F1 Business Maths Aug 06

BUSINESS MATHEMATICS & QUANTITATIVE METHODS

FORMATION 1 EXAMINATION - AUGUST 2006 NOTES

Answer 5 questions. (Only the first 5 questions answered will be marked). All questions carry equal marks.

STATISTICAL FORMULAE TABLES ARE PROVIDED DEPARTMENT OF EDUCATION MATHEMATICS TABLES ARE AVAILABLE ON REQUEST

TIME ALLOWED:

3 hours, plus 10 minutes to read the paper.

INSTRUCTIONS:

During the reading time you may write notes on the examination paper but you may not commence writing in your answer book. Marks for each question are shown. The pass mark required is 50% in total over the whole paper. Start your answer to each question on a new page.

You are reminded that candidates are expected to pay particular attention to their communication skills and care must be taken regarding the format and literacy of the solutions. The marking system will take into account the content of the candidates' answers and the extent to which answers are supported with relevant legislation, case law or examples where appropriate.

The Institute of Certified Public Accountants in Ireland, 9 Ely Place, Dublin 2.

THE INSTITUTE OF CERTIFIED PUBLIC ACCOUNTANTS IN IRELAND

BUSINESS MATHEMATICS & QUANTITATIVE METHODS FORMATION 1 EXAMINATION - AUGUST 2006. Time Allowed: 3 hours, plus 10 minutes to read the paper

1.

Answer 5 questions Only the first five questions answered will be marked. All questions carry equal marks.

As a financial consultant you are providing advice to an investor on the following proposal. He wishes to invest €25,000 in a managed fund for 5 years with an expected rate of return of 5%. If he leaves the investment, or any part of it, in a particular plan for a further 4 years the rate of return will increase to 7%. He wants, however, the option to withdraw half the original sum of €25,000 after the 5 year period. You are required to advise the investor on (i)

The total value of the investment after the 9 year period if nothing is withdrawn (8 Marks)

(ii)

The total value of the investment if half the money is withdrawn after the 5 year period (8 Marks)

(iii)

Write a note on the ‘time value of money’

(4 Marks) [Total: 20 Marks]

2.

Trainees for DIB Manufacturing Co. Ltd are paid a salary during training based on both their experience and stage of training. The number of trainees and the allowances paid are set out in the following table. Payment €

Number of Trainees

100 and less 110

1

110 and less 120

4

120 and less 130

7

130 and less 140

13

140 and less 150

7

150 and less 160

3

160 or more

1

You are required to (i) (ii) (iii)

Present the data on a cumulative frequency graph Estimate the median allowance from the graph Compare the median with the mean and modal allowances.

(8 Marks) (4 Marks) (8 Marks) [Total: 20 Marks]

1

3.

The SECO hotel, a small family owned enterprise, is attempting to estimate the expected daily profit from the business. Although the hotel has a prestigious restaurant the daily profit depends on room occupancy. As the adviser to the hotel you have estimated the probability of the number of guests arriving each night. The data is set out in the following table. No guests

1

2

3

4

5

Probability

0.1

0.2

0.4

0.2

0.1

Profit / room €

35

75

125

175

225

As part of your report to the hotel owner you are asked to include details on the following. (i)

Explain how the expected profit is measured and what it represents ?

(8 Marks)

(ii)

Calculate the expected daily profit from the data provided

(8 Marks)

(iii)

Write a note on probability and ‘expectation’

(4 Marks)

[Total: 20 Marks]

4.

On the international commodities market the movement in the price of particular foods will have a major impact on local prices to consumers in Ireland. The following data from the EU Bureau of Statistics indicates the trend in prices for three high consumption products over the past 3 years. Price (€ per 1000 kgs) Commodity

2003

2004

2005

Coffee

20

25

35

Bananas

12

14

18

Tea

6

8

12

As an analyst for the Consumers Association you are required to (i)

Develop a simple price index for each product

(8 Marks)

(ii)

Develop a simple aggregate price index

(8 Marks)

(iii)

Explain the principles underlying your calculations for both indexes and the results obtained. (4 Marks) [Total: 20 Marks]

2

5.

