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Y12 Easter IL 2020: Statistics 10 Exam QuestionsStudent name:1Three bags, A, B and C, each contain 1 red marble and some green marbles.Bag A contains 1 red marble and 9 green marbles onlyBag B contains 1 red marble and 4 green marbles onlyBag C contains 1 red marble and 2 green marbles onlySasha selects at random one marble from bag A. If he selects a red marble, he stops selecting. If the marble is green, he continues by selecting at random one marble from bag B. If he selects a red marble, he stops selecting. If the marble is green, he continues by selecting at random one marble from bag C.(a)??Draw a tree diagram to represent this information.(2)(b)??Find the probability that Sasha selects 3 green marbles.(2)(c)??Find the probability that Sasha selects at least 1 marble of each colour.(2)(d)??Given that Sasha selects a red marble, find the probability that he selects it from bag B.(2) (Total for question = 8 marks)?Q2.?In a large company,78% of employees are car owners, 30% of these car owners are also bike owners, 85% of those who are not car owners are bike owners.(a) Draw a tree diagram to represent this information.(3)An employee is selected at random.(b) Find the probability that the employee is a car owner or a bike owner but not both.(2)Another employee is selected at random.Given that this employee is a bike owner,(c) find the probability that the employee is a car owner.(3)Two employees are selected at random.(d) Find the probability that only one of them is a bike owner.(3)(Total 11 marks)Q3.?A midwife records the weights, in kg, of a sample of 50 babies born at a hospital. Her results are given in the table below.[You may use ∑ fx2 = 611.375]A histogram has been drawn to represent these data.The bar representing the weight 2 ≤ w < 3 has a width of 1 cm and a height of 4 cm.(a)??Calculate the width and height of the bar representing a weight of 3 ≤ w < 3.5(3)(b)??Use linear interpolation to estimate the median weight of these babies.(2)(c)??(i)??Show that an estimate of the mean weight of these babies is 3.43 kg.(ii)??Find an estimate of the standard deviation of the weights of these babies.(3)Shyam decides to model the weights of babies born at the hospital, by the random variable W, where W ~ N(3.43, 0.652)(d)??Find P(W < 3)(3)(e)??With reference to your answers to (b), (c)(i) and (d) comment on Shyam's decision.(3)A newborn baby weighing 3.43 kg is born at the hospital.(f)??Without carrying out any further calculations, state, giving a reason, what effect the addition of this newborn baby to the sample would have on your estimate of the(i)??mean,(ii)??standard deviation.(3)?(Total for question = 17 marks)?Q4.?Charlie is studying the time it takes members of his company to travel to the office. He stands by the door to the office from 08 40 to 08 50 one morning and asks workers, as they arrive, how long their journey was.(a)??State the sampling method Charlie used.(1)(b)??State and briefly describe an alternative method of non-random sampling Charlie could have used to obtain a sample of 40 workers.(2)Taruni decided to ask every member of the company the time, x minutes, it takes them to travel to the office.(c)??State the data selection process Taruni used.(1)Taruni's results are summarised by the box plot and summary statistics below.(d)??Write down the interquartile range for these data.(1)(e)??Calculate the mean and the standard deviation for these data.(3)(f)??State, giving a reason, whether you would recommend using the mean and standard deviation or the median and interquartile range to describe these data.(2)Rana and David both work for the company and have both moved house since Taruni collected her data.Rana's journey to work has changed from 75 minutes to 35 minutes and David's journey to work has changed from 60 minutes to 33 minutes.Taruni drew her box plot again and only had to change two values.(g)??Explain which two values Taruni must have changed and whether each of these values has increased or decreased.(3)?(Total for question = 13 marks)?Q5.?A potter believes that 20% of pots break whilst being fired in a kiln. Pots are fired in batches of 25.(a)??Let X denote the number of broken pots in a batch. A batch is selected at random. Using a 10% significance level, find the critical region for a two tailed test of the potter's belief. You should state the probability in each tail of your critical region.(4)The potter aims to reduce the proportion of pots which break in the kiln by increasing the size of the batch fired. He now fires pots in batches of 50. He then chooses a batch at random and discovers there are 6 pots which broke whilst being fired in the kiln.(b)??Test, at the 5% level of significance, whether or not there is evidence that increasing the number of pots in a batch has reduced the percentage of pots that break whilst being fired in the kiln. State your hypotheses clearly.(5)?(Total for question = 9 marks)?Q6.?The time taken for a randomly selected person to complete a test is M minutes, where M ~ N (14, σ2)Given that 10% of people take less than 12 minutes to complete the test,(a)??find the value of σ(3)Graham selects 15 people at random.(b)??Find the probability that fewer than 2 of these people will take less than 12 minutes to complete the test.(3)Jovanna takes a random sample of n people.Using a normal approximation, the probability that fewer than 9 of these n people will take less than 12 minutes to complete the test is 0.3085 to 4 decimal places.(c)??Find the value of n.(8)?(Total for question = 14 marks)?Q7.?(a) State the conditions under which the normal distribution may be used as an approximation to the binomial distribution.(2)A company sells seeds and claims that 55% of its pea seeds germinate.(b) Write down a reason why the company should not justify their claim by testing all the pea seeds they produce.(1)To test the company's claim, a random sample of 220 pea seeds was planted.(c) State the hypotheses for a two-tailed test of the company's claim.(1)Given that 135 of the 220 pea seeds germinated,(d) use a normal approximation to test, at the 5% level of significance, whether or not the company's claim is justified.(7)(Total 11 marks)Q8.?The time taken, in minutes, by children to complete a mathematical puzzle is assumed to be normally distributed with mean μ and standard deviation σ. The puzzle can be completed in less than 24 minutes by 80% of the children. For 5% of the children it takes more than 28 minutes to complete the puzzle.(a) Show this information on the Normal curve below.(2)(b) Write down the percentage of children who take between 24 minutes and 28 minutes to complete the puzzle.(1)(c) (i) Find two equations in μ and σ.(ii) Hence find, to 3 significant figures, the value of μ and the value of σ.(7)A child is selected at random.(d) Find the probability that the child takes less than 12 minutes to complete the puzzle.(3)(Total 13 marks)Q9.?The Venn diagram shows three events A, B and C, where p, q, r, s and t are probabilities.P(A) = 0.5, P(B) = 0.6 and P(C) = 0.25 and the events B and C are independent.(a)??Find the value of p and the value of q.(2)(b)??Find the value of r.(2)(c)??Hence write down the value of s and the value of t.(2)(d)??State, giving a reason, whether or not the events A and B are independent.(2)(e)??Find P(B | A ∪ C).(3)?(Total for question = 11 marks)?Q10.The Venn diagram in Figure 1 shows the number of students in a class who read any of 3 popular magazines A, B and C.One of these students is selected at random.(a)??Show that the probability that the student reads more than one magazine is (2)(b)??Find the probability that the student reads A or B (or both).(2)(c)??Write down the probability that the student reads both A and C.(1)Given that the student reads at least one of the magazines,(d)??find the probability that the student reads C.(2)(e)??Determine whether or not reading magazine B and reading magazine C are statistically independent.(3)(Total 10 marks)Total 117 Marks