additional mathematics projectwork 4 2010

8/9/2019 additional mathematics projectwork 4 2010

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Additional

Mathematics

Project Work4

/XNPDQXOKDNLPDZDOXGGLQ

6PNDJDPDNRWDNLQDEDOX

lukman


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Content Acknowledgement..................................................

Objectives...............................................................

Introduction ...........................................................

Part 1......................................................................

Part 2......................................................................

Part 3......................................................................

Further Explorations...............................................

Reflections............................................................

Conclusion..............................................................


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AcknowledgementFirst of all, I would like to say Alhamdulillah, for giving me the strength and health

to do this project work and finish it on time.

Not forgotten to my parents for providing everything, such as money, to buy

anything that are related to this project work, their advise, which isthe most needed for

this project and facilities such as internet, books, computers and all that. They also

supported me and encouraged me to complete thistask so that I will not procrastinate in

doing it.

Then I would like to thank to my teacher, Mdm Fazilah for guiding me throughout

this project. Even I had some difficultiesin doing thistask, butshe taught me patiently

until we knew what to do. She tried and tried to teach me until I understand what Im

supposed to do with the project work.

Besides that, my friends who always supporting me. Even this project is

individually but we are cooperated doing this project especially in disscussion and

sharing ideasto ensure ourtask will finish completely.

Last but not least, any party which involved either directly or indirect incompleting this project work. Thank you everyone.


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ObjectivesThe aims of carrying outthis project work are:

i. To apply and adapt a variety of problem-solving strategiesto solve

problems.

ii. To improve thinking skills.

iii. To promote effective mathematical communication.

iv. To develop mathematical knowledge through problem solving

in a way thatincreasesstudents interest and confidence.

v. To use the language of mathematicsto express mathematical

ideas precisely.

vi. To provide learning environmentthatstimulates and enhances

effective learning.

vii. To develop positive attitude towards mathematics.


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IntroductionA Brief History Of Statistic

By the 18th century, the term " statistics" designated the systematic

collection of demographic and economic data by states. In the early 19th

century, the meaning of "statistics" broadened, then including the disciplineconcerned with the collection, summary, and analysis of data. Today statisticsis

widely employed in government, business, and all the sciences. Electronic

computers have expedited statistical computation, and have allowed statisticians

to develop "computer -intensive" methods.

The term "mathematical statistics" designates the mathematical theories

of probability and statistical inference, which are used in statistical practice. The

relation between statistics and probability theory developed rather late, however.

In the 19th century, statistics increasingly used probability theory, whose initial

results were found in the17th and 18th centuries, particularly in the analysis of

games of chance (gambling). By 1800, astronomy used probability models and

statistical theories, particularly the method of leastsquares, which wasinvented


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by Legendre and Gauss. Early probability theory and statistics wassystematized

and extended by Laplace; following Laplace, probability and statistics have been

in continual development.

In the 19th century, social scientists used statistical r easoning and

probability modelsto advance the new sciences of experimental psychology and

sociology; physical scientists used statistical reasoning and probability modelsto

advance the new sciences ofthermodynamics and statistical mechanics.

The development ofstatistical reasoning was closely associated with the

development of inductive logic and the scientific method. Statisticsis not a field

of mathematics but an autonomous mathematical science , like computerscience

or operations research. Unlike mathematics, statistics had its origins in public

administration and maintains a special concern with demography and economics.

Being concerned with the scientific method and inductive logic, statistical theory

has close association with the philosophy of science ; with its emphasis on

learning from data and making best predictions, statistics has great overlap with

the decision science and microeconomics. With its concerns with data, statistics

has overlap with information science and computerscience .

Statistics Today

During the 20th century, the creation of precise instruments for

agricultural research, public health concerns (epidemiology, biostatistics,

etc.),industrial quality control, and economic and social purposes (unemployment

rate, econometry, etc.) necessitated substantial advancesin statistical practices.

Today the use of statistics has broadened far beyond its origins.

Individuals and organizations use statistics to understand data and make

informed decisions throughout the natural and social sciences, medicine,


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business, and other areas. Statistics is generally regarded not as a subfield of

mathematics but rather as a distinct, albeit allied, field. Many universities

maintain separate mathematics and stati stics departments. Statistics is also

taughtin departments as diverse as psychology, education, and public health.

