> Thursday 7 October 2021 – Afternoon A Level Further Mathematics B (MEI) Y422/01 Statistics Major

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Section A (29 marks) Answer all the questions. 1 When babies are born, their head circumferences are measured. A random sample of 50 newborn female babies is selected. The sample mean head circumfe... rence is 34.711cm. The sample standard deviation head circumference is 1.530cm. (a) Determine a 95% confidence interval for the population mean head circumference of newborn female babies. [4] (b) Explain why you can calculate this interval even though the distribution of the population of head circumferences of newborn female babies is unknown. [2] 2 In a game at a charity fair, a player rolls 3 unbiased six-sided dice. The random variable X represents the difference between the highest and lowest scores. (a) Show that P 0 ( ) X = = 36 1 . [2] The table shows the probability distribution of X. r 0 1 2 3 4 5 P( ) X r = 36 1 36 5 9 2 4 1 9 2 36 5 (b) Draw a graph to illustrate the distribution. [2] (c) Describe the shape of the distribution. [1] (d) In this question you must show detailed reasoning. Find each of the following. • E(X) • Var(X) [5] As a result of playing the game, the player receives 30X pence from the organiser of the game. (e) Find the variance of the amount that the player receives. [1] (f) The player pays k pence to play the game. Given that the average profit made by the organiser is 12.5 pence per game, determine the value of k. [2]3 © OCR 2021 Y422/01 Oct21 Turn over 3 In air traffic management, air traffic controllers send radio messages to pilots. On receiving a message, the pilot repeats it back to the controller to check that it has been understood correctly. At a particular site, on average 4% of messages sent by controllers are not repeated back correctly and so have been misunderstood. You should assume that instances of messages being misunderstood occur randomly and independently. (a) Find the probability that exactly 2 messages are misunderstood in a sequence of 50 messages. [2] (b) Find the probability that in a sequence of messages, the 10th message is the first one which is misunderstood. [1] (c) Find the probability that in a sequence of 20 messages, there are no misunderstood messages. [1] (d) Determine the expected number of messages required for 3 of them to be misunderstood. [3] (e) Determine the probability that in a sequence of messages, the 3rd misunderstood message is the 60th message in the sequence. [3]4 © OCR 2021 Y422/01 Oct21 Section B (91 marks) Answer all the questions. 4 A radioactive source contains 1 000000 nuclei of a particular radioisotope. On average 1 in 200 000 of these nuclei will decay in a period of 1 second. The random variable X represents the number of nuclei which decay in a period of 1 second. You should assume that nuclei decay randomly and independently of each other. (a) Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of X. [3] Use a Poisson distribution to answer parts (b) and (c). (b) Calculate each of the following probabilities. • P(X = 6) • P(X 2 6) [3] (c) Determine an estimate of the probability that at least 60 nuclei decay in a period of 10 seconds. [Show More]

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