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Quantitative Methods Examination
L1AF105 202001
β’ The maximum mark for this paper is 100. Use black ink or ballpoint pen.
β’ You should complete ten of the twelve questions only.
β’ All necessary working should be shown, otherwise marks for method may be lost.
β’ The duration of this exam is two hours.
β’ Useful formulae and tables are given at the end of this paper
ILSC Study Skills Incorporated and Tested
Skills
Yes/No Comments
Presentation Skills
yes
Students should present their written work neatly and in
an organized manner.
Self  Directed Study
yes
Students will have needed to study independently to be
prepared for the examination.
Writing Skills
yes
(Accuracy, Coherence)
Analysis and Problem
and accurately.
yes
Solving
Planning Aspects
(Structure, Content
Development)
Students are tutored to present solutions laid out clearly
Students are required to determine the correct statistical
methods to solve the given problems.
yes
Students need to present solutions in a structured, logical
manner.
1). a) Solve the following pairs of simultaneous equations:
i)
3x + 5 y = 6
5 y β x = 12
ii)
3y β 9x β 5 = 0
3x + 9 y = 10
b) Simplify the following:
3
i) π₯(π₯ 2 ) Γ (βπ₯ 4)
3
ii)
( β2 )
π₯
2
π₯
c) Simplify the following:
i) 3πΏππ5 β 4πΏππ3 + 2πΏππ6
4
2
ii) 12 πΏππ216 + 4πΏππ3 β 4 πΏππ36
[10]
2).
a) The probability that Portsmouth will have a good weather is 0.1. A sample of 40 days within a year is
analysed. Find the following, giving your answers for (ii) and (iii) correct to 3 significant figures:
i) The expected number of days with good weather from the sample.
ii) The probability that exactly 10 days will be days with good weather.
iii) The probability that more than 4 days will have good weather.
b)
π₯
π(π = π₯)
i)
ii)
2
4
6
8
10
0.2
0.2
0.4
a
0.05
Find π and π correct to 1 decimal place.
ii) Find π(π β₯ π)
[10]
3). In a call centre, the duration of a sample of phone calls is recorded. The results are shown below:
Length of call,π, (mins)
Frequency
0.5 < π β€ 6.5
5
6.5 < π β€ 10.5
31
10.5 < π β€ 14.5
30
14.5 < π β€ 18.5
37
18.5 < π β€ 20.5
19
a) Calculate estimates, to two decimal places, for:
i) The mean length of call times.
ii) The standard deviation in the length of call times.
iii) The median length of call times.
b) Using your answers to a) calculate Pearsonβs Coefficient of Skewness. Given your answer, would you say
that the mean or the median would be the best measure to represent the length of call times?
[10]
4). A small cafΓ© is selfservice and there is only one till to serve customers. Customers select the items they
want, then join a queue at the till and wait to receive service from the till operator. As there are sometimes
customers waiting during busy periods, an investigation is conducted to see if it is worth installing a second
till. A survey is carried out to find the time between customer arrivals (interarrival time) and how long it
takes for each customer to be served at the till (service time), to the nearest minute. The results are
below:
Interarrival time (nearest minute)
1
2
3
4
5
Probability (%)
15
27
31
21
6
Service time (nearest minute)
2
3
4
5
6
Probability (%)
12
39
26
15
8
a) Perform a simulation for 10 customers, with the assumption that the first customer arrives at time zero.
Copy and complete the 3 tables below to show the simulation:
Interarrival time (nearest minute)
1
2
3
4
5
Probability (%)
15
27
31
21
6
Service time (nearest minute)
2
3
4
5
6
Probability (%)
12
39
26
15
8
Random numbers
Random numbers
Cust RN Interarrival
time
1
0
Customer
Arrives
Start
service
RN Service
time
Service
end
Wait in
queue
Queue
length
(max)
35
2
42
92
3
22
78
4
03
98
5
80
16
6
19
06
7
37
31
8
62
34
9
93
82
10
55
17
b) In your simulation, what is the maximum number of customers waiting and the maximum time spent
queuing? Do you think installing a second till to serve customers is justified? Why/why not?
