AC – CT3 – P MA – 18Mock Exam A ActEd Study Materials: 2018 Examinations Subject CT3 Contents Mock Exam A Questions If you think that any pages are missing from this pack, please contact ActEd’s admin team by email at
[email protected]. How to use Mock Exam A Guidance on how and when to use Mock Exam A is set out in the Study Guide for the 2018 exams. The recommended date and deadline date for submission of Mock Exam A are listed on a summary page at the back of this pack. We strongly recommend that you work to the recommended date. Please note that we only accept the current version of Mock Exam A for marking, ie you can only submit this mock exam in the sessions leading to the 2018 exams. However, if you wish to submit your script for marking in 2018, but do not have the latest version of Mock Exam A, please let us know and we will send, free of charge, an up-to-date version for you to attempt and submit. Important: Copyright Agreement This study material is copyright and is sold for the exclusive use of the purchaser. You may not hire out, lend, give out, sell, store or transmit electronically or photocopy any part of it. You must take care of your material to ensure that it is not used or copied by anybody else. By opening this pack you agree to these conditions. The Actuarial Education Company © IFE: 2018 Examinations All study material produced by ActEd is copyright and is sold for the exclusive use of the purchaser. The copyright is owned by Institute and Faculty Education Limited, a subsidiary of the Institute and Faculty of Actuaries. Unless prior authority is granted by ActEd, you may not hire out, lend, give out, sell, store or transmit electronically or photocopy any part of the study material. You must take care of your study material to ensure that it is not used or copied by anybody else. Legal action will be taken if these terms are infringed. In addition, we may seek to take disciplinary action through the profession or through your employer. These conditions remain in force after you have finished using the course. © IFE: 2018 Examinations The Actuarial Education Company Subject CT3: Mock Exam A 2018 Examinations Time allowed: 3 hours, plus 15 minutes reading time Instructions to the candidate 1. Please: – attempt all of the questions, as far as possible under exam conditions – begin your answer to each question on a new page – leave at least 2cm margin on all borders – write in black ink using a medium-sized nib because we will be unable to mark illegible scripts – note that mock exam marking is not included in the price of the Course Materials. Please purchase a Marking Voucher before submitting your script. – note that we only accept the current version of mock exams for marking, ie you can only submit this mock exam in the sessions leading to the 2018 exams. 2. Please do not: – use headed paper – use highlighting in your script. At the end of the mock exam If your script is being marked by ActEd, please follow the instructions on the reverse of this page. In addition to this paper, you should have available actuarial tables and an electronic calculator. The Actuarial Education Company © IFE: 2018 Examinations Submission for marking You should aim to submit this script for marking by the recommended submission date. The recommended and deadline dates for submission of this mock exam are listed on the summary page at the back of this pack and on our website at www.ActEd.co.uk. Scripts received after the deadline date will not be marked, unless you are using a Marking Voucher. It is your responsibility to ensure that scripts reach ActEd in good time. If you are using Marking Vouchers, then please make sure that your script reaches us by the Marking Voucher deadline date to give us enough time to mark and return the script before the exam. When submitting your script, please: complete the cover sheet, including the checklist scan your script, cover sheet and Marking Voucher to a pdf document, then email it to:
