f1 - business maths august 07

8/14/2019 f1 - Business Maths August 07

1/13

NOTESAnswer 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.

List on the cover of each answer booklet, in the space provided, the number of each question(s)

attempted.

BUSINESS MATHEMATICS &QUANTITATIVE METHODS

FORMATION 1 EXAMINATION - AUGUST 2007

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


2/13

THE INSTITUTE OF CERTIFIED PUBLIC ACCOUNTANTS IN IRELAND



Time Allowed: 3 hours, plus 10 minutes to read the paper Answer 5 Questions

Only the first five questions answered will be marked.

All questions carry equal marks.

1. Superior Products Ltd. is considering a project to purchase additional equipment for its miniature batterymanufacturing plant. The equipment will cost 52,000. The Company Accountant estimates the following cashflows:

Year 1 2 3 4

Cash Flows 21,000 26,000 18,000 1,000

At the end of year 4 the equipment will have a disposal value of 5,000. The company uses a cost of capital of16% for investment proposals.

(i) Calculate the Net Present Value (NPV) and the Internal Rate of Return (IRR) on the project.(8 Marks)

The company also wishes to determine the best method of financing the project.

The following options are available

- Bank Loan of 52,000 repayable at the end of year 4 with interest of 14% payable each year.

- Lease the equipment for 4 years at a rental of 16,000 per year, payable at the end of each year.

- Hire-purchase with an initial payment of 14,000 and 4 annual payments of 14,000.

(ii) Advise the company on the best method of financing the project.(12 Marks)

[Total: 20 Marks]

2. The Auditor of CPA Accountants selected 100 invoices and examined them for errors. He found a range of errors

in both the companys favour, which he marked with a plus, and in the suppliers favour which he marked with aminus. The results of his audit are set out as follows:

Error () Frequency

-50 to - 40 1-40 to -30 3-30 to -20 4-20 to -10 14-10 to 0 45

0 to +10 20

+10 to +20 6+20 to +30 4+30 to +40 2+40 to +50 1

1


3/13

You are required to

(i) Find the mean and standard deviation of the data. (8 Marks)

(ii) Estimate the 95% confidence limits within which you would expect the mean errorof all invoices to lie. (6 Marks)

(iii) If the company processed 25,000 invoices for 2006, estimate the most favourable,least favourable and most likely errors for that year based on 95% confidence limits. (6 Marks)

[Total: 20 Marks]

3. GAirways specialises in executive travel and uses top range aircraft to attract its customers to exotic locations. Itleases its aircraft from Boeing Corporation and wants to plan for further leases over the next 5 years. It carried outa survey of seat bookings and customer income for the past 7 years as a means to forecast future aircraft leases.The data from its survey is given below.

Year Reservations (000s) Customer Income (000s)

1 100 250

2 115 2553 120 258

4 130 267

5 145 270

6 152 272

7 155 273

To assist the company with its planning you are asked to

(i) Determine the strength of the relationship between the data by means of the co-efficient of correlation.(10 Marks)

(ii) Advise the company if the relationship between seat bookings and customer income is a sound basis forpurposes of prediction. Briefly discuss other methods of validation of the forecast. (10 Marks)

[Total: 20 Marks]

4. DIY Ltd. employs three grades of male operatives and three grades of female operatives. The Equality Authoritywishes to compare the increase in the average wage from 2004 to 2006. You compile the data below from thecompany records. With 2004 as the base year, the MD asks you to calculate an index number for the averagewage in 2006. You decide to use

(i) Base weighting (6 Marks)(ii) Current weighting (6 Marks)(iii) And report on the relationship between the two. (8 Marks)

Year 2004 2006

Operator Grade Rate/hr No. of operatives Rate/hr No. of operatives

M1 8.90 35 10.60 40

M2 8.80 15 10.55 10

M3 8.50 15 10.50 15

F1 8.35 40 10.30 45

F2 8.30 20 10.20 15

F3 8.25 25 10.10 30

[Total: 20 Marks]