[10]
5). A study has been carried out to examine the hair length of women within the UK. A large sample of
women was studied, and it was found that the sample was normally distributed with a mean of 30cm and a
standard deviation of 8cm. Give your answers to the following questions correct to 3 decimal places.
a) The participants are selected at random. Find the probability that the womenβs hair is:
i. Longer than 40cm
ii. Shorter than 36 cm
iii. Is between 24 and 32 cm
b) Between what lengths would you expect 68%, 95% and 99.7% of womenβs hair length to be?
[10]
6). Laminate floor boards are to be fitted throughout an office. The precedence table below shows the
tasks involved and their duration.
Activity
A: Remove carpets
Duration (days)
0.75
Immediate Predecessors

B:
C:
D:
E:
Lay polythene
Lay foam
Lay boards
Paint skirting
1
0.9
3
2
A
B
C
A
F: Stain beading
G: Fit beading
H: Fit doorplates
1
1.3
1
D, E, F
D
I: Clean
0.9
G, H
a) Complete an activity network for the project.
b) Find the earliest start time and latest finish time for each activity.
c) Write down the critical activities.
d) Due to workers being ill, Activity E is delayed by 2.5 days and Activity G is delayed by 1 day. What
effect does this have on the completion of the project? Give your reasoning.
[10]
7). GlobalPharmaβs total estimated revenue from the sale of x units of a vaccine is given by
π
(π₯) = π₯ 2 β 5π₯ + 10
Find
a) the average revenue
b) the marginal revenue
c) the marginal revenue at x = 25 units
d) the actual revenue from selling the 51st item.
[10]
8). GlobalPharma sets up a factory to manufacture a vaccine with a fixed cost of Β£150,000. The variable
cost of the vaccine is estimated at 20% of its selling price, when the product sells at the rate of Β£30 per
unit.
a. Find the revenue if 30,000 vaccines are sold.
b. Find the total cost if 30,000 vaccines are sold.
c. Find the breakeven point.
d. If GlobalPharma decreased the selling price to Β£25 per unit, what would the new breakeven point
be?
[10]
9). A small company has two salesmen. The management wants to know if the mean number of sales per
month for these salesmen are different. It is known that the monthly variance in sales for the first
salesman is 1900 and for the second salesman is 12100.
A sample of 12 months for the first salesman gives the following monthly sales numbers:
300
313
350
454
379
300
355
387
368
419
384
488
A sample of 15 months for the second salesman gives the following monthly sales numbers:
499
380
375
490
399
354
427
471
354
280
416
466
327 389
425
Test at the 5% significance level whether the mean daily sales for these two salesmen are different.
You may assume that the sales are normally distributed.
[10]
10). A factory makes two types of car parts, type X and type Y.
Each type X part takes 10 hours to make and each type Y part takes 12 hours to make. In each week there
are 200 hours available to make car parts.
To satisfy customer demand, at least 5 of each type of car part must be made each week.
When completed, the car parts are put into one container for shipping. The volume of the container is 7
m3. A type X car part occupies a volume of 0.5 m3 and a type Y car part occupies a volume of 0.3 m3.
The four inequalities derived from the above information are shown on the graph below, and are
labelled A, B, C and D.
y
A
B
C
D
x
a) If π₯ represents the number of type X car parts produced, and π¦ represents the number of type Y car
parts produced, write the inequalities:
A
B
C
D
b) The profit on each type X car part is Β£100 and on each type Y car part is Β£70.
The weekly profit is to be maximised. Write down the objective function and find the maximum profit.
[10]
11). a) A random sample of 100 students are measured and found to have a mean height of 167.7cm, with
a standard deviation of 7cm. The population can be assumed to be normally distributed.
Find the 95% and 98% confidence intervals for the actual population mean.
b) A sample of monthly rental prices for student accommodation in Portsmouth are shown below.
Assuming that rental prices are normally distributed, find 95% and 99% confidence intervals for the mean
monthly rental cost in the city.
Β£750 Β£755 Β£695 Β£750 Β£770 Β£700 Β£770 Β£760 Β£725 Β£670 Β£780.
