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1. (a) In this question, A,B, and C are random variables with means µA,µB, and µC re- spectively, and a common variance σ2. An independent random sample of size 10 is taken from each of these random variables, with the following results:

ā = 49.617 ∑10i=1 a 2 i = 24635.30 ∑

10 i=1(ai − ā)

2 = 16.832

b̄ = 48.526 ∑10i=1 b 2 i = 23583.16 ∑

10 i=1(bi − b̄)

2 = 35.438

c̄ = 51.031 ∑10i=1 c 2 i = 26100.33 ∑

10 i=1(ci − c̄)

2 = 58.704

i) Explain what assumptions on A,B, and C are required in order to perform a one- way ANOVA test of the null hypothesis µA = µB = µC against the alternative that this is false. [2 marks]

ii) Given that those assumptions hold, perform the ANOVA test at the 0.05 signifi- cance level. [5 marks]

iii) Let SSw denote the total sum of squares random variables in this ANOVA set-up. What is the distribution of SSw/σ2? [3 marks]

iv) Derive a 95% confidence interval for σ2. [4 marks]

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(b) You have performed a simple linear regression with independent variable x, resulting the following plot of the residual at xi against xi: Does this cast any doubt on the assumptions of the simple linear regression? Explain your answer. [2 marks]

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(c) You are given the following paired data

i xi yi

1 1 0.93

2 2 −0.70

3 3 −2.94

4 4 −5.00

5 5 −6.90

6 6 −9.16

7 7 −10.95

8 8 −12.87

9 9 −15.33

10 10 −16.96

for which

x̄ = 5.5 ȳ = −7.988,

10

∑ i=1

(yi − ȳ)2 = 336.62,

10

∑ i=1

(xi − x̄)2 = 82.5,

10

∑ i=1

(xi − x̄)(yi − ȳ) = −166.59.

Assume that the data satisfy the usual simple linear regression model, so that the yi are observations of a random variable Y whose conditional distribution given X = x has the form N(a + bx,σ2).

i) Calculate the predicted mean response if x = 11. [4 marks] ii) Find a 95% confidence interval for the slope parameter b. [5 marks]

Total: 25 marks

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2. i) State the central limit theorem for a sequence of independent and identically dis- tributed random variables X1,X2,… with finite mean µ and variance σ2. [3 marks]

ii) Let X be a random variable with the Bernoulli(p) distribution, so that

P(X = 1) = p, P(X = 0) = 1 − p.

a) Derive the expected value of X . [3 marks] b) Derive the variance of X . [3 marks]

iii) Let B be a random variable with the Binomial(n, p) distribution. Using the central limit theorem and the fact that B has the same distribution as

X1 +···+ Xn,

where X1,…,Xn are independent and each has the Bernoulli(p) distribution, show that

B − np√ np(1 − p)

has approximately a standard normal distribution for large n. [6 marks]

iv) Someone claims to be able to predict the outcome of a coin flip. In n = 100 flips they are correct 55 times. Let B be a random variable counting the number of successful predictions in 100 flips and p be the probability they predict a single flip correctly. Test the hypothesis H0 : p = 1/2 against the alternative H1 : p > 1/2 at significance level α = 0.05 using the test statistic

T = B − 100 p√ 100p(1 − p)

,

which you can assume to have approximately the N(0,1) distribution. [6 marks]

v) What is the minimum number of successful predictions out of 100 flips which would cause you to reject the null hypothesis in the test of part iv)? [4 marks]

Total: 25 marks

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3. (a) Hooke’s law states that the elongation L of a spring subjected to a weight force W is given by W = kL, where W is measured in kg, L is measured in mm, k is called the elastic constant of the spring and is expressed in kg/mm. A group of Physics students wants to measure the elastic constant of a spring. Not having very precise instruments they perform an experiment by applying to the spring weights of increas- ing intensity, from 10 to 50 kg, for five times. The measurements are influenced by the approximation of the reading of the lengths and by the fact that the spring does not behave like a perfect spring and the same weight applied several times does not give the same elongation. The measurements of elongation, in millimeters, for each test done are shown below.

Weight (kg) Measured elongation L (mm)

W L1 L2 L3 L4 L5

10 48.6 47.6 48.8 51.5 49.8

15 78.4 77.5 71.6 77.5 73.6

20 95.7 98.6 100.4 102.4 97.3

25 123.5 131.1 118.9 130.6 128.3

30 150.6 154.5 148.3 146.0 153.3

35 175.5 176.3 173.2 181.8 181.8

40 209.4 199.8 197.8 195.9 203.5

45 230.9 233.2 230.5 218.7 222.6

50 245.2 249.2 257.0 256.7 244.3

i. Write the equation of a simple linear regression model with repeated observa- tions for this dataset. [3 marks]

ii. Perform a statistical test to establish if a linear regression model with repeated observations is a good fit for these data. [6 marks]

iii. Perform a statistical test to establish if Hookes law can be assumed to be valid for this dataset with the 5% significance level. [5 marks]

(b) In this question, X is a continuous random variable with probability density function

f (x) =

{ α −1x(α

−1)−1 0 < x < 1 0 otherwise.

Here α is a positive parameter which we will estimate using an independent random sample from X .

i) Show that if t < α−1 then E(X −t) = α −1

α−1−t . [3 marks]

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ii) Show that the log-likelihood function l(a) based on independent observations x1,…,xn of X is

l(a) = −n log a +(a−1 − 1) n

∑ i=1

log xi

for a > 0. [2 marks] iii) Show that the maximum likelihood estimator of α is

1 n

n

∑ i=1

(−log Xi).

You may assume any critical point of l is a maximum. [2 marks] iv) Show that the maximum likelihood estimator of α is unbiased, and compute its

mean squared error. You can use without proof the fact that if Z ∼ Exponential(λ) then E(Z) = λ−1 and V (Z) = λ−2. [4 marks]

Total: 25 marks

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4. (a) In this question, X is a continuous random variable with probability density function

fX (x) =

{ θxθ−1 0 < x < 1 0 otherwise,

where θ is a positive parameter.

[6 marks]i) Derive E(X) and V (X). ii) Let y be a real number. Show

that

P(X ≥ y) =

 

1 y ≤ 0 1 − yθ 0 < y < 1 0 y ≥ 1

[4 marks]

iii) Use the previous part to show that the density function fX−1 of X −1 is fX−1(r) =

θr−θ−1 if r > 1 and fX−1(r) = 0 otherwise. [5 marks]

(b) In this question, X and Y are jointly distributed discrete random variables. The prob- ability P(X = i,Y = j)for discrete random variable (X,Y ) is given by the entry in column i, row j of the following table:

X

0 1 2

Y 0 pq q − pq 0

1 p − pq p2 − q + pq 1 − 3 p 2

i) Show that P(X = 0) = p, P(X = 1) = p/2 and P(X = 2) = 1 − 3 p/2. [3 marks]

ii) Are the events Y = 0 and X = 0 independent? Justify your answer. [2 marks]

iii) For which values of p and q are the events X = 2 and Y = 1 independent? [1 mark]

iv) Give a formal definition of what it means for two random variables to be inde- pendent. [1 mark]

v) For which values of p and q are the random variables X and Y independent? [3 marks]

Total: 25 marks

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