Cambridge A Level Mathematics 9709 — 2002 May/June Paper 7 · Variant 1

CAMBRIDGE INTERNATIONAL EXAMINATIONS. General Certificate of Education Advanced Subsidiary Level General Certificate of Education Advanced Level HIGHER MATHEMATICS 8719/7 MATHEMATICS 9709/7 PAPER 7 Probability & Statistics 2 {$2} MAY/JUNE SESSION 2002 1 hour 15 minutes Additional materials: Answer paper Graph paper List of Formulae (MF9) TIME 1 hour 15 minutes INSTRUCTIONS TO CANDIDATES Write your name, Centre number and candidate number in the spaces provided on the answer paper/answer booklet. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. INFORMATION FOR CANDIDATES The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 50. Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. This question paper consists of 3 printed pages and 1 blank page. UNIVERSITY ef CAMBRIDGE Local Examinations Syndicate [Turn over © CIE 2002

2 The result of a fitness trial is a random variable X which is normally distributed with mean 4 and standard deviation 2.4. A researcher uses the results from a random sample of 90 trials to calculate a 98% confidence interval for #. What is the width of this interval? [4] The manager of a video hire shop wishes to estimate the proportion of videos damaged by customers. He takes a random sample of 120 videos and finds that 33 of them are damaged. Find a 95% confidence interval for the true proportion of videos that are being damaged when hired from this shop. [4] Mary buys 3 packets of sugar and 5 packets of coffee and puts them in her shopping basket, together with her purse which weighs 350g. Weights of packets of sugar are normally distributed with mean 500 g and standard deviation 20g. Weights of packets of coffee are normally distributed with mean 200 g and standard deviation 12 g. Find the probability that the total weight in the shopping basket is less than 2900 g. [6] The mean time to mark a certain set of examination papers is estimated by the examination board to be 12 minutes per paper. A random sample of 150 examination papers gave Ex = 2130 and Lx? = 37746, where « is the time in minutes to mark an examination paper. (i) Calculate unbiased estimates of the population mean and variance. [2] Gii} Stating the null and alternative hypotheses, use a 10% significance level to test whether the examination board’s estimated time is consistent with the data. {5} To test whether a coin is biased or not, it is tossed 10 times. The coin wil] be considered biased if there are 9 or 10 heads, or 9 or 10 tails. {i) Show that the probability of making a Type 1 error in this test is approximately 0.0215. [4] (ii) Find the probability of making a Type II error in this test when the probability of a head is actually 0.7. [4] Between 7p.m. and 11 p.m., arrivals of patients at the casualty department of a hospital occur at random at an average rate of 6 per hour. @) Find the probability that, during any period of one hour between 7 p.m. and 11 p.m., exactly 5 people will arrive. [2] (ii) A patient arrives at exactly 10.15 p.m. Find the probability that at least one mote patient arrives before 10.35 p.m. [3] {iii} Use a suitable approximation to estimate the probability that fewer than 20 patients arrive at the casualty department between 7 p.m. and 11] p.m. on any particular night. [5] 9709/7AA/ 02

3 A factory is supplied with grain at the beginning of each week. The weekly demand, X thousand tonnes, for grain from this factory is a continuous random variable having the probability density function given by fy = {709 Orel, 0 otherwise. Find ( the mean value of X, [3] (ii) the variance of X, 3) Gi) the quantity of grain in tonnes that the factory should have in stock at the beginning of a week, in order to be 98% certain that the demand in that week will be met. [5] s7ORFTIMLIIOR

CAMBRIDGE INTERNATIONAL EXAMINATIONS JUNE 2002 GCE Advanced Level GCE Advanced Subsidiary Level MARK SCHEME MAXIMUM MARK : 50 SYLLABUS/COMPONENT :9709 /7, 8719 /7 MATHEMATICS (Probability and Statistics 2) RR University of CAMBRIDGE “2” Local Examinations Syndicate