Index Number

Index numbers are today one ofthe most widely used statistical indicators.

Generally used to indicate the state of the economy, index numbers are aptly

called barometers of economic activity. Index numbers are used in comparing

production, sales or changes exports or imports over a certain period of time.

The role-played by index numbers in Indian trade and industry is impossible to

ignore. It is a very well known fact that the wage contracts of workers in our

country are tied to the cost of living index numbers.

By definition, an index number is a statistical measure designed to show

changes in a variable or a group or related variables with respect to time,

geographic location orothercharacteristicssuch asincome, profession, etc.

Characteristics of anIndex Numbers

1. These are expressed as a percentage: Index numberis calculated as a ratio

of the currentvalue to a base value and expressed as a percentage. It must be

clearly understood that the index number for the base year is always 100. An

index numberis commonly referred to as an index.


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2. Index numbers are specialized averages: An index number is an average

with a difference. An index numberis used for purposes of comparison in cases

where the series being compared could be expressed in different units i.e. a

manufactured productsindex (a part ofthe whole sale price index) is constructed

using items like Dairy Products, Sugar, Edible Oils, Tea and Coffee, etc. These

items naturally are expressed in different units like sugarin kgs, milk in liters, etc.

The index numberis obtained as a result of an average of all these items, which

are expressed in different units. On the other hand, average is a single figure

representing a group expressed in the same units.

3. Index numbers measures changes that are not directly measurable: An

index number is used for measuring the magnitude of changes in such

phenomenon, which are not capable of direct measurement. Index numbers

essentiallycapture the changesin the group ofrelated variables over aperiod of

time. For example, ifthe index ofindustrial production is 215.1 in 1992-93 (base

year 1980-81) it meansthatthe industrial production in that year was up by 2.15

times compared to 1980-81. Butit does not, however, mean thatthenetincrease

in the index reflects an equivalent increase in industrial production in all sectors

of the industry. Some sectorsmight have increased their production more than

2.15 times while other sectors may have increased their production only

marginally.

Uses of index numbers

1. Establishes trends

Index numbers when analyzed reveal a general trend ofthe phenomenon under

study. For eg. Index numbers of unemployment of the country not only reflects

the trends in the phenomenon but are useful in determining factors leading to

unemployment.


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2. Helps in policy making

It is widely known thatthe dearness allowances paid to the employeesis linked

to the cost of living index, generally the consumer price index. From time to time

itisthe cost of l iving index, which formsthe basis of many a wages agreement

between the employees union and the employer. Thus index numbers guide

policy making.

3. Determines purchasing power of the rupee

Usually index numbers are used to determine the purchasing powerofthe rupee.

Suppose the consumers price index for urban non-manual employeesincreased

from 100 in 1984 to 202 in 1992, the real purchasing power ofthe rupee can be

found out as follows: 100/202=0.495 It indicates that if rupee was worth 100

paise in 1984 its purchasing poweris 49.5 paise in 1992.

4. Deflates time series data

Index numbers play a vital role in adjusting the original data to reflectreality. For

example, nominal income(income at current prices) can be transformed into real

income(reflecting the actual purchasing power) by using income deflators.

Similarly, assume that industrial production is represented in value terms as a

product of volume of production and price. If the subsequent years industrial

production were to be higher by 20% in value, the increase may not be as a

result ofincrease in the volume of production as one would have it but because

ofincrease in the price. The inflation which has caused the increase in the series

can be eliminated by the usage of an appropriate price index and thus making

the seriesreal.

Types of index numbers

Three are three types of principal indices. They are:

1. Price Index


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The most frequently used form ofindex numbersisthe price index. A price

index compares charges in price of edible oils. If an attempt is being made to

compare the prices of edible oilsthis yearto the prices of edible oils last year, it

involves, firstly, a comparison oftwo price situations overtime and secondly, the

heterogeneity ofthe edible oils given the variousvarieties of oils. By constructing

a price index number, we are summarizing the price movements of each type of

oil in this group of edible oils into a single number called the price index. The

Whole Price Index (WPI). Consumer Price Index (CPI) are some ofthe popularly

used price indices.

