AQA A-Level Further Maths Paper 3 Statistics June 2021 PDF

Please write clearly in block capitals. Centre number Candidate number Surname _________________________________________________________________________ Forename(s) ______________________________________________________...

Please write clearly in block capitals. Centre number Candidate number Surname _________________________________________________________________________ Forename(s) _________________________________________________________________________ Candidate signature _________________________________________________________________________ I declare this is my own work. A-level FURTHER MATHEMATICS Paper 3 Statistics Time allowed: 2 hours Materials For Examiner’s Use l You must have the AQA Formulae and statistical tables booklet for A‑level Mathematics and A‑level Further Mathematics. Question Mark l You should have a scientific calculator that meets the requirements of the specification. (You may use a graphical calculator.) 1 l You must ensure you have the other optional Question Paper/Answer Book 2 for which you are entered (either Discrete or Mechanics). You will have 2 hours to complete both papers. 3 4 Instructions l Use black ink or black ball‑point pen. Pencil should only be used for drawing. 5 l Fill in the boxes at the top of this page. l Answer all questions. 6 l You must answer each question in the space provided for that question. 7 If you require extra space for your answer(s), use the lined pages at the end of this book. Write the question number against your answer(s). 8 l Do not write outside the box around each page or on blank pages. TOTAL l Show all necessary working; otherwise marks for method may be lost. l Do all rough work in this book. Cross through any work you do not want to be marked. Information l The marks for questions are shown in brackets. l The maximum mark for this paper is 50. Advice l Unless stated otherwise, you may quote formulae, without proof, from the booklet. l You do not necessarily need to use all the space provided. (JUN2173673S01) PB/Jun21/E4 7367/3S  2 Do not write outside the box Answer all questions in the spaces provided. 1 The discrete uniform distribution X can take values 1, 2, 3, …, 10 Find P(X  7) Circle your answer. [1 mark] 0.3 0.4 0.6 0.7 2 The random variable X has variance Var (X ) Which of the following expressions is equal to Var (aX þ b), where a and b are non-zero constants? Circle your answer. [1 mark] a Var (X ) a Var (X ) þ b a 2 Var (X ) a 2 Var (X ) þ b (02) Jun21/7367/3S  3 Do not write outside the 3 In a game, it is only possible to score 10, 20 or 30 points. box The probability of scoring 20 points is twice the probability of scoring 30 points. The probability of scoring 20 points is half the probability of scoring 10 points. 3 (a) Find the mean points scored when the game is played once, giving your answer to two decimal places. [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 3 (b) Mina plays the game. Her father, Michael, tells her that he will multiply her score by 5 and then subtract 10 He will then give her the value he has calculated in pence rounded to the nearest penny. Calculate the expected value in pence that Mina receives. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over s (03) Jun21/7367/3S  4 Do not write outside the 4 Oscar is studying the daily maximum temperature in C in a village during the month box of June. He constructs a 95% confidence interval of width 0.8 C using a random sample of 150 days. He assumes that the daily maximum temperature has a normal distribution. 4 (a) Find the standard deviation of Oscar’s sample, giving your answer to three significant figures. [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 4 (b) Oscar calculates the mean of his sample to be 25.3 C He claims that the population mean is 26.0 C Explain whether or not his confidence interval supports his claim. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (04) Jun21/7367/3S  5 Do not write outside the 4 (c) Explain how Oscar could reduce the width of his 95% confidence interval. box [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over for the next question Turn over s (05) Jun21/7367/3S  6 Do not write outside the 5 The continuous random variable X has cumulative distribution function box 8 > 0 x1 > > > > 1 1 > > < x 1 > 1 2 1 > > x þ 6 > 90 10 > : 1 x>9 5 (a) Find the probability density function f (x) [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 6737 5 (b) Show that Var (X ) ¼ 1200 [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (06) Jun21/7367/3S  7 Do not write outside the Turn over for the next question box DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED Turn over s (07) Jun21/7367/3S  8 Do not write outside the 6 Danai is investigating the number of speeding offences in different towns in a country. box She carries out a hypothesis test to test for association between town and number of speeding offences per year. 6 (a) State the hypotheses for this test. [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 6 (b) The observed frequencies, O, have been collected and the expected frequencies, E, have been calculated in an n  m contingency table, where n > 3 and m > 3 One of the values of E is less than 5 6 (b) (i) Explain what steps Danai should take before calculating the test statistic. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 6 (b) (ii) State an expression for the test statistic Danai should calculate. [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (08) Jun21/7367/3S  9 Do not write outside the 6 (c) Danai correctly calculates the value of the test statistic to be 45.22 box The number of degrees of freedom for the test is 25 Determine the outcome of Danai’s test, using the 1% level of significance. [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over for the next question Turn over s (09) Jun21/7367/3S  10 Do not write outside the 7 The random variable X has an exponential distribution with parameter l box 1 7 (a) Prove that E(X ) ¼ l [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (10) Jun21/7367/3S  11 Do not write outside the 1 box 7 (b) Prove that Var (X ) ¼ l2 [7 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over s (11) Jun21/7367/3S  12 Do not write outside the 8 A company records the number of complaints, X, that it receives over 60 months. box The summarised results are X X x ¼ 102 and (x  x)2 ¼ 103:25 8 (a) Using this data, explain why it may be appropriate to model the number of complaints received by the company per month by a Poisson distribution with mean 1.7 [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (12) Jun21/7367/3S  13 Do not write outside the 8 (b) The company also receives enquiries as well as complaints. The number of enquiries box received is independent of the number of complaints received. The company models the number of complaints per month with a Poisson distribution with mean 1.7 and the number of enquiries per month with a Poisson distribution with mean 5.2 The company starts selling a new product. The company records a total of 3 complaints and enquiries in one randomly chosen month. Investigate if the mean total number of complaints and enquiries received per month has changed following the introduction of the new product, using the 10% level of significance. [6 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over s (13) Jun21/7367/3S  14 Do not write outside the 8 (c) It is later found that the mean total number of complaints and enquiries received box per month is 6.1 Find the power of the test carried out in part (b), giving your answer to four decimal places. [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ END OF QUESTIONS (14) Jun21/7367/3S  15 Do not write outside the There are no questions printed on this page box DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED (15) Jun21/7367/3S  16 Do not write outside the box (16) Jun21/7367/3S  17 Do not write outside the box (17) Jun21/7367/3S  18 Do not write outside the box (18) Jun21/7367/3S  19 Do not write outside the box (19) Jun21/7367/3S  20 Do not write outside the There are no questions printed on this page box DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED Copyright information For confidentiality purposes, all acknowledgements of third-party copyright material are published in a separate booklet. This booklet is published after each live examination series and is available for free download from www.aqa.org.uk. Permission to reproduce all copyright material has been applied for. In some cases, efforts to contact copyright-holders may have been unsuccessful and AQA will be happy to rectify any omissions of acknowledgements. If you have any queries please contact the Copyright Team. Copyright ª 2021 AQA and its licensors. All rights reserved. (20) (216A7367/3S) Jun21/7367/3S