This paper is not to be removed from the Examination Halls

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1 ~~ST104B ZA d0 This paper is not to be removed from the Examination Halls UNIVERSITY OF LONDON ST104B ZB BSc degrees and Diplomas for Graduates in Economics, Management, Finance and the Social Sciences, the Diplomas in Economics and Social Sciences and Access Route Statistics 2 Friday, 03 May 2013 : 2.30pm to 4.30pm Candidates should answer all FOUR questions: QUESTION 1 of Section A (40 marks) and all THREE questions from Section B (60 marks in total). Candidates are strongly advised to divide their time accordingly. A list of formulae and extracts from statistical tables are provided after the final question on this paper. A calculator may be used when answering questions on this paper and it must comply in all respects with the specification given with your Admission Notice. The make and type of machine must be clearly stated on the front cover of the answer book. PLEASE TURN OVER University of London 2013 UL13/0210 Page 1 of 30 D1

2 SECTION A Answer all parts of question 1 (40 marks in total). 1. (a) For each one of the statements below say whether the statement is true or false explaining your answer. A and B are events such that 0 < P(A) < 1 and 0 < P(B) < 1. X,Y and Z are random variables with non-zero variance. i. If P(A)+P(B)=P(A B) then A and B can not be independent. ii. If A B, then P(A c ) P(B c ). iii. If Var(X +Y Z) =Var(X)+Var(Y ) Var(Z), it is possible that X,Y and Z are independent. (6 marks) (b) Consider the following testing of hypotheses on the parameter μ. Let {μ = 0} be the null hypothesis and {μ = 1} the alternative hypothesis. T is the test statistic and we will reject the null hypothesis if {T > t 0 }. Let α be the size of the test. Provide a mathematical expression for α using the symbols mentioned in this question. (3 marks) (c) The random variables X 1 and X 2 are each normally distributed with mean 1 and variance 1. Their correlation coefficient is 1 2. Find P(2X 1 > X 2 ). (5 marks) (d) The random variables ε i, i = 1,2,3,4 are independent with mean 0 and variance 1 and α is an unknown parameter. Suppose that you are given observations y 1, y 2 and y 3 such that y 1 = α + ε 1, y 2 = 3α + ε 2, y 3 = 3α + ε 3. Find the least square estimator ˆα, verify it is unbiased and calculate its variance. (9 marks) (e) Continuing from the previous question with the assumptions there, is ˆα the maximum likelihood estimator as well? Explain your answer. (2 marks) Page2of5 UL13/0210 Page 2 of 30 D1

3 (f) Suppose that in (d) you observed y 1 = 1, y 2 = 3 and y 3 = 4. Suppose also that ε i, i = 1, 2, 3, 4 are normally distributed and Provide a 95% prediction interval for y 4. y 4 = 2α + ε 4. (6 marks) (g) Consider two random variables X and Y. X can take the values 0, 1 and 2 and Y can take the values 1 and 2. The joint probabilities for each pair are given by the following table. X = 0 X = 1 X = 2 Y = Y = Find E (Y ), P(Y > X X > 0) and P(Y > X X +Y = 2). (9 marks) Page3of5 UL13/0210 Page 3 of 30 D1

4 SECTION B Answer all three questions in this section (60 marks in total). 2. Seven trainees tried to pass four different tests. Each trainee took each test once. The average score for test A was 75.6, of test B 77.2, of test C 78.0 and of test D The following is the calculated ANOVA table with some entries missing. Source degrees of freedom sum of squares mean square F - value Trainees 3.6 Tests Error Total (a) Complete the table using the information provided above. (7 marks) (b) Is there a significant difference between scores in different tests? Explain your answer. (4 marks) (c) Construct a 90% confidence interval for the difference in scores between tests A and D. Would you say there is a difference? (5 marks) (d) Construct 95% simultaneous confidence intervals for the difference in scores between A and D and the difference between B and C. (5 marks) Page4of5 UL13/0210 Page 4 of 30 D1