CPA office refurbishment is due to commence in September 2006. The contractor has stated that the project will be finished within 10 days. You have been asked to confirm the duration of the work since it will be necessary to vacate the premises for that time. The following list of activities and durations have been provided by the contractor.

Activity

Preceding Activity

Duration (Days)

A

Remove all furniture

-

2

B

Clear all areas

-

1

C

Put in new utilities (gas, water, electricity)

-

6

A,B

5

D

Rewire rooms

E

Add more power points

A

2

F

Add new fittings

D

3

G

Modify wiring layout

A,B

3

H

Change lighting

E

2

I

Test lights, fittings

C,G,H

1

J

Test all systems

F,I

1

You are required to: (i)

Draw a network to illustrate the activities

(10 Marks)

(ii)

Confirm the duration of the project and indicate the critical path.

(10 Marks)

[Total: 20 Marks]

6.

"A number of methods are available to attempt to achieve a representative sample of the population" Outline the principles of four of the following methods of sampling: (i) simple random sampling (ii) stratified random sampling (iii) systematic sampling (iv) multistage sampling (v) quota sampling (vi) cluster sampling [Total: 20 Marks]

END OF PAPER

3

Suggested Solutions

BUSINESS MATHEMATICS & QUANTITATIVE METHODS FORMATION 1 EXAMINATION - AUGUST 2006. SOLUTION 1. (i)

The general formula for calculation of compound interest is Sn

=

P(1 + r)n

where Sn = sum accrued, P = principal, r = interest rate, n = number of time periods.

P = €25,000,

n = 5,

r = 5% = .05

25,000 (1 + 0.05)5

S5

=

=

€31,907.5

=

25,000 (1.2763) 4 Marks

This is reinvested for 4 years S9

=

31,907 (1 + 0.07)4

=

=

€41,823.7

This is the final value after 9 years.

31,907 (1.3108)

4 Marks

(ii)

If half the investment is withdrawn after 5 years, that is, €12,500. the balance remaining is €31,907 - €12,500 = €19,407 (P). S9 = =

(iii)

19,407 (1 + 0.07)4

=

19,407 (1.3108) 8 Marks

€25,438.7

Time value of money. Capital investment decisions are those decisions which involve current outlays of cash in return for a stream of benefits in future years. Projection of company expenditures are made in the expectation of realising future benefits. However, because money can be used to earn interest, waiting for the recoupment of this money has a cost. Since we are waiting for the return of out investment funds so that they can be invested elsewhere we favour projects which give us the earliest cash flows. This implies that money has a time value. The concept of a cost of capital relies on this. In order for us to forgo the use of money for a period an amount should be paid in compensation. That is money received at different points in time is of differing significance. The longer we have to wait for it the less valuable a given sum is. The extent of this is governed by the rate of interest. 4 Marks [Total 20 Marks]

4

SOLUTION 2 (i)

The ogive is constructed from the following data. From the open ended distribution the end point has been set at 180. Class Boundaries

No Trainees (f)

Mid Point (x)

Payment less than €

Cumulative frequency

100

0

fx

100 - 110

1

105

110

1

105

110 - 120

4

115

120

5

460

120 - 130

7

125

130

12

875

130 - 140

13

135

140

25

1755

140 - 150

7

145

150

32

1015

150 - 160

3

155

160

35

465

160 +

1

165

180

36

165

∑

36

4840 2 Marks

Cum Frequency

40

x x x

30 x 20 x 10

Median x x

10 0

110

120

130

140

150

160

180

Payment € / week 6 Marks

(ii)

The median can be derived from the graph and is approx. €137.

(iii)

Mean

=

x

=

∑fx ∑f

=

4820 36

= €134

4 Marks

2 Marks

Mode: found from the payment range with the greatest frequency, €130 - €140. 2 Marks

5

The appropriate value: •

Since the mode is the value that occurs most often, it may be accepted as typical of the data; it is not consistent in that there may be more than one mode in a set of data

•

The median is considered to give a true middle of the data set and is preferable if there are extreme values in the data

•

The mean uses every data equally in its calculation but is influenced by exceptionally high or low values in the data.

