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PRIFYSGOL CYMRU/UNIVERSITY OF WALES
ABERYSTWYTH
INSTITUTE OF MATHEMATICS AND PHYSICS
SEMESTER 2 EXAMINATIONS, MAY 2008
MA113 10 – Statistics
Time allowed – 2 hours
- Full marks will be given for complete answers to ALL questions in Section A and to THREE questions in Section B. In Section B credit will be given for the best three questions.
- Statistical tables will be provided.
- Calculators are permitted, provided they are silent, self-powered, without communications facilities, and incapable of holding text or other material that could be used to give a candidate an unfair advantage. They must be made available on request for inspection by invigilators, who are authorised to remove any suspect calculators.
Information
Information about standard distributions can be found on the back page of the booklet of Statistical Tables.
You may quote without proof the following: 2 3 2 3
1 1 for | | 1 (1 ) (^1) 2! 3!^ x for all real
x x x (^) x x
x x^ x e x
MA11310:Statistics May 2008 Page 2 of 6
Section A
1 The random variable^ V^ has mean -10 and standard deviation (SD) 3. (a) Find the values of (i) E[16-4V] (ii) SD(16-4V) (b) Between what limits would you expect the bulk of values of 16-4V to lie? [5]
2 Z~Pasc(r,p) where E(Z)=33 and Var(Z)= 66. (a) What values do r and p take? (b) Evaluate P(Z = 15) [5]
3 In each of the following cases, decide whether or not the Binomial distribution model would be reasonable. If you think it is, give the values of the parameters n and p; if not, suggest a sensible, possible alternative. (a) The number of motorists who break the speed limit going down Penglais Hill between 12 and 2pm today. (b) The number of times I get double 6 when rolling two dice 10 times. (c) The number of errors made in the electronic transmission of a coded message containing 47 characters if the long-run percentage of errors is 0.1%. [5]
4 If K~Bin(50,0.75) use Binomial tables to find (i) P(K≥40); (ii) P(34<K<39). [5]
5 X and Y each have mean 10 and variance 6; the variance of X+Y is also 6. Find the covariance between X and Y and the mean and the variance of 2 X+3Y. [5]
6 Accidents at a blackspot on a certain road occur randomly at a rate of 2 per month. (i) What distribution is appropriate for the number, N, of accidents in a 2 month period? Use tables to find P(N≥5). (ii) Would you be surprised if there were no accidents in a three month period? Justify your answer. [5]
(^7) The times taken in minutes to complete the various processes in the finishing
department of a production process are independently and Normally distributed and have the following means and standard deviations:
MA11310:Statistics May 2008 Page 4 of 6
Section B
Credit will be given for the best THREE answers from this section.
9B (a) One quarter of the visitors to a general hospital are carriers of a drug- resistant super-bug while the others are non-carriers. When a carrier is tested for the super-bug, the chance of a positive response is φ; when a non-carrier is tested the chance of a negative response is also φ. (i) Show that the chance of a positive test response is (3–2φ)/4. (ii) Suppose n visitors are randomly chosen and tested and that X of these produce positive responses. State the distribution of X and write down its mean and variance. (iii) Show that T 1 = 1.5 – (2X/n) is an unbiased estimator of φ, and find its standard error. (b) In another hospital, a different procedure yields a statistic W that has a Bin(n, 2φ/3) distribution. Deduce a second unbiased estimator T 2 of φ and explain which of T 1 , T 2 is to be preferred, giving your reasons.
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10B (a) The length of time, X days, until the next eruption of an active volcano has probability density function (pdf) given by 4 5 f x ( ) 4 ,x x = θ^ > θ, where θ is the minimum time and is unknown. (i) Calculate E[X] and find an unbiased estimator of θ based on X. (ii) The interval 0.2X to 0.9X is proposed as a confidence interval for θ. Calculate its confidence coefficient. (b) Evaluate P(X=k+1)/P(X=k) and hence find the most likely value of X when P(X=k) = c (5.8)k/k! for k=0, 1, 2, … Show also that c = exp(-5.8).
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11B (^) (a) For R ~ Bin(20, 0.9), evaluate P(R ≤ 13). A drug, when applied at standard strength, has a 90% chance of reducing inflammation in a child suffering from earache. A stock of the drug, which has been held for some time, is applied to 20 children suffering from earache and reduces inflammation in just 13 cases. Copy and complete the following
[2]
MA11310:Statistics May 2008 Page 5 of 6
report of a test of hypotheses concerning p, the probability that the drug reduces inflammation. To measure the evidence that the drug has lost its strength we test the …………. hypothesis H 0 : p = 0.9 versus the alternative H 1 : …….. using the test ……….. R, which is the number of children out of 20 whose inflammation was reduced. Under the conditions of the …….. hypothesis, R has a …….. distribution. Assuming this distribution, we calculate the P-value, i.e. the observed ……… ………., as P(R ≤ 13) = ……... The test is therefore significant at the …..% level implying that there is ……… evidence that the drug ……….
(b) The random variable N is the number of failures occurring prior to the first success in a sequence of Bernoulli trials with success probability α (i) Show that the probability mass function (pmf) of N is pn = α(1–α)n, for n = 0,1,2, …. (ii) Verify that for each r,
∑n^ ∞ =r p n^ =^ (1^ −α)r
and deduce the range of values of α such that P(N≥10) is less than 0.05. (iii) If the first success occurs on trial number 11, would you believe a claim that the probability of success is as high as 0.2? Justify your answer.
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12B (a) Two political parties, the Progressives and the Questioners, are each supported by 30% of the electorate. Let P and Q denote the number of supporters of the two parties among a sample of n members of the electorate, chosen at random. Identify the distributions, means and variances of P, of Q, and of P+Q. Between what limits would you expect values of (P–Q)/n to lie? How large should n be for you to be fairly sure that the percentage supportt for the two parties will be within 2 of each other in the sample? (b) X,Y and Z all have mean 10 and variance 5. The covariance between X and Y is +3 whilst the correlation between Y and Z is –0.2; X and Z are uncorrelated. Find the mean and standard deviation of W = 2X–3Y–Z+20.
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