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Department of Mathematics and Statistics STAT2402: Analysis of Observations

Important: This assignment is assessed, and carries weight 20% towards your final mark

for the STAT2402 unit. Your work for this assignment must be submitted by 12noon on Friday,

25 October 2019.

Attach a properly completed assignment cover sheet (available in the reception area of the

Department of Mathematics and Statistics or on LMS) to the front of your solutions. You may

hand in your work during the computer lab of STAT2402 on 23 October. You may instead

email soft copies of your solutions to my UWA email, and this is the preferred submission

method.

Unless special considerations were granted, any student failing to submit work by the deadline

will receive a penalty for late submission (10% per day late, 0 marks after 7 days). Please

ensure that you write your name and student number on your work.

Plagiarism: You are encouraged to discuss assignments with other students and to solve

problems together. However, the work that you submit must be your sole effort (i.e. not copied

from anyone else). If you are found guilty of plagiarism you may be penalised.

You are reminded of the Faculty of Engineering, Computing and Mathematics’ ‘Policy on

Plagiarism’:

http://www.ecm.uwa.edu.au/students/archived/exams/dishonesty

Task 1. The points on opposite sides of a die add up to seven. Assume that you suspect that

a die is biased towards rolling sixes. One possible model for the outcome X of a roll of this die

would be to use the following probability mass function for X:

x 1 2 3 4 5 6

is unknown.

(a) Assume the die is rolled n = 100 times and we observe y = 32 sixes.

(1) How would you model the outcome of this experiment? Clearly state your statistical

model.

(2) Using your model, find the likelihood function and the log-likelihood function of θ

(given the data). Using the log-likelihood function, also determine the score function.

(3) Find the maximum likelihood estimate ˆθ of θ and its standard error.

(4) Based on your model and the data, what conclusion do you draw about whether the

die is biased or not?

Hint: You may assume that the sampling distribution of your estimator is approximately

normal.

(b) Assume the die is rolled n = 100 times and we observe y = 32 sixes and z = 12 ones.

(1) How would you model the outcome of this experiment? Clearly state your statistical

model.

(2) Using your model, find the likelihood function and the log-likelihood function of θ

(given the data).

(3) Find the maximum likelihood estimate ˆθ of θ and its standard error.

(4) Based on your model and the data, what conclusion do you draw about whether the

die is biased or not?

Hint: You may assume that the sampling distribution of your estimator is approximately

normal.

Semester 2, 2019, page 1 Set 2 Due date: Friday, 2019-10-25, 12:00 noon

Department of Mathematics and Statistics STAT2402: Analysis of Observations

Task 2. A total of 678 women, who got pregnant under planned pregnancies, were asked how

many cycles it took them to get pregnant. The women were classified as smokers and nonsmokers;

it is of interest to compare the association between smoking and probability of pregnancy.

The following table (Weinberg and Gladen, 1986, “The Beta-Geometric Distribution Applied

to Comparative Fecundability Studies”, Biometrics 42(3): 547–560) summarises part of the

data (essentially, women who had used the pill as a contraceptive are excluded).

Observed cycles to pregnancy

Non- NonCycle

Smokers smokers Cycle Smokers smokers

1 29 198 8 5 9

2 16 107 9 1 5

3 17 55 10 1 3

4 4 38 11 1 6

5 3 18 12 3 6

6 9 22 >12 7 12

7 4 7

The data is available on LMS in the file pregnancies.txt. Contact the lecturer immediately

if you have difficulty accessing this data set (and do not want to enter the data yourself into

R).

(a) Fit a geometric model to each group and compare the estimated probability of pregnancy

per cycle.

(b) Is there any evidence that there is an association between smoking and the probability

of pregnancy? Justify your answer.

Task 3. A. Geissler collected data on the distributions of the sexes of children in families in

Saxony during 1876–1885. The data below is the number of girls within the first 12 children

of families with 13 children.

Number of girls in family

0 1 2 3 4 5 6 7 8 9 10 11 12

7 45 181 478 829 1112 1343 1033 670 286 104 24 3

The aim of this analysis is to estimate π, the probability of a girl birth.

(a) How would you model the outcome of this experiment? Clearly state your statistical

model.

(b) Using your model, find the likelihood function and the log-likelihood function of π (given

the data).

(c) Find the maximum likelihood estimate ˆπ of π and its standard error.

(d) Does your model use any assumptions that might open it to criticism? Include any

evidence of model inadequacy in a brief discussion.

(e) As stated above, these data are on families with 13 children and only the sex distribution

among the first 12 children is reported.

What may be the reason for omitting the information on the last child? Discuss briefly.

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