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 dead- line 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 123456 Pr[X=x] 1−θ 1 1 1 1 1+θ
where 0 ≤ θ ≤ 1 is unknown. 6
(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 θˆ 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 approx- imately 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 θˆ 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 approx- imately normal.
6 66666
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 nonsmok- ers; 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- Cycle Smokers smokers
129198 216107 31755 4438 5318 6 9 22 747
Non- Cycle Smokers smokers
859
915 10 1 3 11 1 6 12 3 6
>12 7 12
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 πˆ 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.
Semester 2, 2019, page 2 Set 2 Due date: Friday, 2019-10-25, 12:00 noon