Prob & Stats 2 OCR Past Papers - The Grange School Blogs

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3 Jan 2008 6 The number of house sales per week handled by an estate agent is modelled by the distribution Po(3). (i) Find the probability that, in one randomly chosen week, the number of sales handled is (a) greater than 4, [2] (b) exactly 4. [2] (ii) Use a suitable approximation to the Poisson distribution to find the probability that, in a year consisting of 50 working weeks, the estate agent handles more than 165 house sales. [5] (iii) One of the conditions needed for the use of a Poisson model to be valid is that house sales are independent of one another. (a) Explain, in non-technical language, what you understand by this condition. [1] (b) State another condition that is needed. [1] 7 A continuous random variable X 1 has probability density function given by where k is a constant. kx 0 ≤ x ≤ 2, f(x) ={ 0 otherwise, (i) Show that k = 1 2 . [2] (ii) Sketch the graph of y = f(x). [2] (iii) Find E(X 1 ) and Var(X 1 ). [5] (iv) Sketch the graph of y = f(x − 1). [2] (v) The continuous random variable X 2 has probability density function f(x − 1) for all x. Write down the values of E(X 2 ) and Var(X 2 ). [2] 8 Consultations are taking place as to whether a site currently in use as a car park should be developed as a shopping mall. An agency acting on behalf of a firm of developers claims that at least 65% of the local population are in favour of the development. In a survey of a random sample of 12 members of the local population, 6 are in favour of the development. (i) Carry out a test, at the 10% significance level, to determine whether the result of the survey is consistent with the claim of the agency. [7] (ii) A local residents’ group claims that no more than 35% of the local population are in favour of the development. Without further calculations, state with a reason what can be said about the claim of the local residents’ group. [2] (iii) A test is carried out, at the 15% significance level, of the agency’s claim. The test is based on a random sample of size 2n, and exactly n of the sample are in favour of the development. Find the smallest possible value of n for which the outcome of the test is to reject the agency’s claim. [4] © OCR 2008 4733/01 Jan08
June 2008 2 1 The head teacher of a school asks for volunteers from among the pupils to take part in a survey on political interests. (i) Explain why a sample consisting of all the volunteers is unlikely to give a true picture of the political interests of all pupils in the school. [2] (ii) Describe a better method of obtaining the sample. [3] 2 The annual salaries of employees in a company have mean £30 000 and standard deviation £12 000. (i) Assuming a normal distribution, calculate the probability that the salary of one randomly chosen employee lies between £20 000 and £24 000. [4] (ii) The salary structure of the company is such that a small number of employees earn much higher salaries than the others. Explain what this suggests about the use of a normal distribution to model the data. [2] 3 In a factory the time, T minutes, taken by an employee to make a single item is a normally distributed random variable with mean 28.0. A new ventilation system is installed, after which the times taken to produce a random sample of 40 items are measured. The sample mean is 26.44 minutes and it is given that Σ t2 40 − 26.442 = 37.05. Test, at the 10% significance level, whether there is evidence of a change in the mean time taken to make an item. [8] 4 The random variable U has the distribution N(μ, σ 2 ), where the value of σ is known. A test is carried out of the null hypothesis H 0 : μ = 50 against the alternative hypothesis H 1 : μ > 50. The test is carried out at the 1% significance level and is based on a random sample of size 10. (i) The test is carried out once. The value of the sample mean is 53.0. The outcome of the test is that H 0 is not rejected. Show that σ > 4.08, correct to 3 significant figures. [4] (ii) The test is carried out repeatedly. In each test the actual value of μ is 50. Find the probability that the first test to result in a Type I error is the fifth to be carried out. Give your answer correct to 2 significant figures. [3] © OCR 2008 4733/01 Jun08
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Prob & Stats 2 OCR Past Papers - The Grange School Blogs

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