2


4/13

5. As the Investment Advisor of CPA Consultants, you are asked to deal with the following problems presented byclients.

(i) DIB Ltd. estimates that it will have to purchase a new plant in two years from now at a cost of 450,000. Ithas been quoted a nominal interest rate by New World Banking of 12%, compounded at three monthlyintervals. Advise the company on

- the sum that it should now set aside to purchase the equipment (4 Marks)- the APR (annual percentage rate) of such a loan. (4 Marks)

(ii) DIB Ltd. is considering whether to use straight line or reducing balance methods of depreciation. Itpurchases an asset for 450,000 , expected to last 5 years and to have a scrap value of 50,000. Outlinethe annual depreciation that would be expected with straight line depreciation and the value of the asset

after 5 years with the reducing balance method (assuming a depreciation rate of 20%). (6 Marks)

(iii) Superior Products Ltd. has estimated its fixed costs of production at 2,100 per week and its variable costsat 11 per unit. You are asked to derive the demand function linking costs and quantity demanded. Henceestimate the total costs at a production level of 12,000 per week. (6 Marks)

[Total: 20 Marks]

6. At a recent seminar you have presented a paper on statistical inference and testing. This has created considerableinterest with a number of clients who request you to summarise the key elements of the principles involved. In yoursummary you are asked to include explanations of the following:

- The principles of hypothesis testing.- The key steps in implementing the process.- Type 1 and Type 2 errors.- Typical business applications. [Total: 20 Marks]

END OF PAPER

3


5/13



Solution 1(i) Net Present Value and Internal Rate of Return.

The NPV of the project is 270. (4 Marks)

Using interpolation or the following formula for IRR

N1I2 - N2I1 , where discount rate I1 gives NPV N1 and discount rate

N1 - N2 I2 gives NPV N2.

Where N1 = 270, I1 = 16%; N2 = ( 1,487), I2 = 18%

IRR = 270 x 0.18 - (1487) x 0.16 = 28,652 = 16.6%270 + 1487 1727

The IRR for the project is 16.6%. (4 Marks)

(ii) The best financial option is the one which gives the lowest present value of cost when cash flows are discountedat 14%. most

Loan

Interest payments of 7,280 for 4 years: 7280 x annuity factor at 14% (2.914)

Loan repayment of 52,000 at year 4: 52,000 x discount factor @ 14% (0.592)

Scrap value of 5,000 at end year 4: 5,000 x discount factor @ 14% (0.592)

Present Value: 7280 x 2.914 + 52,000 x 0.592 5000 x 0.592 = 49,037 (4 Marks)

Lease

Rental of 16,000 per year for 4 years: 16,000 x annuity factor @ 14% (2.914)

Present Value: 16,000 x 2.914 = 46,624 (4 Marks)

4

SUGGESTED SOLUTIONS

Year Cash Flow Discount Factor PV Discount Factor PV@ 16% @ 18%

0 ( 52,000) 1.000 ( 52,000) 1.000 ( 52,000)

1 21,000 0.862 18,101 0.847 17,787

2 26,000 0.743 19,318 0.718 18,668

3 18,000 0.641 11,538 0.609 10,962

4 1,000 0.552 552 0.516 516

5,000 0.552 2,760 0.516 2,580

NPV 270 ( 1,487)


6/13

Hire-Purchase

Initial payment of 14,000: 14,000

Payments of 14,000 for 4 years: 14,000 x annuity factor @ 14 % (2.914)

Present Value: 14,000 + 14,000 x 2.914 = 54,796. (4 Marks)

Leasing is the most financially viable option followed by a bank loan. [Total: 20 Marks]

5


7/13

Solution 2(i) The mean and standard deviation of the invoice errors. Assuming that the data is continuous gives the mid

points in the table.

