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Unformatted text preview: Find x, correct to two decimal places. (3) (b) In the past year, 80 % of Karl’s throws have been longer than 56.52 m. If the mean distance of his throws was 59.39 m, find the standard deviation of his throws, correct to two decimal places. (3) (c) This year, Karl’s throws have a mean of 59.50 m and a standard deviation of 3.00 m. Ian’s throws still have a mean of 60.33 m and standard deviation 1.95 m. In a competition an athlete must have at least one throw of 65 m or more in the first round to qualify for the final round. Each athlete is allowed three throws in the first round. (i) Determine which of these two athletes is more likely to qualify for the final on their first throw. (ii) Find the probability that both athletes qualify for the final. (11) (Total 17 marks) 489 729. Consider the differential equation (a) dy y = . dθ e 2 θ + 1 Use the substitution x = eθ to show that ∫ dy dx . = y x( x 2 + 1) ∫ (3) (b) Find dx . 2 + 1) ∫ x( x (4) (c) Hence find y in terms of θ , if y = 2 when θ = 0. (4) (Total 11 marks) 730. Let X be a random variable with a Poisson distribution such that Var(X) = (E(X))2 – 6. (a) Show that the mean of the distribution is 3. (3) (b) Find P(X ≤ 3). (1) Let Y be another random variable, independent of X, with a Poisson distribution such that E(Y) = 2. (c) Find P(X + Y < 4). (2) 490 (d) Let U = X + 2Y. (i) Find the mean and variance of U. (ii) State with a reason whether or not U has a Poisson distribution. (4) (Total 10 marks) 731. A chicken farmer wishes to find a confidence interval for the mean weight of his chickens. He therefore randomly selects n chickens and weighs them. Based on his results, he obtains the following 95 % confidence interval. [2148 grams, 2188 grams] The weights of the chickens are known to be normally distributed with a standard deviation of 100 grams. (a) Find the value of n. (5) (b) Assuming that the same confidence interval had been obtained from weighing 166 chickens, what would be its level of confidence? (3) (Total 8 marks) z3 4 for z ∈ [0, 2] and 0 4 elsewhere. A physicist assumes that the lifetime of a certain particle can be modelled by this random variable. The interval [0, 2] is divided into the following equal intervals: 732. The random variable Z has probability density function f(z) = z – I1 = [0, 0.4[ I2 = [0.4, 0.8[ I3 = [0.8, 1.2[ I4 = [1.2, 1.6[ I5 = [1.6, 2] 491 The physicist carried out 40 experiments and recorded the number of times the value of Z lay in each of the intervals Ik where k = 1, 2, 3, 4, 5 as shown in the following table: I1 I3 I4 I5 2 (a) I2 12 9 8 9 Assuming that the physicist’s assumption is correct, for each value of k find pk = P(Z ∈ Ik). (4) (b) At the 5% significance level can his assumption be accepted? (8) (Total 12 marks) 733. The relation R on is defined as follows z1Rz ⇔ z1 = z2 for z1, z2 ∈ (a) Show that R is an equivalence relation on . . (3) (b) Describe the equivalence classes under the relation R. (2) (Total 5 marks) 734. Conside...
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