Download Junior Sophisters Monetary and Welfare Economics

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UNIVERSITY OF DUBLIN
TRINITY COLLEGE
FACULTY OF SOCIAL AND HUMAN SCIENCES
DEPARTMENT OF ECONOMICS
Junior Sophister/ Senior Sopister
Hilary Term 2008
Faculties
EXAM TITLE
Mathematics and Statistics
DATE
VENUE
TIME
LECTURER
Dr. J. Thijssen
EXAM INSTRUCTIONS
Please answer all questions.
Please motivate all answers. Failure to do so may lead to a deduction of points.
All questions carry equal weight.
Materials Permitted for this Examination
Standard non-programmable calculator.
You may not start this examination until you are instructed to do so by the
Invigilator.
© UNIVERSITY OF DUBLIN 2008
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1. A sample of ten people have followed a particular diet as advertised in a well-known
day-time TV chat show. The weight loss (in kilos) for each person is given below:
7
3
7
3
5
3
7
5
5
3.
(a) Compute the average weight loss.
(b) Compute the standard variation of the weight loss.
(c) Draw a bar chart of the relative frequencies.
2. It is known that 20% of all farms in a particular county exceed 160 acres and that
60% of all farms in that county are owned by persons over 50 years old. Of all farms in
the county exceeding 160 acres, 55% are owned by persons over 50 years old.
(a) What is the probability that a randomly chosen farm in this county both exceeds
160 acres and is owned by a person over 50 years old?
(b) What is the probability that a farm in this county either is bigger than 160 acres
or is owned by a person older than 50 years old (or both)?
(c) What is the probability that a farm in this county, owned by a person older than
50 years old, exceeds 160 acres?
(d) Are size of farm and age of owner in this county independent?
3. It is estimated that 45% of graduates from a particular university graduate with a
II.1 or a I (in the UK these are called “good degrees”). Let X denote the number of
graduates with a “good degree” from a random sample of 7 students.
(a) What distribution does X follow?
(b) Compute the probability that at most two students graduate with a good
degree.
In a particular year, 123 students graduate from the university.
(c) Find the mean and standard deviation of the number of graduates with a
good degree.
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(d) Use a normal approximation to compute the probability that a majority of
these 123 graduates obtain a good degree.
4. Betty’s is the most famous tearoom in York and they sell tea in boxes of (nominally)
100 grams. Because of EU legislation they take extra care and wrap, on average,
101.0 grams in each box. Assume that the weight of the boxes is normally distributed
with a standard deviation of 0.6 grams.
(a) What is the probability that a box does not contain enough tea (round to two
decimals)?
(b) Betty’s want to reduce the probability found in (a) by half by reducing the
standard deviation. This will be achieved by making the wrapping machine
more precise. What standard deviation should Betty’s aim for to achieve its
goal?
5. Suppose you are working for a union and want to raise awareness among politicians
about the inequality in Irish wages. As a first step you issue a report to get some idea
about the difference between the average wage of an Irish worker and the country’s
top salaries. You get the following report.
“A random sample of Irish workers has been obtained by asking 20 Dublin Bus drivers
outside Ringsend terminal about their wages. We found an average gross wage of
22,350 Euros, with a standard deviation of 3,200 Euros. In order to obtain some
feeling about the accurateness of this average, we computed a 95% confidence
interval around the mean. We do not have evidence on the exact shape of the income
distribution, but the Central Limit Theorem tells us that the sampling distribution of the
mean
is
approximately
normal
with
mean
μ
and
standard
deviation
[(3,200)2/20]1/2=715.54. Using the formula μ±zα/2s/√n, we find a 95% confidence
interval of [22350-1.64*715.54,22350+1.64*715.54]=[21176.51,23523.49]. So, 95%
of Irish workers will earn between 21,177 and 23,523 Euros.” Comment on this report.
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