## Transcribed Text

3. (a) A random sample {X1, X2, , Xn} is drawn from the following probability
distribution:
f(x;0) = 0 forAi. Derive the maximum likelihood estimator of 0 and explain why this is a
maximum.
(6 marks)
ii. If n = 4, such that T1 = 0.70, x2 = 0.92, x3 = 0.21 and x4 = 0.34,
determine the maximum likelihood estimate of 0.
(2 marks)
iii. As n
00, would you expect the maximum likelihood estimate of 0 to get
closer to 0? Briefly explain your answer.
(2 marks)
(b) A random sample of size n = 9 drawn from a normal distribution had a sample
variance of s2 = 72.76. Construct a 90% confidence interval for 02.
(5 marks)
(c) Let X be a random variable such that X ~ Exp() Suppose we wish to test
the null hypothesis H0 : l = 2 VS. H1 : l > 2. Consider the test procedure
which rejects H0 if X < 0.75.
i. Compute the probability of a Type I error, expressing your answer in terms
of the exponential constant e.
(2 marks)
ii. Determine the power function of the test, expressing your answer in terms
of the exponential constant e.
(3 marks)
4. (a) Suppose that you are given independent observations y1, y and Y such that:
y1=3a+B+E1 =
y2=a-28+82 -
y = =-a+2B+E3.
The random variables Ei, for i = 1,2,3, are normally distributed with a mean
of 0 and a variance of 4.
i. Find the least squares estimators of the parameters a and B, and verify
that they are unbiased estimators.
(7 marks)
ii. Calculate the variance of the estimator of a.
(3 marks)
(b) A supermarket chain investigated its profit margins over a period of 4 years.
It splits its merchandise into 6 categories and recorded the profit margin for
each category for each year. The investigators constructed an ANOVA table,
which is given below (some entries are missing).
Source
DF
SS
MS
F
Category
62.10
Year
3.31
Error
3.14
Total
i. Complete the table in your answer booklet.
(6 marks)
ii. Is there a significant difference between the profit margins of different
categories? Use a 5% significance level.
(4 marks)

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