F21SA : Statistical Modelling and Analysis - Mathematics Assignment Help

Assignment Task:

Task:

1. In a recent election, 40% of female voters and 55% of male voters voted for party A. All the other voters voted for party B. Moreover, 45% of the voters were female.

(a) Calculate the probability that a randomly selected person voted for party A in that election.
[3 marks]
(b) One voter is selected at random, and it turns out that they voted for party A. Calculate the probability that the voter is female. [3 marks]

(c) In a certain constituency, errors where reported in the election day at a rate of 2 per hour. Assuming that the errors can be modelled as a Poisson random variable, calculate the probability that at least 36 errors were reported during a period of 14 hours. [4 marks]
Hint: You may use a normal approximation to the Poisson distribution.

(d) A poll of 100 male voters finds that 50% intend to vote for party A in an upcoming election. We wish to use these data to test the hypothesis that 55% of male voters vote for party A, against the alternative hypothesis that this proportion is less than 55%.

i) Propose a statistical test for the considered hypothesis testing problem and calibrate it to achieve a Type I error of approximately 5%. [5 marks]
ii) What is the conclusion of the test based on the observed data?

2. Let x = (x1, . . . , xn) denote a sample from a normal distribution with unknown mean θ ∈ R and known variance σ
2
. The likelihood of a single observation xk ∈ R is given by L(θ; xk) = 1√2πσ2exp {− 12σ2 (xk − θ)2}.

Assume that observations are i.i.d. given the value of θ.
(a) Assuming a normal prior for θ with prior mean α ∈ R and variance β 2 > 0, derive the posterior distribution for θ given the observed data x. [4 marks]
(b) Report the posterior mean and variance for θ. [2 marks]
(c) Suppose that your prior beliefs lead you to adjust the values of α and β such that E(θ) = 0 and Var(θ) = 100. Would you describe the resulting prior as informative? Explain your answer. [2 marks]

(d) Show that the posterior mean is the Bayesian estimator that minimises the expected quadratic loss. [5 marks]

(e) Assuming the prior specified in (a), derive the predictive probability density function p(y|x) for a future observation Y . [5 marks]
Hint: The integral R ∞−∞ exp{−(s − a) 2/2b 2}ds =√2πb2 for all a ∈ R and b > 0.

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