2. Quantity Index

A quantity index measures the changes in quantity from one period to

another. If in the above example, instead of the price of edible oils, we are

interested in the quantum of production of edible oilsin those years, then we are

comparing quantities in two different years or over a period of time. It is the

quantity index that needs to be constructed here. The popular quantity index

used in this country and elsewhere isthe index ofindustrial production (HP). The

index of industrial production measures the increase or decrease in the level ofindustrial production in a given period compared to some base period.

3. Value Index

The value index is a combination index. It combines price and quantity

changes to present a more spatial comparison. The value index as such

measures changes in net monetary worth. Though the value index enables

comparison ofvalue of a commodity in a yearto the value ofthat commodity in abase year, it has limited use. Usually value index is used in sales, inventories,

foreign trade, etc. Its limited use is owing to the inability of the value index to

distinguish the effects of price and quantity separately.


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Calculating index number

y Index numberIs a measure used to show the change of a certain quantity for a

stated period oftime by choosing a specific time asthe base year. In general an

index number is the comparison of a quantity at two different times and is

expressed as a percentage.

I = index number

Q1

= quantity atspecific time

Qo

= quantity at base time

y The composite index isthe weighted mean forall the itemsin a certain situation.

=

= Composite index

W = weightage


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= index number


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3DUW


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The prices of good sold in shops are vary from one shop to another. Shoppers

tend to buy goods which are not only reasonably priced but also give value for their

money. I had carried out a survey on four different items based on the following

categories which is food, detergent and stationery. The survey was done in three

differentshops. Informations below showsthe results from my research.

Question (a)

Picture

Stationery

Food


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Detergent

Question (b)

Data

Category Item Price

Giant Servay khidmat

Food 1.self-raising flour 2.70 3.70 3.30

2.sugar 1.80 1.60 1.35

3.butter 3.60 2.90 3.00

4.Eggs(grade A) 3.60 2.90 3.00

Total price 11.70 12.00 12.15

Detergent 1.Washing powder 19.00 21.00 20.50

2.dish washer 4.00 3.20 2.10

3.liquid bleach 6.00 5.50 4.90

4.tile cleaner 10.20 9.80 9.50

Total price 39.20 39.50 38.00

Stationary 1.pencil(shaker) 8.90 9.20 8.20

2.highlighter 3.50 2.90 3.80

3.permenent marker 3.50 2.90 3.80

4.card indexing 14.70 15.00 16.00

Total price 30.60 30.50 32.00

GRAND TOTAL 81.50 82.00 82.15


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0

2

4

6

8

10

12

14

giant servay khidmat

Food

Self Raising Flour

Sugar

Butter

Eggs

0

5

10

15

20

25


Detergent

washing powder

dish washer

liquid bleach

tile cleaner


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0

2

4

6

8

10

12

14

16


Stationery

pencil

highlighter

permenant marker

card indexing

0

5

10

15

20

25

30

35

40

45

food detergent stationary

giant

servay

khidmat


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Question (D)

Based on all the graph in question 1(C) , we can conclude that giant hypermarket

offersthe lowest price fortheir customers. Then followed by servayl and Khidmat. This

is because the supplier of the giant gives the special price for it as it buy by bulk.

servay offerthe normal price fortheir customer asit does not getspecial price from the

supplier. While, khidmat have to sold the items atthe higherprice because the shop buy

the items by bulk from Giant.

0

5

10

15

20

25

30

35

40

food detergent stationary

giant

servay

khidmat


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Other factors that influenced the prices of goods in the shops is such as the

location ofthe shop, the population ofthe customers, the status ofthe shop, the size of

the shop, and the rent forthe shop.

Giant can offer the lowest price because it is situated at stratergic place so

indirectly this factor can attract customer buy at the mall. When there are many

customers, the demand ofthe items will be high and the mall can buy by bulk directly

with the supplier to get the special price. The status of the shop also influenced the

price ofthe goodssold. As example the shop with status mall will offerthe lowest price

than the shop with status mini market. The size of the shop also will influenced the

price. When the size ofthe shop is biggerits mean it can sell many differentitemsin the

shop. Indirectly the shop will known as one stop center and it will attract many

customers as the people nowadays are very busy. Giant is a bigmall and it provides

many itemsthat we need in our life. Eventhough Giant have to pay rent forthe place,

butit not givestoo much effectsto the price of goodssold asit has many buyers.