Lower Upper Freq Mid point x fx (x x) (x x)2 f(x x)2

boundary boundary (f)

- 50 - 40 1 - 45 - 45 - 42.2 1780.8 1780.8

- 40 - 30 3 - 35 - 105 - 32.2 1036.8 3110.5

- 30 - 20 4 - 25 - 100 - 22.2 492.8 1971.4

- 20 - 10 14 - 15 - 210 - 12.2 148.8 2083.8

- 10 0 45 - 5 - 225 - 2.2 4.8 217.8

0 10 20 5 100 7.8 60.8 1216.8

10 20 6 15 90 17.8 316.8 1901.6

20 30 4 25 100 27.8 772.8 3091.4

30 40 2 35 70 37.8 1428.8 2857.7

40 50 1 45 45 47.8 2284.8 2284.8

- 280 20,516.6

Mean = x = fx = -280 = - 2.8 (4 Marks)f 100

Standard deviation = = f (x - x)2 = 20,516.6f 100

= 205.166 = 14.32 (4 Marks)

(ii) Taking a 95% confidence level, the interval estimate for the population mean is

- 2.8 1.96 x 14.32/100

= - 2.8 2.81.

that is, 95% of the means of such samples lie in the range 0.01 to - 5.61 (6 Marks)

(iii) If the company processes 25,000 invoices per year,- the most favourable error is: 25,000 x 0.01 = 250 (2 Marks)

- the least favourable error is: 25,000 x - 5.61 = - 140,250 (2 Marks)

- the most likely error is: 25,000 x - 2.8 = - 70,000. (2 Marks)

The method used has a 95% chance of success in the long run; 95% of the interval estimates contain the truevalue of the population mean, that is, it will be right on 95% of the occasions. As the sample mean is an estimateof the population mean, to retain 95% confidence the estimate should be as large as possible. But since theaccountant has taken a sample of 100 (greater than 30) and is considered statistically large, the result can beaccepted as accurate. The errors in the invoices will favour the suppliers.

(2 Marks)

[Total: 20 Marks]

6


8/13

Solution 3(i) Since the number of reservations depends on family income, Reservations is the dependent variable (y) and

Income is the independent or causal variable (x).

The strength of the relationship is represented by the correlation co-efficient, r

where r = xy/n - x/ny/n,[x2/n - (x/n)2][y2/n - (y/n)2]

Year Customer Income Reservations( 000s) (000s)

x y x2 y2 xy

1 250 100 62,500 10,000 25,000

2 255 115 65,025 13,225 29,325

3 258 120 66,564 14,400 30,960

4 267 130 71,289 16,900 34,710

5 270 145 72,900 21,025 39,150

6 272 152 73,984 23,104 41,344

7 273 155 74,529 24,025 42,315

1845 917 486,791 122,679 242,804

r = 242,804/7 1845/7 x 917/7

[486,791/7 (1845/7)2][122,679/7 (917/7)2]

= 34,686 - 34,527

(69,541 69,469)(17,525 17,161)

= 159 = 159 = 0.9826,208 161.88

Since the value of r is close to +1, this suggests a strong positive relationship between the variables. This canbe confirmed prior to the calculation of r by drawing a scatter graph which will show the form and strength of therelationship.

[10 Marks]

(ii) In considering the shape and strength of the relationship between the variables, there are a number of factorswhich have to be considered.

G Based on the shape of the data from a scattergraph, there appears to be a positive correlation.

G The co-efficient of correlation of 0.98 indicates a strong relationship between the data. The co-efficient ofdetermination, r2, indicates a measure of accuracy of fit.

G However, to have confidence in a regression relationship, it is necessary to have a large number ofobservations, The number of observations with which we are dealing (7) may not be statisticallysignificant.

G Confidence limits can be produced for forecasts produced by a regression equation. This will requirefurther analysis.

G Any form of extrapolation must be carefully done. Outside the observed data, relationships and conditionsmay change drqmatically.

G Regression is not considered to be an adaptive forecasting system. It is not suitable for producingforecasts which adapt to current conditions this is better done by time series forecasting.

7


9/13

G In many conditions it is not accurate to assume that y depends on only one independent variable. Often avalue may depend on two or more factors in which multiple regression is used. This is the case in manybusiness and economic situations and forecasts.

