Further maths + large data set, Exams of Mathematics

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Pearson Edexcel GCE Mathematics (9FM0)

Paper 3B Further Statistics 1

Central Limit Theorem - Questions

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Q1.

The number of defects per metre in a roll of cloth has a Poisson distribution with mean 0. Find the probability that (a) a randomly chosen metre of cloth has 1 defect, (2) (b) the total number of defects in a randomly chosen 6 metre length of cloth is more than 2 (3) A tailor buys 300 metres of cloth. (c) Using a suitable approximation find the probability that the tailor's cloth will contain less than 90 defects. **(5) (Total 10 marks)

Q2.** An online shop sells a computer game at an average rate of 1 per day. (a) Find the probability that the shop sells more than 10 games in a 7 - day period. (3) Once every 7 days the shop has games delivered before it opens. (b) Find the least number of games the shop should have in stock immediately after a delivery so that the probability of running out of the game before the next delivery is less than 0. (3) In an attempt to increase sales of the computer game, the price is reduced for six months. A random sample of 28 days is taken from these six months. In the sample of 28 days, 36 computer games are sold. (c) Using a suitable approximation and a 5% level of significance, test whether or not the average rate of sales per day has increased during these six months. State your hypotheses clearly. (7) (Total 13 marks) ___________________________________________________________________________

Q3.

The number of houses sold by an estate agent follows a Poisson distribution, with a mean of 2 per week. (a) Find the probability that in the next 4 weeks the estate agent sells, (i) exactly 3 houses, (ii) more than 5 houses. (5) The estate agent monitors sales in periods of 4 weeks. (b) Find the probability that in the next twelve of these 4-week periods there are exactly nine periods in which more than 5 houses are sold. (3) The estate agent will receive a bonus if he sells more than 25 houses in the next 10 weeks. (c) Use a suitable approximation to estimate the probability that the estate agent receives a bonus. **(6) (Total 14 marks)

Q4.** The probability that a sunflower plant grows over 1.5 metres high is 0.25. A random sample of 40 sunflower plants is taken and each sunflower plant is measured, and its height recorded. (a) Find the probability that the number of sunflower plants over 1.5 m high is between 8 and 13 (inclusive) using (i) a Poisson approximation, (ii) a Normal approximation. (10) (b) Write down which of the approximations used in part (a) is the most accurate estimate of the probability. You must give a reason for your answer. (2) (Total 12 marks) ___________________________________________________________________________

Q7.

As part of a selection procedure for a company, applicants have to answer all 20 questions of a multiple choice test. If an applicant chooses answers at random the probability of choosing a correct answer is 0. and the number of correct answers is represented by the random variable X. (a) Suggest a suitable distribution for X. (2) Each applicant gains 4 points for each correct answer but loses 1 point for each incorrect answer. The random variable S represents the final score, in points, for an applicant who chooses answers to this test at random. (b) Show that S = 5 X − 20 (2) (c) Find E( S ) and Var( S ). (4) An applicant who achieves a score of at least 20 points is invited to take part in the final stage of the selection process. (d) Find P( S 20) (4) Cameron is taking the final stage of the selection process which is a multiple choice test consisting of 100 questions. He has been preparing for this test and believes that his chance of answering each question correctly is 0. (e) Using a suitable approximation, estimate the probability that Cameron answers more than half of the questions correctly. (5) (Total 17 marks) ______________________________________________

TOTAL FOR PAPER: 94 MARKS