[email protected] do not submit a photograph of your script do not include the question paper in the scan. In addition, please note the following: Please title the email to ensure that the subject and mock exam are clear eg “CT3 Mock Exam A No. 12345”, inserting your ActEd Student Number for 12345. The mock exam should be scanned the right way up (so that it can be read normally without rotation) and as a single document. We cannot accept individual files for each page. Please set the resolution so that the script is legible and the resulting PDF is less than 3 MB in size. The file size cannot exceed 4 MB. Do not protect the PDF in any way (otherwise the marker cannot return the script to ActEd, which causes delays). Please include the “feedback from marker” sheet when scanning. Before emailing to ActEd, please check that your scanned mock exam includes all pages and conforms to the above. © IFE: 2018 Examinations The Actuarial Education Company Subject CT3: Mock Exam A 2018 Examinations Please complete the following information: Name: Number of following pages: _______ Please put a tick in this box if you have solutions and a cross if you do not: Please put a tick in this box if you used the allowed reading time: ActEd Student Number (see Note below): Please tick here if you are allowed extra time or other special conditions in the profession’s exams: Time to do mock Note: Your ActEd Student Number is printed on all (see Note below): _____ hrs _____ mins personal correspondence from ActEd. Quoting it will help us to process your scripts quickly. If you do not know your Under exam conditions ActEd Student Number, please email us at (delete as applicable): yes / nearly / no
[email protected]. Note: If you take more than 3 hours, you should Your ActEd Student Number is not the same as your indicate how much you completed within this time IFoA Actuarial Reference Number or ARN. so that the marker can provide useful feedback on your chances of success in the exam. Score and grade for this mock exam (to be completed by marker): Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Total =_____% 3 4 4 5 5 5 5 6 6 7 15 17 18 100 Grade: A B C D E Marker’s initials: ________ Please tick the following checklist so that your script can be marked quickly. Have you: [ ] Checked that you are using the latest version of the mock exam, ie 2018 for the sessions leading to the 2018 exams? [ ] Written your full name in the box above? [ ] Completed your ActEd Student Number in the box above? [ ] Recorded your attempt conditions? [ ] Numbered all pages of your script (excluding this cover sheet)? [ ] Written the total number of pages (excluding the cover sheet) in the space above? [ ] Included your Marking Voucher? Please follow the instructions on the previous page when submitting your script for marking. The Actuarial Education Company © IFE: 2018 Examinations Feedback from marker Notes on marker’s section The main objective of marking is to provide specific advice on how to improve your chances of success in the exam. The most useful aspect of the marking is the comments the marker makes throughout the script, however you will also be given a percentage score. Based on this score, the marker assigns a grade representing how this script would have fared in an exam. The grades are as follows: A = Clear Pass B = Probable Pass C = Borderline D = Probable Fail E = Clear Fail We aim for our ActEd Mocks to be realistic and consistent with the IFoA examinations, as such the notional pass mark for this Mock Exam is around 60%. Please note that you can provide feedback on the marking of this mock exam at: www.ActEd.co.uk/marking © IFE: 2018 Examinations The Actuarial Education Company CT3: Mock Exam A – Questions Page 1 1 The random variable, X , has a normal distribution with mean and variance 100. In a test of: H 0 : 20 vs H1 : 30 it is decided to reject H 0 in favour of H1 if the sample mean, X , is greater than 25. Determine the smallest sample size required to make the probability of a Type I error less than 0.02. [3] 2 (i) Define the F distribution in terms of the ratio of two other distributions. [1] (ii) Deduce that if random samples of size n1 and n2 are taken from independent normal populations with variances 12 and 22 then: S12 12 ~ Fn1 1, n2 1 S 22 22 where S1 and S 2 denote the respective sample standard deviations. [1] (iii) Given that the samples are of size n1 11 and n2 19 and that they come from normal populations with equal variances, calculate approximately: S2 P 12 2.9 [2] S2 [Total 4] 3 A single sample value of 2 is obtained from a Poisson distribution with mean . Determine a 90% confidence interval for , using tables of “Probabilities for the Poisson distribution” or otherwise. [4] The Actuarial Education Company © IFE: 2018 Examinations Page 2 CT3: Mock Exam A – Questions 4 The random variables U and V have joint density function given by: 1 f (u , v) (3u 5v), 0 u 1, 0 v 1 4 (i) State, with reasons, whether or not the random variables U and V are independent. [1] (ii) Determine the conditional density function of U given V v . [2] (iii) Hence determine the conditional expectation E (U | V v) . [2] [Total 5] 5 A set of data consists of thirty values of 8 and twenty values of 12. (i) Calculate the mean and standard deviation of this data set. [3] (ii) Explain, without calculation, what