Servay and khidmat cannot offer the prices as giant because they are situated

outside the urban area like giant . So the population ofthe customer will not be as many

as customerin giant. These shops getthe supply fortheir goods from giant. Even they

buy by bulk with giant but their prices still will be higher than giant. The size of these

shop also small and cannot provide too much goods fortheir customers. They justsold

basic needed for their customers. As they not have too much customers, so the rent

thatthey have to pay will influenced the price ofthe goodssold.

As a conclusion, there are many factorsthat affectthe price ofthe goodssoldsin

a shop. So, we must be a smart customerto ensure we can getthe lowest price. The

graph below will show the conclusion of the difference among the shops based upon

the shops grand total.


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Question (e)

81.1

81.2

81.3

81.4

81.5

81.6

81.7

81.8

81.9

8282.1

82.2


grand total

grand total


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The item that has large price different among the shops is marker. Mydin Mall

sold it at RM 3.00, Si Comel sold it at RM3.90 while Embat Shop sold it at RM 3.60.

Calculate the mean

Calculate the standard deviation

Or

0.8498

The difference of the price of the marker in these three shops is maybe due to the

price given by the supplier to the shops. giant can sold it at lowest prices because the

demand of the buyers for the the item is high so it can buy by bulk with the supplier. So the

shop can get the special price. The demand of the item in servay and Khidmat are low. This

is because the customers are more interested to buy the stationery items in mall orstationery shops as there are more options to choose. So servay and khidmat cannot buy by

bulk the stationery items with their supplier.


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3DUW


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Every yearmy school organises a carnival to raise funds forthe school. This year

my school plans to install air conditioners in the school library. Last year, during the

carnival, my class made and sold butter cakes. Because of the popularity of butter

cakes, my class has decided to carry outthe same project forthis years carnival.

Question (a)

From the data in Part 1, I would go to Giantto purchase the ingredients forthe butter

cakes. This is because giant offers the lowest price among the shops for the items I

wantto buy. So my class will able to sold the buttercakes atthe low price and getsome

profits form the sale. Futhermore, giantis located not far from my school. So itis easier

to my friends and I to go there.

Ingredient Quantity

per cake

Price in

2009 (Rm)

Price in

2010 (Rm)

Price index 2010 based 2009

Self-raising flour 250g 0.90 0.675

75

Sugar 200g 0.35 0.36

102.86

Butter 250g 3.30 3.60109.10

Eggs(grade A) 5 (300g) 1.20 1.80

144

(i) Calculate Price Index


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Self raising-four

100090 100

11111 Sugar

036035 100

=102.86

Butter

=106.06

Eggs (Grade A)

137125 100

=109.60

(ii) Composite index

=

=107.74


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To calculate composite index firstly u se the formula of composite index. Get

the value for the formula. Lets quantity per cake be as weightage, W. Obtain the

price index from the calculation in question (i). Then, calculate by using the

calculator.

(iii)

On 2009, RM 15.00

On 2010, suitable price is :

Thus, the suitable price forthe buttercake forthe year2010 is RM 16.20. The

increase in price is also suitable becaus e ofthe rise in the price ofthe ingredients.


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Question (c)

(i) To determine suitable capacity of air conditioner to be installed based on

volume/ size of a room

For common usage, air conditioner is rated according to horse power(1HP), which is approximately 700W to 1000W of electrical power. It is

suitable for a room size 1000ft which is around 27m ofvolume. If we buy an

airconditionerwith 3HP, itissuitabl e fora room around 81m.

(ii) Estimate the volume of school library

By using a measuring tape, the dimension forthe library is:

Height=3.6m

Width=9.0m

Length=20.12m

Volume ofthe room=3.6 x 9.0 x 20.12

=651.90One unit of airconditionerwith 3HP is for81 For

8.048This means ourschool library needs 8 unit of airconditioner.

(iii) My classintendsto sponsorone airconditioner forthe school library. The

calculation below isto find how many buttercakes we mustsell in orderto

buy the airconditioner.

1 unit of 3 HP airconditioner= RM 1800Cost fora cake = RM 6.23Selling price = RM 16.20Profit =RM 16.20- RM6.23

= RM 9.97

Numberof cakesto buy 1 unit of airconditioner=


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3DUW


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As a committee member forthe carnival, I am required to prepare an estimated

budgetto organise this years carnival. I hastaken into consideration the increases

in expenditur from the previous yeardue to inflation The price of food, transportation

and tents hasincreased by 15%. The cost of games, prizes and decorationsremains

the same,whereasthe cost of m iscellaneousitems hasincrease by 30%.

