BUSINESS MATHEMATICS &
QUANTITATIVE METHODS
FORMATION 1 EXAMINATION - AUGUST 2007
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.
List on the cover of each answer booklet, in the space provided, the number of each question(s)
attempted.
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 2007
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.
Superior Products Ltd. is considering a project to purchase additional equipment for its miniature battery
manufacturing plant. The equipment will cost €52,000. The Company Accountant estimates the following cash
flows:
Year
Cash Flows €
1
21,000
2
26,000
3
18,000
4
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 of
16% for investment proposals.
(i)
Calculate the Net Present Value (NPV) and the Internal Rate of Return (IRR) on the project.
The company also wishes to determine the best method of financing the project.
(8 Marks)
The following options are available
-
Bank Loan of €52,000 repayable at the end of year 4 with interest of 14% payable each year.
-
Hire-purchase with an initial payment of €14,000 and 4 annual payments of €14,000.
-
(ii)
2.
Lease the equipment for 4 years at a rental of €16,000 per year, payable at the end of each year.
Advise the company on the best method of financing the project.
(12 Marks)
[Total: 20 Marks]
The Auditor of CPA Accountants selected 100 invoices and examined them for errors. He found a range of errors
in both the companyʼs favour, which he marked with a plus, and in the suppliersʼ favour which he marked with a
minus. The results of his audit are set out as follows:
Error (€)
-50 to - 40
-40 to -30
-30 to -20
-20 to -10
-10 to 0
0 to +10
+10 to +20
+20 to +30
+30 to +40
+40 to +50
Frequency
1
1
3
4
14
45
20
6
4
2
1
You are required to
(i)
(ii)
(iii)
3.
Find the mean and standard deviation of the data.
Estimate the 95% confidence limits within which you would expect the mean error
of all invoices to lie.
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.
(8 Marks)
(6 Marks)
(6 Marks)
[Total: 20 Marks]
GAirways specialises in executive travel and uses top range aircraft to attract its customers to exotic locations. It
leases its aircraft from Boeing Corporation and wants to plan for further leases over the next 5 years. It carried out
a 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)
3
120
258
1
100
2
115
4
250
255
130
5
267
145
6
270
152
7
272
155
273
To assist the company with its planning you are asked to
(i)
(ii)
4.
Determine the strength of the relationship between the data by means of the co-efficient of correlation.
(10 Marks)
Advise the company if the relationship between seat bookings and customer income is a sound basis for
purposes of prediction. Briefly discuss other methods of validation of the forecast.
(10 Marks)
[Total: 20 Marks]
DIY Ltd. employs three grades of male operatives and three grades of female operatives. The Equality Authority
wishes to compare the increase in the average wage from 2004 to 2006. You compile the data below from the
company records. With 2004 as the base year, the MD asks you to calculate an index number for the average
wage in 2006. You decide to use
(i)
(ii)
(iii)
Year
Base weighting
Current weighting
And report on the relationship between the two.
Operator Grade
Rate/hr €
M3
8.50
M1
M2
F1
F2
F3
8.90
8.80
8.35
8.30
8.25
2004
No. of operatives
Rate/hr€
15
10.50
35
15
40
10.60
10.55
10.30
20
10.20
25
10.10
2
2006
(6 Marks)
(6 Marks)
(8 Marks)
No. of operatives
40
10
15
45
15
30
[Total: 20 Marks]
5.
As the Investment Advisor of CPA Consultants, you are asked to deal with the following problems presented by
clients.
(i)
DIB Ltd. estimates that it will have to purchase a new plant in two years from now at a cost of €450,000. It
has been quoted a nominal interest rate by New World Banking of 12%, compounded at three monthly
intervals. Advise the company on
-
the sum that it should now set aside to purchase the equipment
the APR (annual percentage rate) of such a loan.
(4 Marks)
(4 Marks)
DIB Ltd. is considering whether to use straight line or reducing balance methods of depreciation. It
purchases an asset for €450,000 , expected to last 5 years and to have a scrap value of €50,000. Outline
the 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)
(ii)
(iii)
Superior Products Ltd. has estimated its fixed costs of production at €2,100 per week and its variable costs
at €11 per unit. You are asked to derive the demand function linking costs and quantity demanded. Hence
estimate 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 considerable
interest with a number of clients who request you to summarise the key elements of the principles involved. In your
summary 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
SUGGESTED SOLUTIONS
Solution 1
BUSINESS MATHEMATICS &
QUANTITATIVE METHODS
FORMATION 1 EXAMINATION - AUGUST 2007
(i) Net Present Value and Internal Rate of Return.
Year
Cash Flow
0
( €52,000)
2
€21,000
€26,000
4
€ 1,000
1
3
€18,000
NPV
€ 5,000
Discount Factor
@ 16%
PV
1.000
( €52,000)
0.743
€18,101
0.862
0.641
€11,538
0.552
€ 2,760
€19,318
0.552
€
552
Discount Factor
@ 18%
PV
1.000
( €52,000)
0.718
€17,787
0.847
0.609
€10,962
0.516
€ 2,580
0.516
€270
The NPV of the project is €270.
