BU52018 Page 1 of 7

Module – BU52018

Level - 5

April/May 2020

2 HOURS

ANSWER QUESTION 1 IN SECTION A AND
THREE QUESTIONS IN SECTION B

ALL QUESTIONS CARRY EQUAL MARKS

The use of approved calculators (FX83, FX85 or FX 115) is
permitted in this examination

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SECTION A

ANSWER ALL PARTS OF THE QUESTION IN THIS SECTION

1. A travel agency would like to forecast the number of UK citizen visits abroad so that they can
develop a growth plan for their business. The agency’s statistician developed the following
model, to be estimated using quarterly data for the past 10 years (40 observations):

ܸܫܵܫܶܵ௧ ൌ ߚଵ ൅ ߚଶܴܶܧܰܦ௧ ൅ ߚଷܩܦܲ_ܲܥ௧ ൅ ߚସܳ2௧ ൅ ߚହܳ3௧ ൅ ߚ଺2ܳ4௧ ൅ ߝ௧
where:
• ܸܫܵܫܶܵ௧ is the number of UK citizen visits abroad during period ݐ, measured in millions,
• ܴܶܧܰܦ௧ is an index indicating the period to which the data point corresponds (1, 2, 3,
… 40),
• ܩܦܲ_ܲܥ௧ is the Gross Domestic Product (GDP) per capita in the UK for period ݐ,
• ܳ2௧ is a dummy variable equal to one if the observation corresponds to quarter 2 (April-
May-June),
• ܳ3௧ is a dummy variable equal to one if the observation corresponds to quarter 3 (July-
August-September), and
• ܳ4௧ is a dummy variable equal to one if the observation corresponds to quarter 4
(October-November-December).

The results of the model estimation are presented in the following table.

Dependent Variable: VISITS
Method: Least Squares
Date: --/--/-- Time: --:--
Sample: 1 40
Included observations: 40
Variable Coefficient Std. Error t-Statistic Prob.
C 95.19166 45.45449 2.094219 0.0438
TREND 0.503173 0.152674 3.295738 0.0023
GDP_PC -2.077089 1.158584 -1.792783 0.0819
Q2 1.650664 0.576868 2.861422 0.0072
Q3 2.674084 0.578551 4.622037 0.0001
Q4 0.192040 0.579254 0.331531 0.7423
R-squared 0.859137 Mean dependent var 19.58068
Adjusted R-squared 0.838421 S.D. dependent var 3.207040
S.E. of regression 1.289129 Akaike info criterion 3.483291
Sum squared resid 56.50300 Schwarz criterion 3.736623
Log likelihood -63.66583 Hannan-Quinn criter. 3.574888
F-statistic 41.47372 Durbin-Watson stat 0.915586
Prob(F-statistic) 0.000000

(a) Report the estimated model in standard form, including standard errors.
(10)

(b) Interpret the estimates of all parameters (including the constant term), paying special
attention to the units of measurement of the dependent and independent variables.
(25)

(c) Interpret the number “0.0819” that appears under the “Prob.” column and on the row
corresponding to the “GDP_PC” variable on the table above.
(10)
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(d) Interpret the F-statistic of the model (clearly state the null and alternative hypotheses
and whether you reject the null or not and at what level of significance).
(15)

(e) Interpret the R2 of the model.
(10)

(f) Test at the 5% significance level the hypothesis that the parameter associated with the
“Q4” variable is greater than 1.0.
(15)

(g) The residuals from the model are plotted against the time index in the figure below.
What deviation(s) from the Gauss-Markov assumptions do you identify from this plot?
What are the consequences of this deviation for the Ordinary Least Squares estimator,
in general?
(15)

-3
-2
-1
0
1
2
3
5 10 15 20 25 30 35 40

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SECTION B

ANSWER THREE QUESTIONS IN THIS SECTION

2. SeaShellsByTheSeaShore Investments is a small retirement fund that usually invests in low-
risk, low-tech stocks. However, the fund’s CEO recently learned about a new start-up, high-
tech company with great potential for profitability, but also high risk:

 the probability of such a start-up surviving the first year,  SP , is only 10%
 the probability of such a start-up failing in the first year,  SP , is 90%

If the start-up survives the first year then it will turn highly profitable.

Because SeaShellsByTheSeaShore Investments’ analysts are unfamiliar with the hi-tech
market, the CEO decides to hire an external consultant, HappyDay Analysts, to predict whether
the start-up company will survive the first year. HappyDay Analysts are not always correct in
their predictions: they are known to be overly optimistic about the prospects of start-ups. In their
website they provide the following data:
 “out of 100 start-ups that survived we predicted correctly on 80!”
 “out of 900 start-ups that failed we predicted correctly on 540!”

