Exam 2017, questions and answers PDF

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2017 Exam for Quantitative Methods...

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SCHOOL OF MATHEMATICAL AND COMPUTER SCIENCES

Actuarial Mathematics and Statistics

F78QT Quantitative Methods Semester 1

2017/18

SECTION A (64 marks) Students are encouraged to answer ALL questions in this section. It consists of 16 multiple choice questions. Four marks will be awarded for each correct answer. One mark will be deducted for each wrong answer. There is no penalty for failing to answer a question. Answers should be returned on the special sheet provided. SECTION B (36 marks) Students are required to answer TWO of the questions in this section. It consists of 3 questions. Answers should be written in the standard exam answer book.

PROVIDED: Section A Standard exam answer book for rough working Special answer sheet for Section A Formula sheet New Cambridge Statistical Tables Section B Standard exam answer book 2 sheets graph paper

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F78QT

Section A. Students are encouraged to answer ALL questions in this section Questions A1-A3 refer to the following data. The number of guests that can be accommodated in a random sample of 30 Airbnb properties in Brighton is shown below. 1 5 4

4 10 4

2 9 2

1 4 4

5 2 4

2 3 4

2 4 2

12 4 2

6 2 2

4 2 8

A1. The median of these data is: A 3

B 3.5

C

4

D 4.5

E 5

D 14.50

E 22.25

A2. The upper quartile of these data is: A 2.50

B 4.25

C 7.75

A3. The 20% trimmed mean of the data set is: (Reminder: To calculate the x% trimmed mean, the biggest x% and the smallest x% are removed from the sample). A 3.255

B 3.833

C 4.000

D 3.333

E 3.552

Questions A4 - A6 refer to the following data. A study of how people book holiday accommodation classified 329 recent bookings by duration of holiday and by various ways of booking. The results are shown in the table below. Booking Method Standard Website* Airbnb Website Tourist Information Centre

Number of People Booking Accommodation 7 days or more Less than 7 days 85 56 50 75 27 36

*Such as hotels.com or booking.com

A4. The probability that a randomly selected person from the study booked a holiday of 7 days or more on Airbnb is: A 0.245

B 0.306

C 0.168

D 0.231

E 0.152

A5. If 3 people in the survey were chosen at random, without replacement, the probability that two booked any duration of holiday on a standard website and the other booked any duration of holiday at a tourist information centre is: A 0.106

B 0.038

C 0.172

D 0.005 1

E 0.214

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A6. Given that a person chosen at random has booked on Airbnb, the probability that the holiday is for less than 7 days is: A 0.621

B 0.600

C 0.431

D 0.340

E 0.282

Questions A7 – A8 refer to the following data. Some agricultural data for a particular country in appropriate units is shown below. Commodity Wheat Rice Cotton

Price 14 12 8

2016 Quantity 40 25 60

Price 22 18 11

2017 Quantity 60 35 55

A7. Taking 2016 as the base year, which of the following statements is true for 2017: A The Laspeyres price index is 152.4 and the Paasche price index is 146.1 B The Laspeyres price index is 148.5 and the Paasche price index is 146.1 C The Laspeyres price index is 152.4 and the Paasche price index is 150.3 D The Laspeyres price index is 148.5 and the Paasche price index is 150.3 E None of these

A8. Wheat quantities increased by 50% from 2016 to 2017. The percentage increase for the transformed wheat quantities when measured on a log10 scale is: A 13%

B 5%

C 7%

D 9%

E 11%

A9. The average weekly pay for a certain industry in Edinburgh in December 2015 when the RPI was 267.1 was £423.62. If the RPI rises to 280.2 in December 2017 and no pay increases have taken place, the purchasing power of the same pay in December 2017 is: A £413.17

B £452.71

C £401.21

D £443.49

E £403.81

Questions A10 – A11 refer to the following data. A retailer buys 5 boxes of perishable goods at £30 per box and sells them at £40 per box. He can sell 3, 4 or 5 boxes with probabilities 0.2, 0.3 and 0.5 respectively before they become unfit for sale. A10. The variance of the probability distribution for the number of boxes sold is: A 2.33

