Assignment 2 MAS183 Statistical Data Analysis

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MAS183 Statistical Data Analysis
Semester 2, 2020
Assignment 2 – Due 11:00 pm Wednesday 16 September 2020
This assignment covers material in chapters 5-9 of the Unit Notes. R is not required in this
assignment, but may be used if you wish. If software is used, attach relevant output to support your
answers. No data files are required.
Total marks = 42.
1. [9 marks]
A new diagnostic test has been proposed for the SARS-COV-2 virus, which causes the COVID-19
illness. Clinical trials indicate that the test detects SARS-COV-2 in 96% of people who are
actually infected by the virus. Among people who do not have an infection, the test returns a
negative result in 92% of cases. We are considering using the test in a population where the
prevalence of SAR-COV-2 is estimated to be 1%.
(a) Structure the above information using a contingency table. Take care to complete all
marginal totals as well as the body of the table. [3]
(b) What proportion of people that get a positive test result would actually have a SARS-COV-2
infection? [2]
(c) Consider those who get a negative test result. If we chose one of these people at random,
what is the chance that the selected person doesn’t have SARS-COV-2? [2]
(d) In this population, what does the test do better: show who has SARS-COV-2, or show who
doesn’t have it? Justify your answer. [2]
2. [3 marks]
Calculate –
(a) the mean, and [1]
(b) the standard deviation [2]
of the random variable Y, which has the following probability distribution:
Y 1 2 3
P( Y=y) 0.22 0.63 0.15
3. [3 marks]
A forest is surveyed in order to estimate the percentage of trees affected by a soil-borne disease
which we will call “Disease Q”. The forest is divided into many thousands of 10m × 10m
squares. One hundred of these squares are randomly selected, using the Random Digits table
on p. 32 of the Tables and Formulae book. In each selected square, every tree is examined to
see whether or not it has Disease Q, and the number of diseased trees in each square is
recorded. If X is the number of diseased trees in a square –
(a) Does X follow a binomial distribution? [1]
(b) Justify your answer. [2]
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4. [12 marks]
In Australia, 10% of the population has blood type B. Consider X, the number having type B
blood among 25 randomly-selected Australians.
(a) What is the probability distribution of X? [3]
(b) Calculate:
(i) the mean and standard deviation of X. [3]
(ii) P( X ≥ 4) [2]
(iii) P( 3 ≤ X < 9) [2]
(c) If, in such a sample, you found 5 people with blood type B, would this be an unusual
sample? Justify your answer using an appropriate probability calculation. [2]
5. [8 marks]
The weights of a population of 16-year old girls are approximately normally distributed with
mean 60.4 kg and standard deviation 6.2 kg.
(a) What proportion of these girls weigh between 52 kg and 62 kg? [3]
(b) What is the chance that one of these girls, chosen at random, would weigh over 55 kg? [2]
(c) What are the lower and upper limits of the middle 80% of weights in this population? [3]
6. [7 marks]
An agricultural researcher is investigating hoof diseases in cattle within a particular agricultural
region. In order to prioritise treatment strategies, the researcher wants to obtain a reliable
indication of the prevalence of various hoof conditions among the region’s cattle. Cattle in the
region are held on 247 properties of varying size, which have a range of soil and vegetation
types and varying levels of water access. The distribution of the number of cattle per property
is positively skewed (i.e., some big herds, but most are smaller). All assessments of hoof
condition are to be made on site (i.e., where the cattle usually live) by the researcher or a
trained assistant. Funds are available to assess hoof condition on about 500 cattle.
The assistant has suggested the following sampling method: Randomly select 50 properties by
means of computer-generated random numbers. The number of cattle assessed on each
selected property will be in proportion to the number of cattle on the property, and so that the
total sample size will be 500. Assess hoof conditions during the annual drenching program on
each property. On each property, select every kth animal for assessment as it goes through the
drenching chute, where k is chosen to give the correct sample size for the property.
For the suggested sampling method:
(a) Identify any elements of simple random sampling, convenience sampling, cluster sampling,
systematic sampling, or stratified sampling. [3]
(b) Briefly assess how well the method will meet the researcher’s objective. [2]
(c) Briefly describe the procedure’s practical advantages and disadvantages compared with
using strict simple random sampling. [2]
END OF ASSIGNMENT

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