(a)Table 3 has been completed based on the above information .

Expenditure Ammount in 2009

(RM)

Amount in 2010

(RM)

Index Weightage

Food 1200 1.15 x 1200 =1380 115 12

Games 500 1 x 500 =500 100 5

Transportation 1300 1.15 x 1300 =345 115 3

Decoration 200 1 x 200 =200 100 2

Prizes 600 1 x 600 =600 100 6

Tonts 800 1.15 x800 =920 115 8

miscellaneous 400 1.3 x400 =520 130 4


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Composite Index

=

=

The total price forthe year2010 increase by 111.625%. Thisis because some price

in the year2009 increased in the year2010.

(a) The change in the composite index forthe estimate budget for the carnival

from the year 2009 to the year 2010 isthe same as the change from the

year

2010 to the year 2011. Below are the calculation to d etermine the

composite index ofthe budget forthe year2011 based on the year2009.

Composite index forthe year2009 to the year2010

=111.625

Composite index forthe year2010 to the year2011

=111.625


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Further Explorations

History of early price indices

No clearconsensus has emerged on who created the first price index. The

earliestreported research in this area came from Welshman Rice Vaughan

who examined price level change in his 1675 book A Discourse of Coin and

Coinage. Vaughan wanted to separate the inflationary impact ofthe influx of

precious metals brought by Spain from the New World from the effect due

to currency debasement. Vaughan compared laborstatutes from his own time

to similarstatutes dating back to Edward III. These statutesset wages for

certain tasks and provided a good record ofthe change in wage levels.

Vaughan reasoned thatthe market forbasic labord id not fluctuate much with

time and that a basic laborerssalary would probably buy the same amount of

goodsin differenttime periods, so that a laborer'ssalary acted as a basket of

goods. Vaughan's analysisindicated that price levelsin England had ris en six

to eightfold overthe preceding century. [1]

While Vaughan can be considered a forerunnerof price index research, hisanalysis did not actually involve calculating an index.[1] In 1707

Englishman William Fleetwood created perhapsthe firsttrue price index. An

Oxford student asked Fleetwood to help show how prices had changed. The

studentstood to lose his fellowship since a fifteenth century stipulation barred

students with annual incomes over five pounds from receiving a fellowship.

Fleetwood, who already had an interestin price change, had collected a large

amount of price data going back hundreds of years. Fleetwood proposed an

index consisting of averaged price relatives and used his methodsto show

thatthe value of five pounds had changed grea tly overthe course of 260years. He argued on behalf ofthe Oxford students and published his findings

anonymously in a volume entitled Chronicon Preciosum.[2]


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Formal calculation

Further information: List of price index formulas

Given a setCof goods and services, the total marketvalue oftransactions

in Cin some period twould be

where

representsthe prevailing price ofcin period t

representsthe quantity ofcsold in period t

If, acrosstwo periodst0 and tn, the same quantities of each goodorservice were sold, but underdifferent prices, then

and

would be a reasonable measure ofthe price ofthe setin one period relative to

thatin the other, and would provide an index measuring relative prices overall,

weighted by quantitiessold.

Of course, forany practical purpose, quantities purchased are rarely if ever

identical across any two periods. Assuch, thisis not a very practical index

formula.

One might be tempted to modify the formula slightly to

This new index, however, doesn't do anything to distinguish growth or

reduction in quantitiessold from price changes. To see thatthi sisso, consider

what happensif all the prices double between t0and tn while quantitiesstay

the same: Pwill double. Now considerwhat happensif all

the quantities double between t0 and tn while all thepricesstay the

same: Pwill double. In either case the change in Pisidentical. Assuch, Pis

as much a quantityindex asitis apriceindex.


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Variousindices have been constructed in an attemptto compensate forthis

difficulty.

Paasche and Laspeyres price indices

The two most basic formulas used to calculate price indices are the Paasche

index (afterthe German economist Hermann Paasche[pa]) and

the Laspeyres index (afterthe German economistEtienne

Laspeyres[laspejres]).

The Paasche index is computed as

while the Laspeyresindex is computed as

where Pisthe change in price level, t0isthe base period (usually the first

year), and tnthe period forwhich the index is computed.

