulations are introduced and in the next 20 weeks he drives a total of 21 800 km.
  1. Stating any assumption(s), test, at the 1% significance level, whether his mean weekly driving distance has decreased. [6]
  2. A similar test at the 1% significance level was carried out using the data from another 20 weeks. State the probability of a Type I error and describe what is meant by a Type I error in this context. [2]
OCR S2 2012 January Q9
11 marks Standard +0.3
It is desired to test whether the average amount of sleep obtained by school pupils in Year 11 is 8 hours, based on a random sample of size 64. The population standard deviation is 0.87 hours and the sample mean is denoted by \(\bar{H}\). The critical values for the test are \(\bar{H} = 7.72\) and \(\bar{H} = 8.28\).
  1. State appropriate hypotheses for the test, explaining the meaning of any symbol you use. [3]
  2. Calculate the significance level of the test. [4]
  3. Explain what is meant by a Type I error in this context. [1]
  4. Given that in fact the average amount of sleep obtained by all pupils in Year 11 is 7.9 hours, find the probability that the test results in a Type II error. [3]