Chapter 2: The Poisson distribution
11 In the past, the number of house sales completed per week by a building company has been modelled by a random variable which has the distribution Po(0.8). Following a publicity campaign, the builders hope that the mean number of sales per week will increase. In order to test at the 5% significance level whether this is the case, the total number of sales during the first 3 weeks after the campaign is noted. It is assumed that a Poisson model is still appropriate. i
Given that the total number of sales during the 3 weeks is 5, carry out the test.
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ii During the following 3 weeks the same test is carried out again, using the same significance level. Find the probability of a Type I error.
[3]
iii Explain what is meant by a Type I error in this context.
[1]
iv State what further information would be required in order to find the probability of a Type II error.
[1]
12 A hospital patient’s white blood cell count has a Poisson distribution. Before undergoing treatment the patient had a mean white blood cell count of 5.2. After the treatment a random measurement of the patient’s white blood cell count is made, and is used to test at the 10% significance level whether the mean white blood cell count has decreased. i
State what is meant by a Type I error in the context of the question, and find the probability that the test results in a Type I error.
[4]
ii Given that the measured value of the white blood cell count after the treatment is 2, carry out the test.
[3]
iii Find the probability of a Type II error if the mean white blood cell count after the treatment is actually 4.1.
[3]
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Cambridge International AS & A Level Mathematics 9709 Paper 73 Q7 November 2010
Cambridge International AS & A Level Mathematics 9709 Paper 71 Q7 June 2010
13 Major avalanches can be regarded as randomly occurring events. They occur at a uniform average rate of 8 per year. i
Find the probability that more than 3 major avalanches occur in a 3-month period.
[3]
ii Find the probability that any two separate 4-month periods have a total of 7 major avalanches.
[3]
iii Find the probability that a total of fewer than 137 major avalanches occur in a 20-year period.
[4]
Cambridge International AS & A Level Mathematics 9709 Paper 71 Q3 June 2009
Original material Š Cambridge University Press 2017
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