Note thatthe only difference in the formulasisthatthe formeruses period n

quantities, whereasthe latteruses base period (period 0) quantities.

When applied to bundles ofindividual consumers, a Laspeyresindex of 1

would state that an agentin the current period can afford to buy the same

bundle as he consumed in the previous period, given thatincome has not

changed; a Paasche index of 1 would state that an agent could have

consumed the same bundle in the base period asshe is consuming in the

current period, given thatincome has not changed.

Hence, one may think ofthe Laspeyresindex as one where the numeraireis

the bundle of goods using base yearprices but currentquantities. Similarly,

the Paasche index can be thought of as a price index taking the bundle of

goods using current prices and currentquantities asthe numeraire.

The Laspeyresindex systematically overstatesinflation, while the Paasche

index understatesit, because the indices do not account forthe factthatconsumerstypically reactto price changes by changing the quantitiesthat

they buy. Forexample, if prices go up forgood cthen, ceteris paribus,

quantities ofthat good should go down.

Fisher index and Marshall-Edgeworth index


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A third index, the Marshall-Edgeworth index (named foreconomistsAlfred

Marshall and Francis Ysidro Edgeworth), triesto overcome these problems of

under- and overstatement by using the arithmethic means ofthe quantities:

A fourth, the Fisher index (afterthe American economist Irving Fisher), is

calculated asthe geometric mean ofPPand PL:

Fisher'sindex is also known asthe ideal price index.

However, there is no guarantee with eitherthe Marshall -Edgeworth index or

the Fisherindex thatthe overstatement and understatement will thus exactly

one cancel the other.

While these indices were introduced to provide overall measurement of

relative prices, there is ultimately no way of measuring the imperfections of

any ofthese indices (Paasche, Laspeyres, Fisher, orMarshall -Edgeworth)

againstreality.

Normalizing index numbers

Price indices are represented asindex numbers, numbervaluesthatindicate relative change but not

absolute values (i.e. one price index value can be compared to anotherora base, butthe number

alone has no meaning). Price indices generally select a base yearand make thatindex value equal to

100. You then express every otheryearas a percentage ofthat base year. In ourexample above,

let'stake 2000 as ourbase year. The value of ourindex will be 100. The price

2000: original index value was $2.50; $2.50/$2.50 = 100%, so ournew index value is 100




When an index has been normalized in this manner, the meaning ofthe number108, forinstance, is

thatthe total cost forthe basket of goodsis 4% more in 2001, 8% more in 2002 and 12% more in

2003 than in the base year(in this case, year2000).

Relative ease of calculating the Laspeyres index


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As

s

fr

the

efinitions

o

e, ifone

lreadyhas

riceand q

antitydataor,

alternati

ely,

riceandexpendituredata) forthe

aseperiod, thencalculatingthe

aspeyres index for

anewperiod requires onlynewpricedata. Incontrast, calculating

anyotherindices

e.g., the

Paascheindex) foranewperiod requires

othnewpricedataandnew quantitydataor, alternati

ely,

othnewpricedataandnewexpendituredata) foreachnewperiod.

ollectingonlynewpricedataisofteneasierthancollecting

othnewpricedataandnew quantitydata, socalculatingthe

aspeyres

index foranewperiodtends to require less timeandeffortthancalculatingtheseotherindices fora

newperiod.[

]

Cal lati i i m iture ata

Sometimes, especially foraggregatedata,expendituredatais more readilya

ailablethan quantity

data.[

] Forthesecases, wecan formulatetheindices interms of relati

eprices andbaseyear

expenditures, ratherthan quantities.

ereis a reformulation forthe

aspeyres index:

et bethetotal expenditureongoodcinthebaseperiod, then

bydefinition)we

ha

e andthereforealso .

ecan substitutethese values

intoour

aspeyres formulaas follows:

A similartransformationcanbemade foranyindex.

Chai ed hai ed al ulati

So far, inourdiscussion, wehavealways hadourpriceindices relativeto some fixedbaseperiod. An

alternativeis totakethebaseperiod foreachtimeperiodtobetheimmediatelyprecedingtime

period.!

his canbedonewithanyoftheaboveindices, buthere"s anexamplewiththe

#

aspeyres

index, wheret$ is theperiod forwhichwewishtocalculatetheindexandt0is a referenceperiodthat

anchors the valueofthe series:

Eachterm


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answers the question% bywhat factorhaveprices increasedbetweenperiodt&

1andperiodt'(

.