5 3. Let x 1,x 2,...,x n be an independent sample from a continuous distribution with density 1 ( θ exp x ) θ where x > 0 and θ is a positive parameter to be estimated. (a) Derive the log-likelihood function of the data. (b) Show that the maximum likelihood estimator of θ is given by (c) Show that ˆθ is unbiased. ˆθ = n i=1 x i n (d) Calculate Var(ˆθ) and show that ˆθ is consistent. (5 marks) (6 marks) (4 marks) (5 marks) You may find the following integral useful ( x n exp x ) dx = n!θ n+1. θ 0 4. Football team A plays in a Sunday league. When team A s star player plays they win with probability 0.5 and draw with probability 0.2. If she does not play, the team loses with probability 0.4 or draws with probability 0.3. The probability that she will turn up on any Sunday is 0.4. Events during each Sunday are independent of events on any other Sunday. (a) Suppose that the team gets awarded 3 points for a win and 1 for a draw. Find the expected value and the variance of the number of points they will earn over the next 10 matches. (8 marks) (b) Given that the last match was a win, what is the probability the star player was playing? (4 marks) (c) Given that the team collected 4 points during the last two matches, what is the probability the star player played in both? (7 marks) END OF PAPER Page5of5 UL13/0210 Page 5 of 30 D1

6 Formulae for Statistics Discrete distributions Distribution p X (x) E(X) Var(X) 1 Uniform, for x =1, 2,...,n n+1 n 2 n Bernoulli p x (1 p) 1 x, for x =0, 1 p p(1 p) Binomial ( n ) x p x (1 p) n x, for x =0, 1,...,n np np(1 p) Geometric (1 p) x 1 p, for x =1, 2, 3,... 1 p 1 p p 2 Poisson e λ λ x x!, for x =0, 1, 2,... λ λ Continuous distributions Distribution f X (x) F X (x) E(X) Var(X) 1 x a Uniform, for a<x<b b a b+a, for a<x<b b a 2 (b a) 2 12 Exponential λe λx, for x>0 1 e λx, for x>0 1/λ 1/λ 2 Normal 1 2πσ 2 e (x μ)2 /2σ 2, for all x μ σ 2 Sample Quantities Sample Variance s 2 = i (x i x) 2 /(n 1) = ( i x2 i n x 2 )/(n 1) Sample Covariance i (x i x)(y i ȳ)/(n 1) = ( i x iy i n xȳ)/(n 1) Sample Correlation ( i x iy i n xȳ)/ ( i y2 i nȳ2 )( i x2 i n x2 ) UL13/0210 Page 6 of 30 D1

7 Inference Variance of Sample Mean σ 2 /n One-sample t statistic x μ s/ n with (n 1) degrees of freedom Two-sample t statistic ( x 1 x 2 ) (μ 1 μ 2 ) [1/n1 +1/n 2 ]{[(n 1 1)s 2 1 +(n 2 1)s 2 2]/(n 1 + n 2 2)} Variances for differences of binomial proportions Pooled where ( 1 ˆp(1 ˆp) + 1 ), n 1 n 2 ˆp = n 1p 1 + n 2 p 2 n 1 + n 2 Separate p 1 (1 p 1 )/n 1 + p 2 (1 p 2 )/n 2 Estimates for y = α + βx fitted to (y i,x i ) for i = 1, 2,...,n are â =ᾱ ˆβ x and i ˆβ = (x i x)(y i ȳ) i (x. i x) 2 Least Squares The estimate of the variance is ˆσ 2 = i (y i ȳ) 2 ˆb 2 i (x i x) 2. n 2 The variance of ˆb is σ 2 i / (x i x) 2 Chi-square Statistic (Observed Expected) 2 /Expected, with degrees of freedom depending on the hypothesis tested. UL13/0210 Page 7 of 30 D1

8 STATISTICAL TABLES Cumulative normal distribution Critical values of the t distribution Critical values of the F distribution Critical values of the chi-squared distribution New Cambridge Statistical Tables pages C. Dougherty 2001, 2002 (c.dougherty@lse.ac.uk). These tables have been computed to accompany the text C. Dougherty Introduction to Econometrics (second edition 2002, Oxford University Press, Oxford), They may be reproduced freely provided that this attribution is retained. UL13/0210 Page 8 of 30 D1