€18,668
€
516
( €1,487)
(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%.
(ii)
(4 Marks)
The best financial option is the one which gives the lowest present value of cost when cash flows are discounted
at 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
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
(4 Marks)
(4 Marks)
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.
Leasing is the most financially viable option followed by a bank loan.
5
(4 Marks)
[Total: 20 Marks]
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
boundary
- 50
- 40
- 30
- 20
- 10
0
10
20
30
40
∑
Upper
boundary
- 40
- 30
- 20
- 10
0
10
20
30
40
50
Mean = x = ∑fx
∑f
Freq
(f)
1
3
4
14
45
20
6
4
2
1
fx
- 45
- 35
- 25
- 15
- 5
5
15
25
35
45
- 45
- 105
- 100
- 210
- 225
100
90
100
70
45
- 280
(x – x)
-
42.2
32.2
22.2
12.2
2.2
7.8
17.8
27.8
37.8
47.8
σ = √ ∑ f (x - x)2 =
∑f
=
√ 205.166
(x – x)2
1780.8
1036.8
492.8
148.8
4.8
60.8
316.8
772.8
1428.8
2284.8
f(x – x)2
1780.8
3110.5
1971.4
2083.8
217.8
1216.8
1901.6
3091.4
2857.7
2284.8
20,516.6
(4 Marks)
= -280 = - €2.8
100
Standard deviation =
(ii)
Mid point x
√ 20,516.6
=
100
€14.32
(4 Marks)
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
(iii)
If the company processes 25,000 invoices per year,
-
-
-
the most favourable error is: 25,000 x 0.01 =
€250
the least favourable error is: 25,000 x - 5.61 = - €140,250
the most likely error is: 25,000 x - 2.8 = - €70,000.
(6 Marks)
(2 Marks)
(2 Marks)
(2 Marks)
The method used has a 95% chance of success in the long run; 95% of the interval estimates contain the true
value of the population mean, that is, it will be right on 95% of the occasions. As the sample mean is an estimate
of the population mean, to retain 95% confidence the estimate should be as large as possible. But since the
accountant has taken a sample of 100 (greater than 30) and is considered statistically large, the result can be
accepted as accurate. The errors in the invoices will favour the suppliers.
(2 Marks)
6
[Total: 20 Marks]
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
Customer Income
( €000s)
x
250
255
258
267
270
272
273
1845
Year
1
2
3
4
5
6
7
∑
r
=
=
=
(ii)
∑xy/n - ∑x/n∑y/n,
√[∑x2/n - (∑x/n)2][∑y2/n - (∑y/n)2]
=
Reservations
(000s)
y
100
115
120
130
145
152
155
917
x2
62,500
65,025
66,564
71,289
72,900
73,984
74,529
486,791
y2
10,000
13,225
14,400
16,900
21,025
23,104
24,025
122,679
xy
25,000
29,325
30,960
34,710
39,150
41,344
42,315
242,804
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
√26,208
=
159
161.88
=
0.98
Since the value of r is close to +1, this suggests a strong positive relationship between the variables. This can
be confirmed prior to the calculation of r by drawing a scatter graph which will show the form and strength of the
relationship.
[10 Marks]
In considering the shape and strength of the relationship between the variables, there are a number of factors
which have to be considered.
●
●
●
●
●
●
Based on the shape of the data from a scattergraph, there appears to be a positive correlation.
The co-efficient of correlation of 0.98 indicates a strong relationship between the data. The co-efficient of
determination, r2, indicates a measure of accuracy of fit.
However, to have confidence in a regression relationship, it is necessary to have a large number of
observations, The number of observations with which we are dealing (7) may not be statistically
significant.
Confidence limits can be produced for forecasts produced by a regression equation. This will require
further analysis.
Any form of extrapolation must be carefully done. Outside the observed data, relationships and conditions
may change drqmatically.
Regression is not considered to be an adaptive forecasting system. It is not suitable for producing
forecasts which adapt to current conditions – this is better done by time series forecasting.
7
●
●
In many conditions it is not accurate to assume that y depends on only one independent variable. Often a
value may depend on two or more factors in which multiple regression is used. This is the case in many
business and economic situations and forecasts.
The company therefore should be aware of the deficiencies in using a linear regression model for
forecasts which could have a major impact on the future of the business. In such important decisions it
should carry out further analyses and tests to confirm the accuracy of a forecast. Otherwise it should be
aware of the incumbent dangers of accepting the forecast outlined.
(10 Marks)
[Total: 20 Marks]
8
4.
(i)
Using Laspeyres Index as the base weighted index and for (ii) using Paasche Index as the current
weighted index.
The data for both indexes is compiled in the following table.