would happen to the mean, standard deviation and skewness if ten 3’s were added to this data set. [2] [Total 5] 6 A random sample of 20 observations from a normal distribution gives x 52 and ( xi x )2 48.2 . Obtain a 95% confidence interval for: (i) , the true population mean [3] (ii) 2 , the true population variance. [2] [Total 5] © IFE: 2018 Examinations The Actuarial Education Company CT3: Mock Exam A – Questions Page 3 7 A random sample of 4 children is taken from a large population that consists of an equal number of boys and girls. (i) Calculate the probability that the sample contains 2 or more boys. [1] One hundred random samples, each of 4 children, are taken independently from a large population that consists of an equal number of boys and girls. (ii) Use a suitable approximation to calculate the probability that at least 60 of the samples contain 2 or more boys. [4] [Total 5] 8 A man commutes to London by train every working day. He can catch the “early” train or the “late” train to get to work. The probability that the “early” and “late” trains arrive in London on time for work are 0.6 and 0.55, respectively. The probability that he makes it to the station on time to catch the early train is 0.7. (i) Given that he arrives in London on time, calculate the probability that he caught the “late” train. [3] A new “mid-morning” service is added that arrives in London on time for work with probability 0.57. The probability that the man catches the “early”, “mid-morning” or “late” trains are now 0.7, 0.2 and 0.1, respectively. (ii) Given that he arrives in London on time, calculate the probability that he caught the “mid-morning” train. [3] [Total 6] 9 An insurer knows from past experience that, for a particular risk, claim sizes have a gamma distribution with parameters 74 and 0.5 , and the number of claims per month has a Poisson distribution with mean 17. It is known that the claim size and number of claims are independent and that claims are independent of each other. (i) By writing down an expression for the moment generating function of S (the aggregate claims in a year), show that the cumulant generating function is: 204 (1 2t ) 74 1 [2] (ii) Hence, calculate the coefficient of skewness of S . [4] [Total 6] The Actuarial Education Company © IFE: 2018 Examinations Page 4 CT3: Mock Exam A – Questions 10 (i) Show, using moment generating functions, that if X1 , X 2 , , X n are independent and identically distributed random variables with X i ~ Exp( ) , n then X i ~ gamma(n, ) . i 1 (You may assume the moment generating function of X .) [3] (ii) Claims arrive at an insurance office at an average rate of 10 per day, and the arrival of claims is modelled as a Poisson process. (a) State the distribution of the waiting time (in days) to the arrival of the first claim. (b) Calculate the probability that the time until the arrival of the fourth claim is more than half a day. [4] [Total 7] © IFE: 2018 Examinations The Actuarial Education Company CT3: Mock Exam A – Questions Page 5 11 A random variable X has the probability density function shown in the diagram below. The function is a decreasing linear function on the interval (0, ) . f(x) x 0 (i) Show that the probability density function for X is given by: 2( x) f ( x) , 0 x [2] 2 (ii) Derive the mean and variance of X in terms of . [3] (iii) A random sample of size 5 from this distribution yields the values: 0.8, 1.3, 1.4, 1.7, 10.0 Derive a method of moments estimate for based on these data. [2] (iv) Explain carefully why the method of moments estimate for derived in part (iii) above cannot be a maximum likelihood estimate. [2] (v) Write down the likelihood function for based on the sample given above. Hence show that the maximum likelihood estimate for lies between 12 and 12.5, and determine its value correct to one decimal place. [6] [Total 15] The Actuarial Education Company © IFE: 2018 Examinations Page 6 CT3: Mock Exam A – Questions 12 The scores (as percentages) of ten students in their CT3 and CT1 mocks are as follows: Student 1 2 3 4 5 6 7 8 9 10 Subject CT3 ( x) 27 35 47 56 58 73 79 83 89 92 Subject CT1 ( y ) 35 26 67 45 64 89 80 49 93 74 x 639 , x 2 45,507 , y 622 , y 2 43,358 , xy 43, 205 A plot of this data set is shown below: Mock scores 100 80 CT1 score 60 40 20 0 0 20 40 60 80 100 CT3 score (i) Comment on the suitability of fitting a model of the form yi xi ei . [1] (ii) Calculate estimates of the intercept and gradient parameters. [3] (iii) Carry out a test to establish if , the gradient parameter for the whole population, exceeds 0.5. [3] (iv) Explain in general terms what the coefficient of determination tells us about a regression