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https:
help.open.ac.uk/documents/policies/code-of-practice-student-discipline/files/15/student-discipline.pdf
Question 1, which covers topics in Unit 7, and Question 2, which covers
topics in Unit 8, form M248 TMA 04. Question 1 is marked out of 23;
Question 2 is marked out of 27.
Question 1 – 23 marks
You should be able to answer this question after working through Unit 7.
(a) Let X and Y be independent random variables both with the same
mean µ ̸= 0. Define a new random variable W = aX + bY , where a and
are constants.
(i) Obtain an expression for E(W ). [2]
(ii) What constraint is there on the values of a and b so that W is an
unbiased estimator of µ? Hence write all unbiased versions of W as
a formula involving a, X and Y only (and not b). [3]
(b) An otherwise fair six-sided die has been tampered with in an attempt to
cheat at a dice game. The effect is that the 1 and 6 faces have a
different probability of occu
ing than the 2, 3, 4 and 5 faces.
Let θ be the probability of obtaining a 1 on this biased die. Then the
outcomes of rolling the biased die have the following probability mass
function.
Table 1 The p.m.f. of outcomes of rolls of a biased die
Outcome XXXXXXXXXX
Probability θ 14(1− 2θ)
1
4(1− 2θ)
1
4(1− 2θ)
1
4(1− 2θ) θ
(i) By consideration of the p.m.f. in Table 1, explain why it is
necessary for θ to be such that 0 < θ < 1/2. [2]
(ii) The value of θ is unknown. Data from which to estimate the value
of θ were obtained by rolling the biased die 1000 times. The result
of this experiment is shown in Table 2.
Table 2 Outcomes of 1000 independent rolls of a biased die
Outcome XXXXXXXXXX
Frequency XXXXXXXXXX 190
Show that the likelihood of θ based on these data is
L(θ) = C θ395 (1− 2θ)605,
where C is a positive constant, not dependent on θ. [5]
(iii) Show that
L′(θ) = C θ394(1− 2θ XXXXXXXXXX− 2000 θ). [4]
(iv) What is the value of the maximum likelihood estimate, θ̂, of θ
ased on these data? Justify your answer. What does the value of
θ̂ suggest about the value of θ for this biased die compared with
the value of θ associated with a fair, unbiased, die? [4]
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(c) Studies of the size and range of wild animal populations often involve
tagging observed individual animals and recording how many times each
is caught in a trap (from which it is then released back into the wild).
The dataset presented in Table 3 consists of the numbers of times each
of n = 334 wood mice were caught in a particular trap (over a two-yea
time period). The data are also provided in the Minitab file
wood-mice.mwx.
Table 3 Numbers of trappings of wood mice
Times trapped XXXXXXXXXX
Frequency XXXXXXXXXX19 12 9
Times trapped XXXXXXXXXX XXXXXXXXXX
Frequency XXXXXXXXXX
The geometric distribution with parameter p is a good model for these
data.
(i) What is the maximum likelihood estimator of p for a geometric
model? [1]
(ii) What is the maximum likelihood estimate of p for the data in
Table 3? You are recommended to use Minitab to help you to
answer this part of the question. [2]
Question 2 – 27 marks
You should be able to answer this question after working through Unit 8.
(a) In this part of the question, you should calculate the required
confidence interval by hand, using tables, and show your working. (You
may use Minitab to check your answers, if you wish.)
Modern aircraft cockpit windscreens are complex items, comprising
several layers of material and a heating system. Such windscreens are
eplaced upon damage to any of their components. A dataset was
collected on the times to replacement of n = 84 windscreens of a
particular modern airliner. The sample mean windscreen replacement
time was XXXXXXXXXXhours of flight. The sample standard deviation of
windscreen replacement times was 5168 hours of flight.
(i) Obtain an approximate 90% confidence interval for the mean
eplacement time of this type of aircraft windscreen. What
property of the dataset justifies using this type of confidence
interval, and why? [6]
(ii) Interpret the particular confidence interval that you found in
part (a)(i) in terms of repeated experiments. [3]
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(b) In this part of the question, you should calculate the required
confidence interval by hand, using tables, and show your working. (You
may use Minitab to check your answers, if you wish.)
In a large study of patients who were being treated for hypertension
(high blood pressure), 148 out of 5493 patients receiving the
conventional treatment for hypertension later suffered a stroke. Also,
192 out of 5492 patients receiving an alternative drug to treat thei
hypertension later suffered a stroke.
(i) Obtain an approximate 95% confidence interval for the difference
in proportions between the number of conventionally treated
hypertension patients who later suffered a stroke and the numbe
of hypertension patients treated with the alternative drug who
later suffered a stroke. (You are advised to work with proportions
ounded to four decimal places throughout; also, you may assume
that the numbers involved are large enough that you
approximation is a good one.) [5]
(ii) Some clinicians had suggested that the proportions of hypertension
patients who suffered a stroke would not depend on which
treatment they were being given. Are the data consistent with that
suggestion? Justify your answer
iefly. [2]
(c) In various places in this module, data on the silver content of coins
minted in the reign of the twelfth-century Byzantine king Manuel I
Comnenus have been considered. The full dataset is in the Minitab file
coins.mwx. The dataset includes, among others, the values of the
silver content of nine coins from the first coinage (variable Coin1) and
seven from the fourth coinage (variable Coin4) which was produced a
number of years later. (For the purposes of this question, you can
ignore the variables Coin2 and Coin3.) In particular, in Activity 8 and
Exercise 2 of Computer Book B, it was argued that the silver contents
in both the first and the fourth coinages can be assumed to be normally
distributed. The question of interest is whether there were differences in
the silver content of coins minted early and late in Manuel’s reign. You
are about to investigate this question using a two-sample t-interval.
(i) Using Minitab, find either the sample standard deviations of the
two variables Coin1 and Coin4, or their sample variances. Hence
check for equality of variances using the rule of thumb given in
Subsection 4.4 of Unit 8. [3]
(ii) Whatever the outcome of part (c)(i), use Minitab to obtain a 90%
two-sample t-interval for the difference E(X1)− E(X4), where X1
denotes the silver content in coins of the first coinage, and X4
denotes the silver content in coins of the fourth coinage. State that
interval and comment
iefly on what it tells us about the silve
content of coins in the earlier and later coinages. [3]
(iii) Name the distribution used in constructing the confidence interval
in part (c)(ii), state the value of its parameter, and show why the
parameter takes the value that it does. [2]
(iv) What would have been the outcome if you had obtained a 90%
two-sample t-interval for E(X4)− E(X1) instead of fo
E(X1)− E(X4)? Justify your conclusion in terms of the derivative
of the parameter transformation involved. [3]
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