This question is intended to assess your understanding of point estimation.
You should be able to answer this question after working through Unit D1.
(a) The data in Table 4 relate to the classification of 134 recorded crimes(occurring during a month in a certain UK postcode area) into five crime categories.

Table 4    Classification of crimes
 Crime categories    1    2    3    4    5
Observed frequency    25    14    42    11    42

A possible model for these data is the one indexed by a parameter θ, where 0 < θ < 1, with the following probabilities of categories 1,2,3,4,5, respectively:

(i) Show that the likelihood of θ for these data has the form
,
where c is a number and does not involve θ. (You should show how
c is formed, but you do not need to evaluate its value.)
(ii) Ignoring c, the log-likelihood is [4]
.
Use MINITAB to evaluate l(θ) at θ = 0.05,0.10,0.15,... ,0.95.
Give the values of l(θ) in a table, and produce a graph in which
l(θ) is plotted against θ for each of these values.
(iii) Correct to two decimal places, the value of θ that maximizes l(θ) is 0.90. Find θb, the maximum likelihood estimate of θ, correct to three decimal places. Include sufficient detail in your answer to [6]