model. Calculate the coefficient for this model and comment on its value. [3] (v) Calculate the correlation coefficient and test to see if this sample could have come from a population with correlation coefficient 0.9. [3] (vi) Calculate a 95% confidence interval for the score in the Subject CT1 mock for an individual student who achieved 50% in their Subject CT3 mock. [4] [Total 17] © IFE: 2018 Examinations The Actuarial Education Company CT3: Mock Exam A – Questions Page 7 13 Four different insurance companies were asked to provide data regarding the number of claims of a particular type received each month by their claims departments. The results were as follows: Company 1 6, 11, 7, 3 Company 2 4, 6, 5 Company 3 4, 6, 5, 7, 8, 4 Company 4 8, 10, 9, 8, 6 Consider the model: Yij i eij i 1, 2, 3, 4; j 1, 2, , ni eij ~ N (0, 2 ) where Yij is the j th result for the i th company, ni is the number of responses available for Company i , the eij are independent and identically distributed and ni i 0 . i (i) Calculate the least squares estimates of and i , i 1, 2, 3, 4 . [3] (ii) Perform an analysis of variance on these results stating clearly your assumptions and your conclusion. [7] (iii) Calculate a point estimate and a 95% confidence interval for the underlying common variance, 2 , of the monthly claim numbers in each of the four companies. [3] (iv) State the values of yi for i 1, 2,3, 4 , list the means in order and, using a 5% significant level, illustrate the non-significant pairs using suitable underlining. [5] [Total 18] END OF MOCK EXAM The Actuarial Education Company © IFE: 2018 Examinations All study material produced by ActEd is copyright and is sold for the exclusive use of the purchaser. The copyright is owned by Institute and Faculty Education Limited, a subsidiary of the Institute and Faculty of Actuaries. Unless prior authority is granted by ActEd, you may not hire out, lend, give out, sell, store or transmit electronically or photocopy any part of the study material. You must take care of your study material to ensure that it is not used or copied by anybody else. Legal action will be taken if these terms are infringed. In addition, we may seek to take disciplinary action through the profession or through your employer. These conditions remain in force after you have finished using the course. © IFE: 2018 Examinations The Actuarial Education Company Assignment deadlines v2 For the session leading to the April 2018 exams – CT Subjects Marking vouchers Subjects Assignments Mocks CT1, CT2, CT3, CT6, CT7 21 March 2018 27 March 2018 CT4, CT5, CT8 27 March 2018 4 April 2018 Series X Assignments Recommended Final deadline Subjects Assignment submission date date CT1, CT2, CT3, CT6, CT7 22 November 2017 17 January 2018 X1 CT4, CT5, CT8 29 November 2017 24 January 2018 CT1, CT2, CT3, CT6, CT7 13 December 2017 7 February 2018 X2 CT4, CT5, CT8 20 December 2017 14 February 2018 CT1, CT2, CT3, CT6, CT7 24 January 2018 28 February 2018 X3 CT4, CT5, CT8 31 January 2018 7 March 2018 CT1, CT2, CT3, CT6, CT7 21 February 2018 14 March 2018 X4 CT4, CT5, CT8 28 February 2018 21 March 2018 Mock Exams Recommended Final deadline Subjects submission date date CT1, CT2, CT3, CT6, CT7 14 March 2018 27 March 2018 CT4, CT5, CT8 21 March 2018 4 April 2018 We encourage you to work to the recommended submission dates where possible. If you submit your mock on the final deadline date you are likely to receive your script back less than a week before your exam. The Actuarial Education Company © IFE: 2018 Examinations Assignment deadlines v2 For the session leading to the September 2018 exams – CT Subjects Marking vouchers Subjects Assignments Mocks CT1, CT4, CT6, CT7 22 August 2018 29 August 2018 CT2, CT3, CT5, CT8 29 August 2018 5 September 2018 Series X Assignments Recommended Final deadline Subjects Assignment submission date date CT1, CT4, CT6, CT7 30 May 2018 27 June 2018 X1 CT2, CT3, CT5, CT8 6 June 2018 4 July 2018 CT1, CT4, CT6, CT7 20 June 2018 18 July 2018 X2 CT2, CT3, CT5, CT8 27 June 2018 25 July 2018 CT1, CT4, CT6, CT7 11 July 2018 1 August 2018 X3 CT2, CT3, CT5, CT8 18 July 2018 8 August 2018 CT1, CT4, CT6, CT7 25 July 2018 15 August 2018 X4 CT2, CT3, CT5, CT8 1 August 2018 22 August 2018 Mock Exams Recommended Final deadline Subjects submission date date CT1, CT4, CT6, CT7 15 August 2018 29 August 2018 CT2, CT3, CT5, CT8 22 August 2018 5 September 2018 We encourage you to work to the recommended submission dates where possible. If you submit your mock on the final deadline date you are likely to receive your script back less than a week before your exam. © IFE: 2018 Examinations The Actuarial Education Company