In the present case there is very little difference between the various values. 4 Marks [Total 20 Marks]

6

SOLUTION 3. (i)

Many quantitative techniques treat situations with certainty. However, in many business situations it is not possible to treat outcomes with certainty. This can be done by developing a probability model and a probability distribution can be used. This gives a distribution of profits with their associated probabilities. In this case there is no single deterministic value for the profit. The expected value is an average of the possible range of outcomes weighted according to their probability of occurance, that is, expected value E(X) =

X1P(X1) + X2P(X2) + ------------------ XnP(Xn)

The probability is expressed as a value between 0 and 1. The total probability must equal 1. Since the total distribution of values is used then the expected value is equal to the mean of the x values. The expected values themselves represent a long-term average and, as individual values, they are unlikely to occur. However, in business situations they give a single value for comparing alternatives and represent a ‘best’ single estimate of an uncertain situation. 8 Marks (ii) No guests

1

2

3

4

5

Probability

0.1

0.2

0.4

0.2

0.1

Profit / room €

35

75

125

175

225

The expected profit is E(X) =

X1P(X1) + X2P(X2) + X3P(X3) + X4P(X4) + X5P(X5)

where X is the profit per occupied room. E (X) =

0.1 x 35 + 0.2 x 75 + 0.4 x 125 + 0.2 x 175 + 0.1 x 225

=

3.5 + 15 + 50 + 35 + 22.5

=

€126

The average expected daily profit is €126 – this is the average amount that will be generated per day over a long period. 8 Marks (iii)

Probability and expectation. This process is part of decision analysis and involves using a range of techniques to assist the manager in choosing the most appropriate decisions in given circumstances. There are a number of practical decision making techniques using probability. Such methods are necessary since there are many circumstances where relevant information is not known with any degree of certainty. There may be probabilities associated with the likelihood of an event occurring and this will allow an ‘expected’ value to be determined. In many examples the expected value is obtained by multiplying a probability by the total number of values. For example, if the probability of an employee being satisfied with their job is 0.75 and there are 200 employees in the company, the expected number of satisfied employees would be 0.75 x 200 = 150. This process can be extended to relate to more complex problems. In general the expected value of a variable is obtained by multiplying each probability by the corresponding value and obtaining the sum of these products. The expected value can be regarded as being an estimate of the average value for the variable. 4 Marks

[Total 20 Marks]

7

SOLUTION 4. (i)

Commodity

2003

2004

2005

Coffee

20

25

35

Bananas

12

14

18

6

8

12

Tea

To construct a simple price index calculate the ratio of the new price to the base year price for each commodity and then multiply by 100. the ratio of new price to vase year price is the ‘price relative’. Simple price index = Pn/Po x 100 where Pn = new price (year n), Po = base year price (year 0). Simple price index for each of the commodities.

Coffee

Year

Price

Pn/Po

Index

2003

20

1.00

100

2004

25

1.25

125

2005

35

1.75

175 2 Marks

Bananas

Year

Price

Pn/Po

Index

2003

12

1.00

100

2004

14

1.16

116

2005

18

1.50

150 2 Marks

Tea

Year

Price

Pn/Po

Index

2003

6

1.00

100

2004

8

1.33

133

2005

12

2.00

200 2 Marks

From the above it can be seen that the price of coffee has risen by 75%, the price of bananas by 50% and the price of tea by 100% over the whole time period. 2 Marks (ii)

To derive an index for the overall change in price of all three types of commodity, a ‘simple aggregate price index’ is required. To include all three types of commodity their respective prices in each of the three years is summed, that is, Simple Aggregate Price Index = ∑ Pn x 100 ∑ Po 2 Marks Where Pn = new prices (year n) and Po = base year prices (year 0). Coffee

Bananas

Tea

∑ Pn / ∑ Po

P2003

20

12

6

38/38 = 1.00

P2004

25

14

8

47/38 = 1.23

P2005

35

18

12

65/38 = 1.71 6 Marks

This price index shows that the aggregate prices of the three drinks has risen by 71%. However, this index has ignored the relative quantities consumed in each of the three years. If consumption is taken into account the fact that tea experienced the greatest % increase in price over the period should be taken into account by calculating a weighted aggregate price index. 4 Marks [Total 20 Marks] 8

SOLUTION 5. (i)

The network for the activities is outlined below. It illustrates the sequence of activities and the steps involved.

