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NAME: HOLIDAY EXERCISE S.6 MATHEMATICS PAPER 425/2 TIME: 2 Hours 14TH MAY 2013. INSTRUCTIONS: Answer all questions in section A and B. Where necessary use, g , = 9.8 SECTION A: 1. A discrete random variable X has the following probability distribution. x 3 P( x = x) 1 2 3 /16 ½ ¼ 4 1 /16 Find the mean and the variance of x 2. Two events A and B are neither independent nor mutually exclusive. Given that P(B) = 1/3 , P(A) = ½ and P(A∩B’ ) = 1/3 , find (i) 3. P( A’ U B’ ) (ii) P( A’/B’ ) A Mathematics student measured the times taken in seconds for a trolley to run down slopes of varying gradients and obtained the following results; 35.2 , 34.5 , 33.5 , 29.3 , 30.9 , 31.8 Calculate; (i) the mean times (ii) the standard deviation 4. The equation x 3 x 16 0 has a root between x = 2 and x = 3 Use linear interpolation to obtain the root. 5. Given that the values x = 2 , y = 1 each has been approximated to the nearest integer. Determine the greatest possible value for each of the following; (i) x y (ii) y 7y x 6. A farmer keeps two breeds A, B of chicken, 70% of the egg production is from birds of breed A. Of the eggs laid by the A hens, 30% are large, 50% medium and the remainder small: For the B hen the corresponding values are 40%, 30% and 30%. Egg color (brown or white) is ‘’MATHEMATICS + DETERMINATION = DISTINCTION’’ Page 1 manifested independently of size in each breed, 30% of A eggs and 40% of B eggs are brown. Find: (a) The probability that an egg laid by an A hen is large and brown. (02 marks) (b) The probability that an egg is large and brown. (03 marks) 7. Given the following data: f(0.9) = 0.226, f(1.0) = 0.242, f(1.1) = 0.218 and f(1.2) = 0.192. Use linear interpolation to find the value of f(1.04) and the value of x for which f(x) = 0.25. (05 marks) 8. A continuous random variable x has pdf given by k x;0 x 4 f x 0 ; otherwise Determine; (a) the value of k, (b) P(1 < X < 2). (02 marks) (03 marks) SECTION B: 9. 10. (a) A total population of 700 students sat an examination for which the pass mark was 50. The marks were normally distributed. 28 students scored below 40 marks while 35 scored above 60 marks. Determine; (i) (ii) (iii) the mean and standard deviation of the students marks the probability that a student chosen at random passed the examination. The more students who passed when the pass mark was lowered by 2 marks. X is a continuous random variable with p.d.f. f (x) where f (x) = 11. x 3 ; 0 ≤ x ≤ 2 –2 x+2 3 ; 2 ≤ x ≤ 3 0 ; else where (a) (b) (c) (d) Sketch f (x) Find the cumulative distribution function F(x) Find P( 1 ≤ x ≤ 2.5 ) Find the median and mode of the distribution. (a) There are 3 black and 2 white balls in each of the two bags. A ball is taken from the first bag and put in the second bag, then a ball is taken from the second into the first. What is the probability that there are now the same number of black and white balls in each bag as there were to begin with ? ‘’MATHEMATICS + DETERMINATION = DISTINCTION’’ Page 2 (b) The germination time for a certain species of seeds is known to be normally distributed. If , for a given batch of the seeds 20% take more than 6 days to germinate and 10% take less than 4 days to germinate, (i) (ii) 12. determine the mean and the standard deviation of the germination time. find the 99% confidence limits of the germination time. The table below gives the cumulative distribution of the heights of 400 children in a certain school Height (cm) < 100 < 110 < 120 < 130 < 140 < 150 < 160 < 170 (a) Draw the cumulative frequency curve and use it to determine the estimated; (i) (ii) (iii) (b) Cumulative frequencies 0 27 85 215 320 370 395 400 median interquartile range the 10 to 90 percentage range 8 students took an examination in Mathematics and Physics and their grades were as follows; Maths A A D O D B F Physics C C E E C A F Calculate the rank correlation coefficient between the grades and comment on your results. END ‘’MATHEMATICS + DETERMINATION = DISTINCTION’’ Page 3