G The company therefore should be aware of the deficiencies in using a linear regression model forforecasts which could have a major impact on the future of the business. In such important decisions itshould carry out further analyses and tests to confirm the accuracy of a forecast. Otherwise it should beaware of the incumbent dangers of accepting the forecast outlined.

(10 Marks)

[Total: 20 Marks]

8


10/13

4. (i) Using Laspeyres Index as the base weighted index and for (ii) using Paasche Index as the currentweighted index.

The data for both indexes is compiled in the following table.

2004 2006

Grade P0 Q0 Pn Qn P0Q0 PnQ0 PnQn P0Qn

M1 8.90 35 10.60 40 311.5 371.0 424.0 356.0

M2 8.80 15 10.55 10 132.0 158.25 105.5 88.0

M3 8.50 15 10.50 15 127.5 157.5 157.5 127.5

F1 8.35 40 10.30 45 334.0 412.5 463.5 375.75

F2 8.30 20 10.20 15 166.0 204.0 153.0 124.5

F3 8.25 25 10.10 30 206.25 252.5 303.0 247.5

(2 Marks)

Laspeyres Index = PnQ0 x 100 = 1555.25 x 100 = 121.77

P0Q0 1277.25(4 Marks)

(ii) Paasche Index = PnQn = 1606.5 x 100 = 121.77

P0Qn 1319.25(4 Marks)

(iii) Laspeyres index uses base year statistics as weights. This implies that the quantities do not vary over timeand in many cases this is not correct, as in the present case. This index tends to overstate increases in P.To attempt to overcome this current year quantities are used in the Paasche index. This tends to understatethe affects of P. The Laspeyres index generally exceeds the Paasche index, In the present case bothindexes are similar. However, the overall quantities the number of operatives have increased over the3 year period giving greater weights to the Paasche index. Over such a short time period, the change isinsignificant and either index is equally valid.

(8 Marks)

[Total: 20 Marks]

9


11/13

Solution 5(i) Sum to be invested:sum should accrue to 450,000 after 8 three monthly periods.

That is, 450,000 = P(1 + 0.12)8, Where P = sum to be invested.

Therefore, P = 450,000/1.128

= 450,000/2.476 = 181,745 (4 Marks)

Actual Percentage Rate.

APR = (1 + i/n)

n

- 1 where i = nominal interest rate,n = number of compounding periods= [1 + (0.12/8)]

8- 1

= 1.0158

- 1

= 1.126 - 1 = 12.6% (4 Marks)

(ii) The straight line method reduces the value by the same absolute amount each year, that is,

Annual depreciation = (initial value scrap value)/estimated life of asset.

Annual depreciation = ( 450,000 - 50,000)/5 = 80,000pa. (3 Marks)

The reducing balance method is the converse of compound interest. Therefore the depreciation at the end of 5years is

[original book value][1 - i]n, where i = interest (depreciation) rate, 20%, n = no. of years, 5.

Depreciation = 450,000(1 - 0.20)5 = 450,000 x 0.327 = 147,000.

This represents the fact that assets depreciate more in the early years. (3 Marks)

(iii) As the relationship is linear the demand function it will be of the form y = a + bx,where a = fixed costs, b = variable costs, x = units of production. (3 Marks)

Total costs of production at 12,000 units per week.

y = 2,100 + 12,000 x 11 = 134,100. (3 Marks)

[Total : 20 Marks]

10


12/13

Solution 6Hypothesis Testing.

