; y OA3009,. UNIVERSITY OF CAMBRIDGE LOCAL EXAMINATIONS SYNDICATE ""'

Size: px

Start display at page:

Download "; y OA3009,. UNIVERSITY OF CAMBRIDGE LOCAL EXAMINATIONS SYNDICATE ""'"

Sydney Gilbert
5 years ago
Views:

1 ~ ; y OA3009 ;ZZ,. UNIVERSITY OF CAMBRIDGE LOCAL EXAMINATIONS SYNDICATE ""' '"'" Joint Examination for the Higher School Certificate and General Certificate of Education Advanced Level MATHEMATICS 9200/4 PAPER 4 Thursday 14 NOVEMBER hours Additional materials: Answer paper. List of Formulae Graph paper ~}j;:~~.,c'~"' 'c. I TIME 3 hours INSTRUCTIONS TO CANDIDATES Write your name, Centre number and candidate number in the spaces provided on the answer paper/answer booklet. There is no restriction on the number of questions which you may attempt. If a numerical answer cannot be given exactly, and the accuracy required is not specified in the question, then in the case of an angle it should be given to the nearest degree, and in other cases it should be given correct to 2 significant figures. INFORMATION FOR CANDIDATES The number of marks is given in brackets [ ] at the end of each question or part question. Within each section of the paper, questions are printed in the order of their mark allocations and candidates are advised, within each section, to attempt questions sequentially. The use of an electronic calculator is expected, where appropriate. This question paper consists of 6 printed pages and 2 blank pages. UCLES 1996 [Turn over

2 "'-- c - ~c~ :~; -, ;- Section (a): Pure Mathematics 2 1 The variables x and yare related by the equation x3y + 3y3 = 2. Find, in terms of x and y, an.dy expression for -.[3] dx Hence find the value of ~ at the point (-I, 1 ). [2] dy -~ In a competition, a long-jumper takes three jumps. The probability that her second jump is longer than her first jump is 0.6. If her second jump is longer than her first then the probability that her third jump is the longest of the three is 0.2. If her first jump is longer than her second jump then the probability that her third jump is the longest of the three is 0.3. The information concerning the results of her second and third jumps is shown in the following tree diagram. l\ )\}~.\'\~ \\-.e. \\1: ~\,:.e.co~\\-.'3.\} \0\}~e.~\).6 (i) Find the probability that, after her first jump, she improves with each subsequent jump. [1] (ii) Find the probability that she improves on her first jump. [2] (iii) Find the conditional probability that her second jump is longest, given that she improves on her first jump. [3] ' "4,,~ ~ ")" :..-J ~~,. i ~\,,~:j~'5 I;.,~ r LI...lt. 'i~".;i~~~y;. 9200/4/W96

3 3 3 A test was set to a group of students and marked out of 70. The minimum mark for a Grade A is 48. and the minimum mark for a Grade B is 36. Marks of 35 or less are given Grade C. The results can be represented by the following stem-and-ieaf diagram ;0"' ~.;~{C A'~',7- ~ ~~~ 9 I~I Marks out of 70. [Key: a mark of 39 is shown as 3 I 9.] ic"c"tci'! ","""~ '~J..~"' ;c (i) How many students took the test? [1] (ii) Sketch a diagram illustrating the percentages of students in the various grades. [2] (iii) Grouping the data with classes 10 to 19,20 to 29,...,60 to 69, determine the frequencies for these classes and illustrate this grouped frequency distribution with a suitable diagram. [4] (iv) State the median mark and find the interquartile range. [3] (v) The test was set to another group of students, and the results were classified as follows. Mark Number of students 5 8 I Calculate an estimate of the mean mark for this group of 35 students. [3] Section (b): Statistics (!) State, giving a reason, whether the following procedure will give a random sample from the letters of the alphabet: repeatedly choose a page at random in a dictionary and take the initial letter of the first word defined on the page. [2] 0 The number of passengers on the last bus each night to Brightlingsea has a Poisson distribution. On each of the days Monday to Friday the mean is 5, on Saturday it is 9, and on Sunday it is 2. State the distribution of the weekly total of passengers on the last bus, giving its mean and variance. [3] 9200/4IW96 [Thrn over