)

henyoumultiplytheseall together, yougettheanswertothe question(

bywhat factorhaveprices

increased sinceperiodt0.

0onetheless, notethat, whenchainindices areinuse, thenumbers cannotbe saidtobe

1inperiodt0"

prices.

Indexnumbertheory

Priceindex formulas canbeevaluatedinterms oftheirmathematical

properties per 2 e. Several differenttests of suchproperties havebeen

proposedinindexnumbertheory literature. .E. Diewert summarizedpast

researchina listofnine suchtests forapriceindex ,

whereP0andPnare vectors givingprices forabaseperiodanda referenceperiodwhile and give quantities fortheseperiods. [ 3 ]

. Identitytest:

heidentitytestbasicallymeans thatifprices remainthe sameand

quantities remaininthe sameproportiontoeachother each quantityof

anitemis multipliedbythe same factorofeither, forthe firstperiod,

or, forthe laterperiod)thentheindex valuewill beone.

. Proportionalitytest:

Ifeachpriceintheoriginal periodincreases bya factor thentheindex

shouldincreasebythe factor .

. Invariancetochanges in scaletest:

hepriceindex shouldnotchangeiftheprices inbothperiods are

increasedbya factorandthe quantities inbothperiods areincreased

byanother factor. Inotherwords, themagnitudeofthe values ofquantities andprices shouldnotaffectthepriceindex.

. ommensurabilitytest:

heindex shouldnotbeaffectedbythechoiceofunits usedtomeasure

prices and quantities.


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5. Symmetric treatment oftime (or, in parity measures, symmetric

treatment of place):

Reversing the orderofthe time periodsshould produce a reciprocalindex value. Ifthe index is calculated from the mostrecenttime period

to the earliertime period, itshould be the reciprocal ofthe i ndex found

going from the earlierperiod to the more recent.

6. Symmetric treatment of commodities:

All commoditiesshould have a symmetric effect on the index.

Differentpermutations ofthe same set ofvectorsshould not change the

index.

7. Monotonicity test:

A price index for lower laterpricesshould be lowerthan a price index

with higher laterperiod prices.

8. Mean value test:

The overall price relative implied by the price index should be between

the smallest and largest price relatives forall commodities.

9. Circularity test:

Given three ordered periodstm, tn, tr, the price index for

periodstm and tntimesthe price index forperiods tn and trshould be

equivalentto the price index forperiodstm and tr.


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Quality change

Price indices often capture changesin price and quantities forgoods and services, butthey often fail

to account forimprovements (oroften deteriorations) in the quality of goods and services. Statistical

agencies generally use matched-modelprice indices, where one model of a particulargood is priced

atthe same store atregulartime intervals. The matched-model method becomes problematic when

statistical agenciestry to use this method on goods and services with rapid turnoverin quality

features. Forinstance, computersrapidly improve and a specific model may quickly become obsolete.

Statisticians constructing matched-model price indices must decide how to compare the price ofthe

obsolete item originally used in the index with the new and improved item thatreplacesit. Statistical

agencies use several different methodsto make such price comparisons.[6]

The problem discussed above can be represented as attempting to bridge the gap between the price

forthe old item in time t, P(M)t, with the price ofthe new item in the latertime period,P(N)t+ 1.[7]

The overlap methoduses prices collected forboth itemsin both time periods, t and t+1. The price

relative P(N)t+ 1/P(N)tis used.

The direct comparison methodassumesthatthe difference in the price ofthe two itemsis not due

to quality change, so the entire price difference is used in the index. P(N)t+ 1/P(M)tis used asthe

price relative.

The link-to-show-no-change assumesthe opposite ofthe direct comparison method; it assumes

thatthe entire difference between the two itemsis due to the change in quality. The price relative

based on link-to-show-no-change is 1.[8]

The deletion methodsimply leavesthe price relative forthe changing item out ofthe price index.

Thisis equivalentto using the average of otherprice relativesin the index asthe price relative for

the changing item. Similarly, class meanimputation usesthe average price relative foritems with

similarcharacteristics (physical, geographic, economic, etc.) to M and N. [9]


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Reflection

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additional mathematics projectwork 4 2010