(i) The principles of hypothesis testing. Using the technique of statistical inference we can draw conclusionsabout a population from a sample. Problems of statistical inference are problems of estimation. This isconcerned with the way sample results are used to estimate or infer values for the population. Samples canbe used for estimating such measures of the population as its mean and proportions of the populationpossessing a given characteristic. A hypothesis is an assumption about a situation. To test an hypothesis itis necessary to know what to expect when an hypothesis is true and we, therefore, often hypothesis the

opposite to what we hope to prove. We wish to test an assumption against one or more alternativeassumptions. We assume that we are testing a main hypothesis (called the Null, Ho, hypothesis) againstone other hypothesis (the Alternative, H1, hypothesis). The null hypothesis challenges us with the view thatthe true mean, , is equal to a specified value, 0, and can be stated as H0: = 0. This view can only berejected on the basis of significant statistical evidence. The rejection of this view necessarily implies theacceptance of another. This is the alternative hypothesis, H1, and is a statement of the view that we areprepared to accept if we reject H0. If the test were concerned only with the view that the population meanwere equal to a certain value (that is, = 0) or different ( 0), then the alternative hypothesis would takethe form H1: 0.

(5 Marks)

(ii) Type 1 and Type 2 errors. In testing the null hypothesis, there are two types of error that might occur. TheNull Hypothesis may be rejected when it is true a Type 1 error. The Null hypothesis may be accepted whenit is false a Type 2 error. Avoiding a Type 1 error is the main concern of problems in this area. When testingthe null hypothesis we state the maximum risk we are willing to accept in committing a Type 1 error, that is,the probability of a Type 1 error. This is the level of significance and is typically either 5% (0.05) or 1% (0.01).With a 95% confidence interval, there is a 95% chance of any sample mean (x) lying within 1.96 standarderrors of the true population mean. However, there is still a 5% chance of a single sample mean lying outsidethis 95% confidence interval. If we are dealing with a Normal distribution, this 5% chance can be split evenlybetween the two tails of the distribution. If the sample means lie outside the confidence limits (x1 or x2), thenthe decision will be to reject H0 - even though we might be wrong in rejecting it. However, the chance ofbeing wrong is less than 5% since there was a less than 5% chance of getting a sample mean (xi) outsidethe confidence limits when is the population mean.

The confidence limits for the population mean, , are regarded as the critical values for tests of hypotheses.

These are the values outside of which we are willing to reject the null hypothesis the values that are criticalto the decision. The size of a Type 2 error can be reduced by careful definition of the rejection region. If thelevel of significance is reduced (say from 5% to 1%) we are less willing to reject the null hypothesis when itis in fact true. If a significance level of 1% is used, it will reduce the chance of a Type 2 error occurring andexcludes 1% of extreme observations. If the z score, that is, [(sample mean hypothesised mean)/standarderror] is calculated for the sample data and found to be outside the z critical value, then H0 is rejected. Inthe case of a two-sided confidence interval the population mean as specified is accepted or the alternative,that the true mean is larger or smaller, is accepted. If we wish to test the null hypothesis that the populationmean is less than the specified value (H1: < 0 ) or greater (H1: > 0 ), the test would be one-sided. Ifwe construct one-sided tests, there is a greater chance that the null hypothesis will be rejected.

(5 Marks)

(iii) The key steps in implementing the process. To test a hypothesis, the summarised steps are:

- state the hypotheses for H0 and H1

- state the significance level

- state the critical values

- calculate the z score, the test statistic [(x )/(/n)], for the sample; the sample standard deviation,s, is used for , providing that the sample size is reasonably large

- compare this z sample score with the z critical value/s

- come to a conclusion: accept or reject H0

- state the conclusion in words, that is, the sample evidence does (or does not) support the null

hypothesis at the stated significance level.(5 Marks)

11


13/13

(iv) Typical business applications. Hypothesis testing is of value in many areas of the business world,particularly in the area of quality control. If a company manufactures components, it is important that thecomponentsare of a particular size or very close to it and comply with the specifications set down. Otherwise they willbe rejected by the customer or in the case of an industry such as confectionary may cause a potential healthhazard. This technique is also frequently adopted in scientific work, for example, testing drugs n

pharmaceutical research. In many applications it can be used to accept or reject claims made by companiesor employees relating claims on pay and conditions of employment. In the accountancy field, the accountantmay wish to confirm that standards are being complied with or that an audit on particular areas of the

company comply with the data provided or claims made by customers or suppliers. He may also wish to testthe variability of errors in the company accounts.(5 Marks)

12

f1 - business maths august 07

Documents