10 Marks The following additional information is provided to give students an appreciation of the process in deriving the network but is not required in the examination. Each activity is represented by an arrow on the diagram. Circles are numbered and drawn to indicate the start and of finish activity. The diagram shows the relationships between the activities as set down in the table. In order to produce a network diagram a list of activities is required and the interdependence between activities, that is, the activities that precede other activities. In the above diagram a ‘dummy’ activity is introduced because activities D and G both follow A and B and E only follows A. The purpose of this dummy is to maintain the logic of the sequence. (ii)

The total project duration is an important factor when managing projects. The overall duration can be calculated from the network providing the duration of each activity is known. In order to calculate the overall duration of the project it is necessary to estimate the earliest and latest event times. The earliest time is determined by the longest route through the network and the largest value is used. This gives the earliest time in which the project can be completed. The latest event times are then calculated. In the network the latest event time equals the earliest event time. Preceding latest event times are calculated by subtracting an activity’s duration from the subsequent latest event time. If two or more activities start from an event, the latest time for each route is calculated and the lowest value is used. Both values are shown in the diagram. It shows that the project can be completed in 11 days. The critical path identifies the route that defines the overall duration. The activities on the critical path have no flexibility if the project is to finish on time. It is 1-2-4-6-7-8 and takes 11 days. 10 Marks [Total 20 Marks]

9

SOLUTION 6. Sampling Methods. There are a wide range of sampling methods depending on the type of sample required and the technique being used. A description of four of the following methods is required. Simple random sampling. In this case every member of the population must have an equal chance of being included in the sample. For large populations each member is normally given a unique identification number and then random numbers are generated or random tables are used. The sample then comprises the population members whose numbers match those generated from the random numbers. This method has minimum bias. In some cases it may be difficult to contact all members of the chosen sample and occasionally an unrepresentative sample may occur. This method is used by large marketing companies to obtain a wide geographical spread in the data. 5 Marks Stratified random sampling. This method is similar to simple random sampling but is used where the population contains distinct groups, that is, groups with different views about the issues under study or of particular interest. The sample can be stratified according to the particular groups so that it has the same proportions, approximately, as the population. For each stratum the sample members are selected randomly as for simple random sampling. It is a unbiased method and gives a representative sample. The process of stratification can be expensive and incurs costs additional to the survey process. 5 Marks Systematic Sampling. This method is similar to simple random sampling. In this case the data are assumed to be random and every nth member is selected where n is determined by (population size) / (sample size). The start point of the sample may be chosen randomly. It is an inexpensive and easy method to use and is useful if the exact population is unknown. Often particular steps must be taken to ensure that the sample is not biased. 5 Marks Multistage sampling. This is a process used for producing a representative sample form a widespread population. The process selects individual sampling units by splitting the sampling process into stages and using the most relevant sampling technique. A typical process is a three-stage survey where 1) stage 1 takes a number of primary sampling units – the population may be divided into a number of regions and then randomly select a number of these 2) Stage 2 where each of the regions is taken and random samples of sub-regions taken (secondary sampling units) 3) stage 3 where individuals could be selected by using systematic sampling (tertiary sampling units). This technique is used for widely spread data. 5 Marks Quota sampling. Used where interviewing is the main method of data collection. It is necessary to ensure that the composition of the sample matches that in the population. The interviewer is given a predetermined sample profile where the number of interviewees in each category is chosen to match the population proportions. The interviewer selects from the population to match the required numbers in each category proportions. This method tends to have a good response rate. However, it is a non-random technique and is reliant on the interviewer. 5 Marks Cluster Sampling. This method is used when the population items of interest are widely spread and it is desirable to ensure that the sample elements are grouped together in some way. A number of groups/areas would be selected and populations within those groups could be interviewed. This is a useful method for widely spread geographical data where the population is not defined exactly. 5 Marks

[Total for four: 20 Marks]

10