Cambridge A Level Mathematics - Further 9231 — 2020 May/June Paper 4 · Variant 3

1 Young children are learning to read using two different reading schemes, A and B. The standards achieved are measured against the national average standard achieved and classified as above average, average or below average. For two randomly chosen groups of young children, the numbers in each category are shown in the table. Standard achieved Above average Average Below average Scheme A 31 35 22 Scheme B 19 50 43 Test at the 5% significance level whether standard achieved is independent of the reading scheme used. [6] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... ....................................................................................................................................................................

Mark scheme: 1 31 E = 22 35 E = 37.4 22 E = 28.6 19 E = 28 50 E = 47.6 43 E = 36.4 M1A1 ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 31 22 35 37.4 22 28.6 19 28 50 47.6 43 36.4 22 37.4 28.6 28 47.6 36.4 − − − − − − + + + + + M1 9.569 (9.57) A1 Tabular value = 5.991 and comparing 9.569 > 5.991 M1 So standard achieved is dependent on reading scheme used A1 6

More questions on χ²-tests

2 A random sample of 40 observations of a random variable X and a random sample of 50 observations of a random variable Y are taken. The resulting values for the sample means, x and y , and the unbiased estimates, sx2 and s 2,y for the population variances are as follows. x = 24 .4 y = 17 .2 s 2x 2 = 10 .2 s y = 11 .1 Find a 90% confidence interval for the difference between the population means of X and Y. [5] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... ....................................................................................................................................................................

Mark scheme: 2 2 10.2 11.1 0.477 40 50 = + = s M1A1 CI = ( ) 24.4 17.2 − ± zs M1 = ( ) 24.4 17.2 1.645 0.477 − ± A1 = [6.06, 8.34] A1 5

More questions on Inference using normal and t-distributions

3 The continuous random variable X has probability density function f given by 1 5 x 0 G x 1 2, - x) 2 G x G 5, f ( x) = 152 ( 5 *0 otherwise. (a) Find the cumulative distribution function of X. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find the median value of X. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (c) Find E ( X2 ) . [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (d) Find P ( 1 G X G 3) . [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 3(a) ( ) ( ) 2 2 , 0 2 10 F 1 2 10 , 2 5 15 3 x x x x x x  <  =   − −     = 0 for x < 0 and = 1 for x > 5 A1 3 3(b) F(m) = ½ so 2 2 20 35 0 − + = m m M1 1 5 30 2.26 2 = − = m A1 2 3(c) E(X2) = ( ) 2 5 2 2 0 2 2 . d 5 . d 5 15 x x x x x x + −   = 4 4 3 2 5 20 15 3 4       + −               x x x M1 = 6.5 A1 2 Question Answer Marks 3(d) F(3) – F(1) M1 = 11 1 19 15 10 30 − = A1 Alternative method for 3(d) ( ) 2 3 2 2 1 2 2 2 d 5 d 5 5 15 10 15 2       + − = + −                 x x x x x x x M1 = 3 1 19 10 3 30 + = A1 2

More questions on Continuous random variables

4 The discrete random variable X has probability generating function G X ( t) given by = 0.2t + 0 .5t + 0 .3t . G X ( t) 2 3 The random variable Y is the sum of two independent observations of X. (a) Find the probability generating function of Y, giving your answer as an expanded polynomial in t. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Use the probability generating function of Y to find E(Y) and Var(Y). [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 4(a) GX (t) = 2 3 0.2 0.5 0.3 + + t t t B1 GY (t) = ( ) 2 3 0.2 0.5 0.3 + + t t t 2 M1 = 2 3 4 5 6 0.04 0.2 0.37 0.3 0.09 + + + + t t t t t A1 3 4(b) G’Y(t) = 2 3 4 5 0.08 0.6 1.48 1.5 0.54 + + + + t t t t t M1 E(Y) = 4.2 A1 G’’Y(t) = 2 3 4 0.08 1.2 4.44 6 2.7 + + + + t t t t M1 Use G’’Y (1) + G’Y(1) – (G’Y(1))2 M1 Var(Y) = 14.42 + 4.2 – 4.22 = 0.98 A1 5

More questions on Probability generating functions

5 Students at two colleges, A and B, are competing in a computer games challenge. (a) The time taken for a randomly chosen student from college A to complete the challenge has a normal distribution with mean n minutes. The times taken, x minutes, are recorded for a random sample of 10 students chosen from college A. The results are summarised as follows. / x = 828 / x 2 = 68622 A test is carried out on the data at the 5% significance level and the result supports the claim that n 2 k . Find the greatest possible value of k. [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) A random sample of 8 students is chosen from college B. Their times to complete the same challenge give a sample mean of 79.8 minutes and an unbiased variance estimate of 9.966 minutes2. Use a 2-sample test at the 5% significance level to test whether the mean time for students at college B to complete the challenge is the same as the mean time for students at college A to complete the challenge. You should assume that the two distributions are normal and have the same population variance. [7] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 5(a) 2 2 828 1 828 82.8 68622 7.0667 10 9 10   = = = − =     x s 2 82.8 10 −k t s  where t = 1.833 (M1 if equality, A1 for inequality and correct t value) M1 A1 81.259 81.3 k k   A1 4 5(b) H0: μ μ = A B H1: μ μ ≠ A B B1 Pooled variance = 9 7.0667 7 9.966 10 8 2 × + × + − = 2 ps M1 = 8.335 A1 82.8 79.8 1 1 10 8 − = + p t s = 2.19 M1A1 Compare with 2.12 (t(16, 0.975)) and reject H0 M1 Population means are not the same A1 7

More questions on Inference using normal and t-distributions

6 A biologist is studying the effect of nutrients on the heights to which plants grow. A random sample of 24 similar young plants is divided into two equal groups A and B. The plants in group A are fed with nutrients and water and the plants in group B are given only water. After four weeks, the height, in cm, of each plant is measured and the results are as follows. Group A 12.3 11.8 12.1 13.2 11.1 10.6 13.8 12.0 12.2 12.4 13.5 13.9 Group B 11.7 10.8 10.9 11.3 11.2 12.6 11.0 10.5 11.9 12.5 10.7 11.6 The biologist decides to carry out a test at the 5% significance level to test whether the nutrients have resulted in an increase in growth. (a) She carries out a Wilcoxon rank-sum test. Give a reason why this is an appropriate choice of test. [1] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Carry out the Wilcoxon rank-sum test for these results. [10] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ Additional Page If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown. ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................

Mark scheme: 6(a) Difference of location test for populations not known to be normal B1 1 Question Answer Marks 6(b) 10.6 2 10.5 1 11.1 7 10.7 3 11.8 12 10.8 4 12.0 14 10.9 5 12.1 15 11.0 6 12.2 16 11.2 8 12.3 17 11.3 9 12.4 18 11.6 10 13.2 21 11.7 11 13.5 22 11.9 13 13.8 23 12.5 19 13.9 24 12.6 20 M1 Total ranks: 109 or 191 A1 H0: = x y m m H1: > x y m m B1 Use normal approximation with attempts at mean and variance M1 mean = 150, variance = 300 A1 Question Answer Marks 6(b) 109.5 150 2.338 300 − = − M1A1 Probability 0.0097 A1 This is less than 0.05, so reject H0 M1 There is evidence of an effect on growth A1 10

More questions on Non-parametric tests