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PRIFYSGOL CYMRU/UNIVERSITY OF WALES
ABERYSTWYTH
INSTITUTE OF MATHEMATICS AND PHYSICS
SEMESTER 2 EXAMINATIONS, MAY 2009
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 2009 Page 2 of 6
Section A
1 The random variable W has mean 34 and variance 81. (a) Find the values of (i) E[150–4W] (ii) SD(150–4W) (b) Between what limits would you expect the bulk of values of 150 – 4W to lie? [5]
2 S~Bin(n,p) where E(S)=50(1+2a) and Var(S)=25(1-4a^2 ). (a) Find n and p in terms of a. (b) Find an unbiased estimator of a. [5]
3 The random time interval, X, has probability density function (pdf) f x( ) = θe −θx
for x>0. Show that 0.005^ X 5.298^ 0. θ θ
P (^) and deduce a 99% confidence interval for θ. [5]
4 If Z ~Bin(100, 0.35) use Binomial tables to find (i) P(Z ≥ 30); (ii) P(Z ≤ 28); (ii) P(37<Z≤42). [5]
5 X, Y each have mean 2 and their respective variances are 1 and 4; the correlation between the two of them is –½. Find the covariance between X and Y and the variance of X–3Y+4. [5]
6 IQ scores vary Normally around a mean of 100 with standard deviation 16. (i) What score is achieved by the top 8%? (ii) Nine people tested this morning recorded an average score of 110. Is this unusual? [5]
7 46% of the electorate support their representative, Ivor Goodlife. Use a suitable approximation to calculate the probability that, in a sample of 150 members of the electorste, the number supporting Ivor is between 65 and 75, both inclusive. [5]
8 X^ and^ Y^ are independent random variables each with mean^ α^ and variance 2α. Find expressions in their simplest form for: (i) the standard deviation of X+Y; (ii) E[X(X+2Y–3)]. [ 5 ]
MA11310:Statistics May 2009 Page 4 of 6
value and state the conclusion of the test. Should she be concerned at the apparent increase?
(b) The discrete random variable N has a probability mass function such that P(N=0) = 0.001 and P( P^ N ( N^ = =n^ n +) 1)^ = 0.9 nn^ ++^31 , for n = 0,1,2,…. Find P(N≥2) and determine the most likely value(s) of N.
[5]
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11B (a)^ In each of the following cases, decide whether or not the Binomial distribution model would be appropriate. If you think it is, give the values of the parameters n and p; if not, suggest a sensible, possible, alternative. (i) The number out of 25 candidates, who pass a test which has a long-run pass rate of 72%. (ii) The number of over 40s who are selected for interview, out of the eleven applicants for a job, seven of whom are over 40. (iii) The total number of grams weighed by 120 similar items whose weights are independently distributed with mean 1g and standard deviation 0.5g.
(b) A statistic S has a cumulative distribution function (cdf) given by 1 2 2 2 (^12)
, for 0 ( ) 1 2 , for 2
s (^) s F s s (^) s
θ θ θ (^) θ θ θ
≤^ ≤
= ^
(i) Calculate P(θ/3 < S < 4θ/3). (ii) Deduce the probability that the interval from ¾SQto 3S contains the true value of θ. (iii) Find the probability density function (pdf) of S and draw a rough sketch. Indicate the median of S on your sketch. How can you tell without further calculation that S is an unbiased estimator of θ?
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12B 60% of the inhabitants of a country belong to the Common class and 20% to the Privileged class. If n members of the population are selected at random, let C and P denote the respective numbers of Common and Privileged people selected.
MA11310:Statistics May 2009 Page 5 of 6
(a) Identify the distributions of C, of P, and of C+P. Quote the mean and variance of each of these distributions. Find the correlation between C and P.
The country is not governed democratically and elections are carried out by selecting a sample of n members of the population. For every member of the Common class in the sample one vote is given to the opposition whereas for every member of the Privileged class selected, four votes are awarded to the government. The government’s majority is thus M=4P–C.
(b) Find the mean and variance of M. (c) Between what limits would you expect M to lie? (d) How large should n be for the government to be confident of survival?
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13B Briefly describe the ideas that lead to the Poisson distribution being used as a model for the numbers of occurrences of random events in a fixed time interval.
(a) Electrical wire is produced by a process that winds the wire onto reels of various sizes. Minor faults occur at the rate of 2.4 per kilometre of wire. (i) What proportion of 250m reels of wire are free of faults? (ii) Reels each of length 100m are boxed in crates of 20 reels per crate. What is the probability that all the reels in such a crate are fault-free? (iii) A 500m coil of cable is known to have at least one fault. What is the chance that it actually has exactly two faults?
(b) Male customers arrive at a supermarket at random at a rate of 20 per hour; female customers also arrive randomly, but at a rate of 30 per hour. If you start observing the entrance to the supermarket at 10.00am, at what time would you expect (i) the first male customer to arrive; (ii) the first female customer to arrive; (iii) the first customer to arrive.
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