Download MSc Regulation and Competition

MSc Regulation and Competition
Quantitative Techniques 1
Exam questions for January 2004
Time allowed: two hours
Attempt ALL questions. The points for each answer are shown in square brackets [like
this]
1.
What are the main sources of data errors? What can be done to reduce errors in
data you did not collect yourself?
[10]
2.
B:
For each of the following statements about the probabilities of outcomes A and
(a) Say whether they are true, false, or uncertain
(b) If uncertain, spell out conditions under which they are true
(i)
(ii)
(iii)
(iv)
(v)
P(A or B) = P(A) + P(B)
P(AB) = P(A) + P(B)
P(AB)=P(A).P(B)
P(AB) = P(A).P(B)
P(A | B)= P (AB)/P(B)
[3]
[3]
[3]
[3]
[4]
3.
Suppose x is a continuous random variable with the probability density function
(pdf):
f(x)= x
for
0x1
2 - 2x for
1 x  2
0
elsewhere
NB This question does not work: see answers file!
a) Draw a graph of this function
[4]
b) Explain how you know this is a valid pdf.
[4]
c) Comment on the relative position of the mean, median and mode.
[4]
d) Calculate the probability that
0.5  x  1.5
[4]
1
4.
You are organising a concert and believe that attendance will depend on the
weather. You believe the following possibilities are appropriate:
Weather
Terrible weather
Mediocre
weather
Good weather
Probability
0.2
Attendance
500
0.6
0.2
1200
2000
a) What is the expected attendance?
[4]
b) Suppose each ticket costs £5 and the fixed costs are £2,000. What are the
expected profits?
[4]
c) Graph the probability distribution for profits
[4]
d) What is the most you could pay for the fixed costs and still have an 80% chance
of making a profit on the event? (to nearest £)
[4]
5. Suppose that heights in a population are normally distributed with a mean of 78
inches and a standard deviation of 5 inches.
a) What is the probability that an individual selected at random will have a height
between 68.2 and 79.8 inches?
[6]
b) Construct a 95% confidence interval for the average height in a random sample of
four individuals.
[6]
6. Suppose we wanted to conduct a survey. It is desired that we produce an interval
estimate of the population mean that is within 5 from the true population mean with
99% confidence. Based on a historical planning value of 15 for the population
standard deviation, how big should the sample be?
[6]
7. Explain in simple terms the differences (and similarities if any) between the following
approaches to estimation;
a) method of moments
[4]
b) maximum likelihood
[4]
c) least squares
[4]
8. Under what conditions will the ordinary least squares estimator be
a) unbiased
b) efficient?
[4]
[4]
c) What does it mean to say an estimate is consistent?
[4]
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