Calculate the Posterior Mean, Mode and Median of the Model - Statistics Assignment Help

Assignment Task:

Task:

QUESTION 1:
Suppose you are given the following information: (x1,...,xn) ~ iid N(μ ,1/Λ) Λ ~ Gamma (α,β) and y = (6.5, 10, 8.5), μ=6,α=2,β=1
a) Using R, calculate the posterior mean, mode and median of the model precision Λ. [3 marks]
b) Produce a graph in R showing the estimates in part (a) and the prior density, prior mean, likelihood and posterior density. [4 marks]
c) Using R, calculate the posterior mean, mode and median of the model variance σ2=1/Λ. [3 marks]
d) Produce a graph in R showing the estimates in part (c) and the prior density, prior mean, likelihood and posterior density. [4 marks]
e) Using R, calculate the posterior mean, mode and median of the model standard deviation σ. [3 marks]
f) Produce a graph in R showing the estimates in part (e) and the prior density, prior mean, likelihood and posterior density. [3 marks]
g) Write a brief report on your findings for parts (a), (c) and (e). [5 marks]
[Total marks: 25]

QUESTION 2:
Suppose a person is interested in learning about the exercise habits of university students in the UK . He hears that doctors recommend at least one hour of physical activity for an average adult per day. What proportion of the university students get at least one hour of physical activity?

A random sample of students from a particular university is taken to learn about the proportion. But the researcher does some initial investigation to learn about the exercise habits of university students. This research will help in constructing a prior distribution. After the investigation, he believes that university students generally get less than an hour of activity and so p (the proportion that activity at least an hour) is likely smaller than 0.5. After some reflection, his best guess at the value of p is 0.3. But it is very plausible that this proportion could be any value in the interval from 0 to 0.5.

A sample of 50 students is taken. In this group, 20 record that they had at least an hour of physical activity the previous day. Based on the prior information, and these observed data, the researcher is interested in estimating the proportion of p. He also believes that the proportion is equally likely to be smaller or larger than p = 0.3 and 90% confident that p is less than 0.5. His beliefs on the median and 90th percentiles of the proportion are 0.3 and 0.5 respectively.

a) By assuming a prior distribution for p, plot the prior, likelihood and posterior densities on the same graph using R and comment on your finding. [8 marks]
b) Use R, to find the probability that the proportion of students doing physical activity is greater than 0.5? Comment on your result. [4 marks]
c) Calculate a 90% interval estimate for p using R. [2 marks]

d) Using R and based on your posterior density, carry out 1000 simulation, display your posterior as a histogram of the simulated values. Calculate a 90% interval estimate for p and compare this estimate with part (c). [6 marks]

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