9 STATISTICAL TABLES 1 TABLE A.1 Cumulative Standardized Normal Distribution A(z) z A(z) is the integral of the standardized normal distribution from to z (in other words, the area under the curve to the left of z). It gives the probability of a normal random variable not being more than z standard deviations above its mean. Values of z of particular importance: z A(z) Lower limit of right 5% tail Lower limit of right 2.5% tail Lower limit of right 1% tail Lower limit of right 0.5% tail Lower limit of right 0.1% tail Lower limit of right 0.05% tail z UL13/0210 Page 9 of 30 D1

10 STATISTICAL TABLES 2. TABLE A.2 t Distribution: Critical Values of t Significance level Degrees of Two-tailed test: 10% 5% 2% 1% 0.2% 0.1% freedom One-tailed test: 5% 2.5% 1% 0.5% 0.1% 0.05% UL13/0210 Page 10 of 30 D1

11 STATISTICAL TABLES 3 TABLE A.3 F Distribution: Critical Values of F (5% significance level) v v UL13/0210 Page 11 of 30 D1

12 STATISTICAL TABLES 4 TABLE A.3 (continued) F Distribution: Critical Values of F (5% significance level) v v UL13/0210 Page 12 of 30 D1

13 STATISTICAL TABLES 5 TABLE A.3 (continued) F Distribution: Critical Values of F (1% significance level) v v UL13/0210 Page 13 of 30 D1

14 STATISTICAL TABLES 6 TABLE A.3 (continued) F Distribution: Critical Values of F (1% significance level) v v UL13/0210 Page 14 of 30 D1

15 STATISTICAL TABLES 7 TABLE A.3 (continued) F Distribution: Critical Values of F (0.1% significance level) v v e e e e e e e e e e e e e e e UL13/0210 Page 15 of 30 D1

16 STATISTICAL TABLES 8 TABLE A.3 (continued) F Distribution: Critical Values of F (0.1% significance level) v v e e e e e e e e e e UL13/0210 Page 16 of 30 D1

17 STATISTICAL TABLES 9 TABLE A.4 2 (Chi-Squared) Distribution: Critical Values of 2 Significance level Degrees of 5% 1% 0.1% freedom UL13/0210 Page 17 of 30 D1

18 Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) Cambridge University Press, reproduced with permission. UL13/0210 Page 18 of 30 D1

19 Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) Cambridge University Press, reproduced with permission. UL13/0210 Page 19 of 30 D1

20 Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) Cambridge University Press, reproduced with permission. UL13/0210 Page 20 of 30 D1

21 Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) Cambridge University Press, reproduced with permission. UL13/0210 Page 21 of 30 D1

22 Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) Cambridge University Press, reproduced with permission. UL13/0210 Page 22 of 30 D1

23 Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) Cambridge University Press, reproduced with permission. UL13/0210 Page 23 of 30 D1

24 Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) Cambridge University Press, reproduced with permission. UL13/0210 Page 24 of 30 D1

25 Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) Cambridge University Press, reproduced with permission. UL13/0210 Page 25 of 30 D1

26 Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) Cambridge University Press, reproduced with permission. UL13/0210 Page 26 of 30 D1

27 Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) Cambridge University Press, reproduced with permission. UL13/0210 Page 27 of 30 D1

28 Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) Cambridge University Press, reproduced with permission. UL13/0210 Page 28 of 30 D1

29 Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) Cambridge University Press, reproduced with permission. UL13/0210 Page 29 of 30 D1

30 Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) Cambridge University Press, reproduced with permission. UL13/0210 Page 30 of 30 D1

GEOMETRIC -discrete A discrete random variable R counts number of times needed before an event occurs

STATISTICS 4 Summary Notes. Geometric and Exponential Distributions GEOMETRIC -discrete A discrete random variable R counts number of times needed before an event occurs P(X = x) = ( p) x p x =,, 3,...