CIMA - C3 Fundamentals Of Business Mathematics

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160 5: Index numbers ⏐ Part B Summarising and analysing data

Index =

8.34

× 100 = 108.75 = 109

The correct answer is therefore C.

The above question is indicative of the way in which index numbers could be assessed using objective test

questions. Make sure you understand how to arrive at the correct answer.

7 The retail prices index for the United Kingdom

The retail prices index (RPI) is used to measure price inflation and can be used to deflate data and for index

linking.

7.1 Items included in the RPI calculation

We will conclude our study of index numbers by looking at the construction of the UK Retail Prices Index (RPI). On

one particular day of each month, data are collected about prices of the following groups of items.

• Food and catering

• Alcohol and tobacco

• Housing and household expenditure

• Personal expenditure

• Travel and leisure

(a) Each group is sub-divided into sections: for example 'food' will be sub-divided into bread, butter,

potatoes and so on. These sections may in turn be sub-divided into more specific items. The groups

do not cover every item of expenditure (for example they exclude income tax, pension fund

contributions and lottery tickets).

(b) The weightings given to each group, section and sub-section are based on information provided by

the

Family Expenditure Survey which is based on a survey of over 10,000 households, spread evenly

over the year.

Each member of the selected households (aged 16 or over) is asked to keep a detailed record of

their expenditure over a period of 14 days, and to provide information about longer-term payments

(such as insurance premiums). Information is also obtained about their income.

time to time using information in the Family Expenditure Survey of the previous year.

Index numbers are a very useful way of summarising a large amount of data in a single series of numbers. They

do, however, have a number of limitations.

7.2 Using the RPI

The RPI is used to measure price inflation. We saw in Section 4 how it can be used to deflate time-related data so

that change in

real values can be measured.

It can also be used for

index linking. For example, pensions can be increased in line with increases in the RPI.

Assessment

focus point

Attention!

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Part B Summarising and analysing data ⏐ 5: Index numbers 161

Question

Index linking

At the start of 20X3, Doris's index-linked pension was $5,200 per year and the RPI was at 360. At the start of the

following year, the RPI had increased to 375. What pension can Doris now expect to receive?

Answe

The percentage increase in the RPI =

valueOriginal

Change

× 100

360

)360375(

−

× 100

= 4.17%

Pension increase = 4.17%

× $5,200

= $216.84

New pension = $(5,200 + 216.84)

= $5,416.84

Chapter Roundup

• An index is a measure, over time, of the average changes in the value (price or quantity) of a group of items.

• An index relative (sometimes just called a relative) is the name given to an index number which measures the

change in a single distinct commodity.

• Index relatives can be calculated using the fixed base method or the chain base method.

• Splicing involves redefining the base year of an index in a particular year and then restating the index values in previous

years.

• The real value of a commodity can only be measured in terms of some 'indicator', such as the rate of inflation

(normally represented by the Retail Prices Index (RPI)).

• Time series deflation is a technique used to obtain a set of index relatives that measure the changes in the real

value of some commodity with respect to some given indicator.

• Composite index numbers cover more than one item.

• Weighting is used to reflect the importance of each item in the index.

• Weighted average of relatives indices are found by calculating indices and then applying weights.

• The retail prices index (RPI) is used to measure price inflation and can be used to deflate data and for index

linking.

162 5: Index numbers ⏐ Part B Summarising and analysing data

Quick Quiz

1 Complete the following equations using the symbols in the box below.

(a)

Price index =

× 100

(b)

Quantity index =

× 100

2 An index relative is the name given to an index number which measures the change in a group of items.

True

False

3 Fixed base method Changes are measured against base period

Chain base method Changes are measured against the previous period

4 In 2006 the retail price index was 198 with 1987 = 100. Convert a weekly wage of $421 back to 1987 prices.

5 The chain base index for an item last year was 130. The price of the item has risen by 10% between last year and

this year.

What is the chain base index for this year?

6 In 2008, a price index based on 1994 = 100 stood at 139. In that year it was rebased at 2008 = 100. By 2010, the

new index stood at 120. For a continuous estimate of price changes since 1994, the new index may be expressed

in terms of the old as (to 1 decimal place)

Answers to Quick Quiz

1 (a) Price index =

× 100

(b) Quantity index =

× 100

2 False. An index relative is an index number which measures the change in a

single distinct commodity.