B 0.61

C 1.52

D 3.18

E 4.25

D £28

E £34

A11. The retailer’s expected profit is: A £8

B £12

C £22 2

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A12. A floor-covering business owner knows there is a probability of 0.6 that she will sell cork flooring to any customer who comes into her shop. For a random sample of 7 people who come into her shop, the probability that more than 2 people will buy a cork floor is: A 0.813

B 0.904

C 0.774

D 0.595

E 0.716

A13. Faults occur at random on lengths of cable at a rate of one every two kilometres. Assuming a Poisson distribution, the probability of 2 or more faults occurring on an 800 metre section is: A 0.041

B 0.062

C 0.083

D 0.104

E 0.125

Questions A14 – A15 refer to the following data. The time taken for an IT call-centre to connect a customer to an expert is normally distributed with mean 4.6 minutes and standard deviation 2.1 minutes. A14. The probability that a customer gets to speak to an expert in less than 3 minutes is: A 0.417

B 0.319

C 0.224

D 0.198

E 0.004

A15. The amount of time that 10% of customers have to wait longer than to speak to an expert is: A 7.3 minutes

B 7.8 minutes

C 8.2 minutes

D 6.1 minutes

E 5.4 minutes

A16. A Linear Programming problem requires the point of intersection of the two equations x + 2y = 11 2x - y = 2 This point is found to be : A (2, 3) B (4,4)

C (5,6)

D (3,4)

3

E (1,5)

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Section B. Students are required to answer TWO of the questions in this section. B1. A door manufacturer makes and sells two types of doors; glass doors and solid doors. Each glass door requires 10 hours of production time and 1 hour of assembly time. Each solid door requires 5 hours of production time and 4 hours of assembly time. Currently there is a maximum of 2000 hours per week available in the production department and 480 hours per week in the assembly department. Each glass door generates £100 profit and each solid door generates £200 profit. The manufacturer aims to maximise weekly profit. You are asked to: (a) Formulate and solve a linear programme to find the optimum production plan for the manufacturer using the existing information. [12 marks] (b) Calculate the profit at your solution. [1 mark] (c) Calculate the shadow price for the assembly time constraint. [5 marks]

[Total 18 marks]

B2. The data below are the scores of 19 randomly chosen candidates in a psychometric test for an accountancy internship. 84 46

92 58

Note:

x

63 94

19 i 1

75 84

81 78

97 40

73 77

69 82

38 88

98

19 i

 1417 and

x

2 i

 111,675 .

i 1

(a) For this sample, calculate the mean, standard deviation and the interquartile range (IQR). Give a brief interpretation of what the IQR measures. [1, 3, 3, 1 marks] (b) Calculate a 95% confidence interval for the population mean based on the sample results. [5 marks] (c) The HR department claims that that the average score for all candidates taking this psychometric test is 85. Carry out a hypothesis test to check whether or not this seems to be the case. [5 marks] [Total 18 marks]

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B3. (a) In a medical experiment concerning 10 patients with a certain type of eye condition, the following measurements were made for blood flow (y) and intraocular pressure (x): x y

10.6 10

9.8 11

8.5 3

For these data  x  107.5 , (i) (ii)

9.4 7

x

10.1 9 2

8.7 5

 1188.29 ,

11.2 8

14.2 17

 y  97 ,  y

2

11.5 14

 1103 ,

13.5 13

 xy  1107 .

Draw a scatter diagram of the medical experiment results [3 marks] Calculate the sample correlation coefficient for the data and, by making reference to the graph, comment briefly on any association between the blood flow and the intraocular pressure. [6 marks]

(b) In a particular university, it is desired to check whether a student’s first language being English is independent of the programme studied. A random sample of 150 students was taken and the results are shown below.

First Language English Other Total

Programme Studied Business Mathematics 48 50 21 31 69 81

Total 98 52 150

Use a chi-squared test to check whether or not there is an association between programme studied and first language being English. Note: The test statistic is

(| O  E | 0.5)2  E

[9 marks] [Total 18 marks]

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