13316.07 R 4 A particle moves along a straight line. Points A, B and C lie on this line. At time t seconds, the particle’s displacement x metres from A is given by x = 1 3 t 3 + 3 4 t 2 + 7t  (0 G t G 2) When t = 2 the particle is at B. (i) Find the displacement of B from A. [2] (ii) Find its velocity at B. [4] [Turn over 13316.07 R For 2 G t < 4 the velocity v m s–1 of the particle can now be modelled as v = 14sec2 (πt 4 – π 2) At t = 3 the particle is at C. (iii) Find the displacement of C from A. [6] 13316.07 R (iv) If the particle models a train on a straight track, what is the problem with using this model as t gets closer to 4? [2] [Turn over 13316.07 R BLANK PAGE DO NOT WRITE ON THIS PAGE (Questions continue overleaf) 13316.07 R SECTION B Statistics 5 An economist studies the correlation between unemployment percentage and average income for a large city. He sets up a hypothesis test on ρ, the population correlation coefficient. His hypotheses are: H0: ρ = 0 H1: ρ < 0 (i) Will the test be one-tailed or two-tailed? [1] The economist uses a random sample of records over an extended period of time. He calculates the product–moment correlation coefficient for his sample and finds the p-value for the test to be 0.00845 (ii) Explain what is meant by the term p-value. [2] [Turn over 13316.07 R (iii) Using a significance level of 1%, complete the hypothesis test. [3] (iv) If instead the economist had tested H1: ρ ≠ 0, what would his conclusion have been? [2] 13316.07 R 6 A brewery produces bottles of cider whose volumes are normally distributed with mean μ and variance σ 2 Given that the upper quartile of the volume of cider in a bottle is 571.71 ml and that 5% of the bottles contain less than 558.95 ml, calculate the values of μ and σ. [10] [Turn over 13316.07 R 13316.07 R 7 It is believed that 1 person in 100 is allergic to milk. A test has been designed to try to detect the allergy. For a person with the allergy, there is a 95% chance that the test will give a positive result. For a person who does not have the allergy, there is a 96% chance that the test will give a negative result. (i) Using a tree diagram, or otherwise, calculate the probability that a randomly selected person gets a positive result on the test. [3] [Turn over 13316.07 R A randomly selected person is tested and the result indicates that they are allergic to milk. (ii) Find the probability that this person is actually allergic to milk. [4] (iii) Comment on your answer to (ii). [2] 13316.07 R 8 The time taken to answer a call arriving at a switchboard follows a normal distribution with mean 12 seconds and standard deviation 3.5 seconds. (i) Calculate the probability that a random call arriving at the switchboard takes between 10 and 15 seconds to be answered. [4] 13316.07 R [Turn over The office manager introduces a bonus scheme designed to reduce the amount of time taken for calls to be answered. He claims that the scheme has been successful. Following the introduction of the scheme, the office manager calculates the average time taken to answer calls in a random sample of 40 to be 10.8 seconds. (ii) Test the office manager’s claim at the 5% level of significance. [7] 13316.07 R 13316.07 R [Turn over BLANK PAGE DO NOT WRITE ON THIS PAGE (Questions continue overleaf) 13316.07 R 9 Daniel took a multiple choice examination in which there were five possible responses to each of the questions. His teacher suspects that Daniel had not revised for the examination and was just guessing his answers. The teacher decides to test her suspicion at the 5% level of significance. (i) Explain why a one-tailed test is appropriate in this context. [1] 13316.07 R (ii) Write down the null and alternative hypotheses for this test. [2] (iii) Describe the meaning of the term level of significance. [2] [Turn over 13316.07 R The examination contained 40 questions and Daniel got 12 of his answers correct. (iv) Complete the hypothesis test, stating clearly whether the teacher’s suspicion is justified. [7] 13316.07 R THIS IS THE END OF THE QUESTION PAPER Permission to reproduce all copyright material has been applied for. In some cases, efforts to contact copyright holders may have been unsuccessful and CCEA will be happy to rectify any omissions of acknowledgement in future if notified. Examiner Number DO NOT WRITE ON THIS PAGE 13316.07 R For Examiner’s use only Question Number Marks 1 2 3 4 5 6 7 8 9 Total Marks Sources: All images © CCEA unless stated