3 Fixed base method Changes are measured against the base period

Chain base method Changes are measured against the previous period

4 Deflated cash flow = Actual cash flow in given year

year given for number Index

year base for numberIndex

= $421

198

100

= $212.63

5 130 × (1 + 10%)

= 130 × 1.1 = 143

[

]

[

212.

Part B Summarising and analysing data ⏐ 5: Index numbers 163

1994 2008 2010

1994 = 100 100 139

2008 = 100 100 120

Between 2008 and 2010, prices increased by 20%. The price index for 2010 with 1994 as the base year

should also show a 20% increase on the 2008 index of 139.

= 139 × 120% = 139 × 1.2

= 166.8

Now try the questions below from the Exam Question Bank

Question numbers Pages

46-48 334-335

66,8

164 5: Index numbers ⏐ Part B Summarising and analysing data

165

Part C

Probability

166

167

Topic list Syllabus references

1 The concept of probability (B), (i), (1)

2 The rules of probability (B), (ii), (iii), (2), (3)

3 Expected values (B), (iv), (4)

4 Expectation and decision making (B), (v), (vi), (vii), (4), (5)

Probability

Introduction

'The likelihood of rain this afternoon is fifty percent' warns the weather report from your radio

alarm clock. 'There's no chance of you catching that bus' grunts the helpful soul as you puff

up the hill. The headline on your newspaper screams 'Odds of Rainbow Party winning the

election rise to one in four'.

'Likelihood' and 'chance' are expressions used in our everyday lives to denote a level of

uncertainty. Probability, is the mathematical term used when we need to calculate the

likelihood of an event happening.

This chapter will therefore explain various techniques for assessing probability and look at

how it can be applied in business decision making.

168 6: Probability ⏐ Part C Probability

1 The concept of probability

1.1 Introducing probability

Probability is a measure of likelihood and can be stated as a percentage, a ratio, or more usually as a number from

0 to 1.

Consider the following.

• Probability = 0 = impossibility

• Probability = 1 = certainty

• Probability = 1/2 = a 50% chance of something happening

• Probability = 1/4 = a 1 in 4 chance of something happening

1.2 Expressing probabilities

In statistics, probabilities are more commonly expressed as proportions than as percentages. Consider the

following possible outcomes.

Probability as a

Probability as

Possible outcome

percentage

a proportion

15.0

0.150

20.0

0.200

32.5

0.325

7.5

0.075

12.5

0.125

12.5

0.125

100.0

1.000

It is useful to consider how probability can be quantified. A businessman might estimate that if the selling price of

a product is raised by 20p, there would be a 90% probability that demand would fall by 30%, but how would he

have reached his estimate of 90% probability?

1.3 Assessing probabilities

There are several ways of assessing probabilities.

• They may be measurable with mathematical certainty

–

If a coin is tossed, there is a 0.5 probability that it will come down heads, and a 0.5

probability that it will come down tails

– If a die is thrown, there is a one-sixth probability that a 6 will turn up

• They may be measurable from an analysis of past experience

• Probabilities can be estimated from research or surveys

It is important to note that probability is a measure of the likelihood of an event happening in the long run, or over

a large number of times.

The rules of probability in Section 2 will go through in detail how to calculate probabilities in various situations.

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Part C Probability ⏐ 6: Probability 169

2 The rules of probability

2.1 Setting the scene

It is the year 2020 and examiners are extinct. A mighty but completely fair computer churns out examinations that

are equally likely to be easy or difficult. There is no link between the number of questions on each paper, which is

arrived at on a fair basis by the computer, and the standard of the paper. You are about to take five examinations.

2.2 Simple probability

It is vital that the first examination is easy as it covers a subject which you have tried, but unfortunately failed, to

understand. What is the probability that it will be an easy examination?

Obviously (let us hope), the probability of an easy paper is 1/2 (or 50% or 0.5). This reveals a very important

principle (which holds if each result is equally likely).

Probability of achieving the desired result =

Number of ways of achieving desired result

Total number of possible outcomes

Let us apply the principle to our example.

Total number of possible outcomes = 'easy' or 'difficult' = 2

Total number of ways of achieving the desired result (which is 'easy') = 1

The probability of an easy examination, or P(easy examination) = 1/2

2.2.1 Example: Simple probability

Suppose that a dice is rolled. What is the probability that it will show a six?

Solution

P(heads) =

outcomes possible of number Total

resultdesiredachievingof ways of Number

or 16.7% or 0.167

2.3 Venn diagrams

A Venn diagram is a pictorial method of showing probability. We can show all the possible outcomes (E) and the

outcome we are interested in (A).

Formula to

learn