Page 1 Mark Scheme Syllabus Paper A & AS Level Examinations - June 2002 9709, 8719 7 For z value of 2.33 Lo ££2.326x—= N90 Ml For expression of correct form involving 490 in 2 denom 2.326 x —=—=x2 Width 90 MI For subtracting lower from upper, or multiplying half-width by 2 Ai 4__ | For correct answer 2 EMTHER 9 (6375x0725 M2 . pia 0275+ L96x Calculation of correct form n 120 Bl (SR M1 if only one side of interval seen} 019S< p< 0355 Use of p= 0.275 Al 4 For correct answer OR 33 1.96V 120x0.275x0.725 Mi Calculation of correct form np+zVnpq (accept just 23.413. < p <42.586 one side of interval) 120 120 Ml Division by 120 (BOTH sides) 0.195 <p <0,355 BL Use of 0.275 Correct answer 3 3 sugar ~ N(1500, 1200) For (norma! dist with) correct means for both 5 coffee ~ N(1000,720) Bi For (nomnal dist with) correct variance for both Total weight ~ N(2850. 1920) M! For adding their variances and means(+ purse)for or ~ N(2500, 1920) coffee and sugar Al For correct mean and variance for their total weight o( 220052859) ie with or without the purse PCW < 2900) = 4/1920 Mi For standardising and use of tables (consistent inclusion/exclusion of purse) ee | = 0.873 Al 6 | For correct answer OrP(W<2550) = v1920 j 30? Bl | For correct mean 40 #142, Pa elas 27) = 50.3(4) |B1 2 | For correct variance (ii) Ho : w= 12 and Hy: p * 22 Bl Both hypotheses correct Test statisticz= 14.2-12 =3.798 MI a 50,34 Al For standardising attempt with se of form vn For 3.80 150 Mi Or comparing (3.798) with 0.95 (or equiv. for one tail test) Signs consistent, Compare with 1.645 or 1,282 for one-tail t 5 | Correct conclusion fi on their z and Hy Reject exam boards claim _| 5 i) PO or 10K) = (0.5)°x (0.5) x yoo + O.5)° For P(9 or 10H) (= 0.01074) For P(9 or 10T) POT or 10T) = 0.01074 P(type 1 error) = 0.0215 AG For identifying outcome for Type I exror For obtaining given answer jegitimately H)P(9 or LOH)= (0.7)° x (0.3) x yoCs + 0.7)" For evaluating P(9 or 10H) with P(HD = 0.7 (=0.1493) For evaluating P(9 or 10T) with P(T) = 0.3 P(9 or 10T) = 0.3)" (0.7) 10Cs + (0.3) = 0.000143 For identifying outcome for Type il error Pétype HT error) = 1 - 0.1493 -0,000143 =0.851 For correct answer (SR 0.851 no working B2)

Page 2 Mark Scheme Syllabus | Paper A& AS Level Examinations — June 2002 9709, 8719 |_7 6 (i) mean=6 PCY = 5) = 0.161 (ii) p=2 P(0) =e? (= 0.135) 1b - P(0) = 0.865 (iii) p= 24, = 24 195-24 V4 = 9.9186 1 - (0.9186) = 0.179 z= 1 7 @ BO) = (2x(l-2) ax L f 2x? - 2x8 de (i) Var) = - (0.3337. 3 4 0.0556 = i (0.3337 fro =x) de Gi? 2x—x? = 0.98 = 0.98 x* ~2x+0.98=0 x= 0.859 859 tonnes CP xq =x) =0.02 Mil Al Mit MI *dep Al 3 For mean 6 and evaluating a Poisson prob For correct answer For p=2 used in a Poisson prob. For 1 - P(O), any mean For correct answer For p=24 For their var=their mean For standardising with or without cc For correct continuity correction For correct answer (SR Using Poisson with no approximation (0.180(26) ) scores Mi Aj only ) For sensible attempt to integrate xf(x) For correct integrand (any form) For correct answer For sensible attempt to integrate x°f(x) For their integral- (their mean)* For correct answer For identifying both sides of equation For correct equation in any form For solving for x {must be sensible attempt) For correct answer For applying concept of continuous rv. For identifying x from a relevant diagram For correct equation For solving for x For correct answer For applying concept of continuous rv.