4 4 6 A test of the null hypothesis of independence of two characteristics is required for the following contingency -tat>le. :::, Characteristic 1 Totals ~~"'~ "j~~~"" I"" j ",, c Characteristic : /' -;.:, \ / cc:~i?~ ~::...~,~.i' Totals '; ;,40,-...". -,, A -, '~~ : " " (i) Find the-expected frequency corresponding to the cell with observed frequency 12. [1] (ii) Given that the value of X2 is 4.80, to 3 significant figures, carry out the test at the 10% level of significance. [3] 7 The random variable X has a geometric distribution with mean 4. Find the probability that X is less than or equal to 3. [4] 8 It is given that X,,-, N(5, 4). Find P(3 < X < 6). [4] 9 Two cards are drawn, at random and without replacement, from a complete pack of 52 cards. Show that the probability that they are both aces is ~.[2] The experiment is carried out 1000 times. Using a suitable approximation, find the probability that two aces are obtained on exactly 5 occasions. [3] 10 The proportion of car-drivers taking lead-free petrol at a particular filling station has been found to be 42%. Following an increase in the tax on other types of fuel, a random sample of 120 cars was observed at the filling station. The number of drivers taking lead-free petrol was found to be 58. Test, at the 5% level, whether there has been an increase in the proportion of drivers taking lead-free petrol at the filling station. [6] 11 For a set of bivariate data, the regression line of yon x has equation y = 1.35x , and the regression line ofx on y has equation x = 0.60y Find.x and y. [3] It is given tbat the product moment correlation coefficient is Give a sketch of a possible scatter diagram, with 8 data points, for the above data. You should indicate clearly the scales on the axes and also show the regression line of yon x. [4] ~ Wooden logs are often sold by the sackful. A Trading Standards Officer buys 50 sacks, and weighs I Q the contents of each sack after the logs have been dried. The results are summarised by r.x = 1174, r.x2 = , where x kg is the mass of dried logs in a sack. Test, at the 10% level, whether the mean mass of dried logs in a sack is less than 25 kg. [8] -,0' 9200/4/W96

5 13 The proportion of the population who are colour-blind is 8%. 5 (i) A random sample of 11 people is taken. Find the probability that at least 2 are colour-blind. [4] (ii) Using a suitable approximation, find the probability that at least 11 in a random sample of 200 people are colour-blind. [5] 114'\ Tw~ g~ades of ~elo?s are imported. Their masses are normally distributed, with means and standard V deviations as given m the table below. Mean (kg) Standard deviatiqp, (kg) Grade A 2.43,:.' c ~, 0.27 c Grade B \, (i) Find the probability that a randomly chosen Grade A melon is more than 40% heavier than a randomly chosen Grade B melon. [6] (ii) Find the probability that 3 randomly chosen Grade A melons weigh more than 4 randomly chosen Grade B melons. [5] 15 A computer is programmed to generate a random integer X in the set { 0, 1, 2,..., 9}.The random variable y is the remainder when X is divided by 4. Give a table showing the probability distribution of Y. [2] Show that the expectation of y is M and find the standard deviation of Y. [5] Three successive observations of Yare made. Find the probability that each observation after the first is greater than the one before. [4] ~ A population of antelope has mean mass denoted by.u kg. The standard deviation is denoted by 0" kg. A random sample of size 57 is taken. State three facts concerning the distribution of the sample mean mass. [3] The actual masses in the sample are summarised by Lx = 5273, Lx2 = , where x kg is the mass of an antelope. Find unbiased estimates of.u and 0"2. [3] Find a 95% confidence interval for the population mean mass, giving the end-points of the interval correct to 1 decimal place. [3] Ten different random samples are taken, and a 95% confidence interval for.u is calculated from each sample. State the expectation of the number of these confidence intervals that will contain.u. [2] 9200/4/W96 [Turn over

6 ~c, 6 17 Six hens are observed over a period of 20 days and the number of eggs laid each day is summarised in the following table. Number of eggs Number of days Show that the mean number of eggs per day is 5. [2] It may be assumed that a hen never lays more than one egg in any day. State one other assumption that needs to be made in order to consider a binomial model, with n = 6, for the total number of eggs laid in a day. [1] Calculate the expected frequencies using a binomial model for the above data and carry out a X2 goodness of fit test, using a 10% significance level. [8] 9200/4IW96

MEI STRUCTURED MATHEMATICS 4767

OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MEI STRUCTURED MATHEMATICS 4767 Statistics 2 Thursday 9 JUNE 2005 Morning