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Unformatted text preview: ≤ v <50 9 50 ≤ v <60 35 60 ≤ v <70 93 70 ≤ v <80 139 80 ≤ v <90 261 90 ≤ v <100 295 100 ≤ v <110 131 110 ≤ v <120 26 120 ≤ v <130
(a) Number of cars 11 For the cars on the road, calculate
(i) an unbiased estimate of the mean speed; (ii) an unbiased estimate of the variance of the speed.
(4) (b) For the cars on the road, calculate
(i) a 95 % confidence interval for the mean speed; (ii) a 90 % confidence interval for the mean speed.
(4) 371 (c) Explain why one of the intervals found in part (b) is a subset of the other.
(2)
(Total 10 marks) 559. Give all numerical answers to this question correct to three significant figures.
Two typists were given a series of tests to complete. On average, Mr Brown made 2.7 mistakes
per test while Mr Smith made 2.5 mistakes per test. Assume that the number of mistakes made
by any typist follows a Poisson distribution.
(a) Calculate the probability that, in a particular test,
(i) Mr Brown made two mistakes; (ii) Mr Smith made three mistakes; (iii) Mr Brown made two mistakes and Mr Smith made three mistakes.
(6) (b) In another test, Mr Brown and Mr Smith made a combined total of five mistakes.
Calculate the probability that Mr Brown made fewer mistakes than Mr Smith.
(5)
(Total 11 marks) 560. A calculator generates a random sequence of digits. A sample of 200 digits is randomly
selected from the first 100 000 digits of the sequence. The following table gives the number of
times each digit occurs in this sample.
digit 0 1 2 3 4 5 6 7 8 9 frequency 17 21 15 19 25 27 19 23 18 16 It is claimed that all digits have the same probability of appearing in the sequence.
(a) Test this claim at the 5 % level of significance.
(7) (b) Explain what is meant by the 5 % level of significance.
(2)
(Total 9 marks) 372 561. The set of all real numbers Runder addition is a group ( , +), and the set real numbers under multiplication is a group (
(
(a) + + + of all positive , ×). Let f denote the mapping of ( , +) to x , ×) given by f(x) = 3 .
Show that f is an isomorphism of ( , +) onto ( + , ×).
(6) (b) Find an expression for f–1.
(1)
(Total 7 marks) a b , where a, b, c and d ∈
562. Let G = c d (a) , and ad – bc ≠ 0 . Show that (G, *) is a group, where * denotes matrix multiplication.
(6) (b) Is this group Abelian? Give a reason for your answer.
(3) Let (H, *) be any subgroup of (G, *) and let M, N be any elements of G.
Define the relation RH on RHG as follows:
MRHN <=> there exists L ∈ H such that M = L * N. (c) Show that RH is an equivalence relation on G.
(5) Let K denote the set of all the elements of G with ad – bc > 0. (d) Show that (K, *) is a subgroup of (G, *).
(5) 373 Let M, N be any 2 elements of G. Define the equivalence relation RK on G as above, i.e.
MRKN <=> there exists L ∈ K such that M = L * N.
(e) (i) Show that there are only two equivalence classes. (ii) Explain how to determine to which equivalence class a given element M of G
belongs.
(4)
(Total 23 marks) 563. (a) Using Euclid’s algorithm,...
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 Fall '13
 Apple
 The Land

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