Grade
M1
M2
M3
F1
F2
F3
P
8.90
8.80
8.50
8.35
8.30
8.25
0
2004
Laspeyres Index
=
=
Q
35
15
15
40
20
25
0
2006
Pn
10.60
10.55
10.50
10.30
10.20
10.10
∑PnQ0 x 100
∑P0Q0
∑PnQn
∑P0Qn
Qn
40
10
15
45
15
30
=
P0Q0
311.5
132.0
127.5
334.0
166.0
206.25
PnQ0
371.0
158.25
157.5
412.5
204.0
252.5
1555.25 x 100
=
1277.25
=
1606.5 x 100
1319.25
=
121.77
121.77
PnQn
424.0
105.5
157.5
463.5
153.0
303.0
P0Qn
356.0
88.0
127.5
375.75
124.5
247.5
(2 Marks)
(4 Marks)
(ii)
Paasche Index
(iii)
Laspeyres index uses base year statistics as weights. This implies that the quantities do not vary over time
and 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 understate
the affects of P. The Laspeyres index generally exceeds the Paasche index, In the present case both
indexes are similar. However, the overall ʻquantitiesʼ – the number of operatives – have increased over the
3 year period giving greater weights to the Paasche index. Over such a short time period, the change is
insignificant and either index is equally valid.
(8 Marks)
(4 Marks)
[Total: 20 Marks]
9
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) , Where P = sum to be invested.
8
(4 Marks)
Therefore, P = 450,000/1.12 = 450,000/2.476 = €181,745
8
Actual Percentage Rate.
APR = (1 + i/n) - 1 where i = nominal interest rate,
n = number of compounding periods
8
= [1 + (0.12/8)] - 1
n
= 1.015 - 1
8
= 1.126 - 1
(ii)
(4 Marks)
= 12.6%
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
=
(3 Marks)
€80,000pa.
The reducing balance method is the converse of compound interest. Therefore the depreciation at the end of 5
years is
[original book value][1 - i] , where i = interest (depreciation) rate, 20%, n = no. of years, 5.
n
Depreciation = 450,000(1 - 0.20)5 = 450,000 x 0.327 = €147,000.
(iii)
This represents the fact that assets depreciate more in the early years.
(3 Marks)
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)
y = 2,100 + 12,000 x 11 = €134,100.
(3 Marks)
Total costs of production at 12,000 units per week.
[Total : 20 Marks]
10
Solution 6
Hypothesis Testing.
(i)
(ii)
The principles of hypothesis testing. Using the technique of statistical inference we can draw conclusions
about a population from a sample. Problems of statistical inference are problems of estimation. This is
concerned with the way sample results are used to estimate or infer values for the population. Samples can
be used for estimating such measures of the population as its mean and proportions of the population
possessing a given characteristic. A hypothesis is an assumption about a situation. To test an hypothesis it
is 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 alternative
assumptions. We assume that we are testing a main hypothesis (called the Null, Ho, hypothesis) against
one other hypothesis (the Alternative, H1, hypothesis). The null hypothesis challenges us with the view that
the true mean, µ , is equal to a specified value, µ0, and can be stated as H0: µ = µ0. This view can only be
rejected on the basis of significant statistical evidence. The rejection of this view necessarily implies the
acceptance of another. This is the alternative hypothesis, H1, and is a statement of the view that we are
prepared to accept if we reject H0. If the test were concerned only with the view that the population mean
were equal to a certain value (that is, µ = µ0) or different (µ ≠ µ0), then the alternative hypothesis would take
the form H1: µ ≠ µ0.
(5 Marks)
Type 1 and Type 2 errors. In testing the null hypothesis, there are two types of error that might occur. The
Null Hypothesis may be rejected when it is true – a Type 1 error. The Null hypothesis may be accepted when
it is false – a Type 2 error. Avoiding a Type 1 error is the main concern of problems in this area. When testing
the 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 standard
errors of the true population mean. However, there is still a 5% chance of a single sample mean lying outside
this 95% confidence interval. If we are dealing with a Normal distribution, this 5% chance can be split evenly
between the two tails of the distribution. If the sample means lie outside the confidence limits (x1 or x2), then
the decision will be to reject H0 - even though we might be wrong in rejecting it. However, the chance of
being wrong is less than 5% since there was a less than 5% chance of getting a sample mean (xi) outside
the confidence limits when µ is the population mean.
The confidence limits for the population mean, µ, 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 critical
to the decision. The size of a Type 2 error can be reduced by careful definition of the rejection region. If the
level of significance is reduced (say from 5% to 1%) we are less willing to reject the null hypothesis when it
is in fact true. If a significance level of 1% is used, it will reduce the chance of a Type 2 error occurring and
excludes 1% of extreme observations. If the z score, that is, [(sample mean – hypothesised mean)/standard
error] is calculated for the sample data and found to be outside the z critical value, then H0 is rejected. In
the 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 population
mean is less than the specified value (H1: µ < µ0 ) or greater (H1: µ > µ0 ), the test would be one-sided. If
we 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 – µ)/(σ/√n)], for the sample; the sample standard deviation,
s, is used for µ, 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
σ
(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 the
components
are of a particular size or very close to it and comply with the specifications set down. Otherwise they will
be rejected by the customer or in the case of an industry such as confectionary may cause a potential health
hazard. 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 companies
or employees relating claims on pay and conditions of employment. In the accountancy field, the accountant
may 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 test
the variability of errors in the company accounts.
(5 Marks)
12