Do the following, showing the steps of your derivations:

(a) Calculate the probability of HappyDay Analysts predicting that a start-up will survive
given that it survived –  SPSP .
(10)

(b) Calculate the probability of HappyDay Analysts predicting that a start-up will fail given
that it failed –  SPSP .
(10)

(c) Use the law of total probability to calculate the probability of HappyDay Analysts
predicting that a start-up will survive –  PSP .
(20)

(d) Use Bayes’ theorem to calculate the probability of a start-up surviving given a prediction
from HappyDay Analysts that it will survive –  PSSP .
(20)

(e) Calculate the probability of the start-up failing given a prediction from HappyDay
Analysts that it will survive –  PSSP .
(10)

(f) Calculate the probability of HappyDay Analysts predicting that the start-up will fail – PSP .
(10)

(g) Calculate the probability of the start-up surviving given a prediction from HappyDay
Analysts that it will fail –  PSSP .
(20)

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3. A health insurance company offers a policy according to which it charges an annual fee
(premium) to a customer (policy holder) and, in return, it covers all costs resulting from visits of
the customer to hospitals or medical doctors during the year. The current level of the premium
is £150 and the cost per visit to a hospital/doctor is £40.

The company’s manager is currently reconsidering the value of the premium because he is
concerned that the expected profit (revenue from the premium minus expected cost from visits
to hospitals/doctors) from this policy may be negative. To make an informed decision, the
manager collects data on the number of visits to hospitals/doctors during the previous year from
100 policy holders. The data appear on the following table.

# of visits to
hospitals/doctors
in the past year
# of policy
holders with
this many visits
0 5
1 19
2 26
3 22
4 13
5 8
6 4
7 2
8 1
Total 100

Please do the following while using appropriate notation and showing your workings.

(a) Calculate the average number of visits to hospitals/doctors per year by a policy holder.
(20)

The insurance company’s manager decides to model the number of visits to hospitals/doctors
per policy holder as a random variable which follows a Poisson distribution with parameter, μ,
equal to the number you identified in (a).

(b) What is the probability of a policy holder visiting a hospital/doctor no more than 2 times
in a year if the number of visits follows a Poisson distribution?
(15)

(c) Assuming that the number of visits to hospitals/doctors per year does indeed follow a
Poisson distribution, what is the probability that a policy holder will visit a hospital/doctor
more than 6 times in a year?
(15)

(d) Assuming that the number of visits to hospitals/doctors per year does indeed follow a
Poisson distribution, what is the probability that a policy holder will visit a hospital/doctor
between 2 and 6 times in a year (margins included)?
(20)

(e) What is the probability that the insurance company makes a loss from a given policy
(15)

(f) What is the expected profit per customer? Please define random variables explicitly
(15)

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4. A large corporation frequently hires data scientists to expand its operations, as well as to fill
posts that become vacant from their own data scientists moving to other companies. For that
reason they need to have good knowledge of the average salary of a data scientist in their
industry. Until now the hiring manager of this corporation has been operating under the
assumption that the average salary of a data scientist is £50,000 and she has been making
offers to job candidates at that rate. However, due to recent trends in the industry, she suspects
that the mean salary has gone up. To examine this possibility she collected data on 49 newly-
hired data scientists and she found that the average salary offered to them was £54,000, with
a standard deviation of £9,000 (sample variance of £281,000,000)

(a) Set up appropriate null and alternative hypotheses to test the claim that the mean
salary of a data scientist in the industry is above £50,000.
(20)

(b) Test the hypothesis that you identified in (a) at the 5% level of significance. Should the
hiring manager alter the salary she has been offering to job applicants?
(30)

Until now the hiring manager has been operating under the assumption that the standard
deviation of data-scientist salaries is £10,000, but given the new data she obtained, she
suspects that the variability in salaries has gone down.

(c) Set up appropriate null and alternative hypotheses to test the claim that the standard
deviation of data-scientist salaries is greater than £10,000 (variance greater than
£2100,000,000).
(20)

(d) Test the hypothesis that you identified in (c) at the 5% level of significance. Can it be
concluded that the variability in salaries is below the £10,000 threshold used until now?
(30)

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5. A class of 12 university students takes a test on Business Analytics. The invigilator assigned to
the class recently read an article which stated that 15% of all people are left handed and he
wants to evaluate whether this claim has any credibility. Let X be the number of students in the
class who are left handed.

(a) What distribution does X follow if 15% of all people are left handed? State the name of
the distribution and the value(s) of its parameter(s)?
(15)

(b) What is the probability of exactly 3 students being left handed if 15% of all people are
left handed?
(10)

(c) What is the probability of no more than 3 students being left handed if 15% of all people
are left handed?
(10)

(d) What is the probability of 4 or more students being left handed if 15% of all people are
left handed?
(15)

(e) What is the probability of 2 to 7 students being left handed (margins included) if 15% of
all people are left handed?
(20)

(f) If 15% of all people are left handed, what is the expected number of left-handed
students in the class?
(15)

(g) If 15% of all people are left handed, what is the expected number of right-handed
students in the class? Please define random variables explicitly and show your working.
(Here you can assume that people are either left or right handed, not both).
(15)

(End of Paper)