[10]
12).
a) GlobalPharma manufactures a vaccine with total cost function given by:
πΆ(π₯) = π₯ 3 β 200π₯ 2 + 10000π₯ + 6000
where x is the number of units produced.
Determine the number of units that must be produced to minimise the total cost.
b) GlobalPharma charges Β£550 per unit for a different product, for an order of 40 units or less. The charge
is reduced by Β£5 for each unit ordered in excess of 40 units. Find the size of the largest order GlobalPharma
should allow, to receive a maximum revenue.
[10]
END OF QUESTIONS
USEFUL FORMULAE AND TABLES
π₯Μ
=
βπ₯
π₯Μ
=
π
β ππ₯
π₯βπ
π‘=π
βπ
β π
β
ππππππβπΏπΆπ΅
Grouped data:
ππΆπ΅βπΏπΆπ΅
=
πππππππ‘πππβπΏπΆπ΅
ππΆπ΅βπΏπΆπ΅
ππππππ πππ ππ‘πππβπΏπΆπ΅ πΆπ’π ππππ
ππΆπ΅ πΆπ’π ππππ βπΏπΆπ΅ πΆπ’π ππππ
=
πππππππ‘πππ πππ ππ‘πππβπΏπΆπ΅ πΆπ’π ππππ
ππΆπ΅ πΆπ’π ππππ βπΏπΆπ΅ πΆπ’π ππππ
ππππππ π£πππππππ: π 2 =
Ungrouped data:
β(π₯βπ₯Μ
)2
πβ1
ππππ’πππ‘πππ ππππππππ: π 2 =
ππππππ π£πππππππ: π 2 =
Grouped data:
ππ’πππ‘πππ πππππππππππ‘ ππ π πππ€πππ π =
πΆππππππππππ‘ ππ π£πππππ‘πππ =
Discrete random variables:
β(π₯βπ)2
π
β π₯2
πβ1
, OR π 2 =
β π(ππππππππ‘βπ₯Μ
)2
πβ1
ππππ’πππ‘πππ ππππππππ: π 2 =
πππππ ππβ² π πππππππππππ‘ ππ π πππ€πππ π =
, OR π 2 =
β
(β π₯)2
π(πβ1)
β π₯2
π
, OR π 2 =
β π(ππππππππ‘βπ)2
π
β π2
β π(ππππππππ‘)2
, OR π 2 =
πβ1
β π(πππππππ‘)2
π
(π3 β π2 ) β (π2 β π1 )
(π3 β π1 )
ππ‘ππππππ πππ£πππ‘πππ
Γ 100
ππππ
ππ΄π
(π) = πΈ(π 2 ) β [πΈ(π)]2
πΈ(π) = β π₯ π(π = π₯)
π₯βπ
π
Confidence intervals:
β π2
3(ππππ β ππππππ)
π π‘ππππππ πππ£πππ‘πππ
π~π΅ππ(π, π): π(π = π) = ππΆπ ππ π πβπ
π~π(π, π 2 ): π§ =
(β ππ₯)2
β π(πβ1)
π = π₯Μ
Β± π§
π
βπ
,
π = π₯Μ
Β± π§
π
βπ
,
π = π₯Μ
Β± π‘
π
βπ
2
2
1
2
π
π
For πΜ
1 β πΜ
2 ~π (0, π1 + π2 ):
π§=
(πΜ
1 βπΜ
2 )β0
2
2
π
π
β 1+ 2
π1 π2
Product moment correlation coefficient:
Least squares regression line:
π=
π¦ = π + ππ₯
β(π₯βπ₯Μ
)(π¦βπ¦Μ
)
β[β(π₯βπ₯Μ
)2 (π¦βπ¦Μ
)2 ]
where π =
OR
β(π₯βπ₯Μ
)(π¦βπ¦Μ
)
β(π₯βπ₯Μ
)2
π=
(β π₯ β π¦)
π
2
(β
(β π¦)2
π₯)
β(β π₯ 2 β
)(β π¦ 2 β
)
π
π
β π₯π¦β
and π = π¦Μ
β ππ₯Μ
P a g e  12
Quantitative Methods L1AF105 Final Examination 202001
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