additional mathematics project work 4 2010-Full version

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AdditionalMathematics

Project Work 4

Lukmanulhakim awaluddin

930423125069

S.m.k agama kota kinabalu

lukman


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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 healthto 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 is the most needed for

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

supported me and encouraged me to complete this task 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 difficulties in doing this task, but she 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 ideas to ensure our task will finish completely.

Last but not least, any party which involved either directly or indirect in

completing this project work. Thank you everyone.


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The aims of carrying out this project work are:

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

problems.

ii. To improve thinking skills.

iii. To promote effective mathematical communication.

iv. To develop mathematical knowledge through problem solving

in a way that increases students interest and confidence.

v. To use the language of mathematics to express mathematical

ideas precisely.

vi. To provide learning environment that stimulates 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 discipline

concerned with the collection, summary, and analysis of data. Today statistics is

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. Therelation 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 least squares, which was invented

by Legendre and Gauss. Early probability theory and statistics was systematized

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 models to advance the new sciences of experimental psychology and

sociology; physical scientists used statistical reasoning and probability models to

advance the new sciences of thermodynamics and statistical mechanics.


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The development of statistical reasoning was closely associated with the

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

of mathematics but an autonomous mathematical science , like computer science

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 computer science .

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 advances in 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,

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 alsotaught in departments as diverse as psychology, education, and public health.


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Index Number

Index numbers are today one of the 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 or other characteristics such as income, profession, etc.

Characteristics of an Index Numbers

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

of the current value 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 number is commonly referred to as an index.

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

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

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

manufactured products index (a part of the 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 sugar in kgs, milk in liters, etc.


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The index number is 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 changes in the group of related variables over aperiod of

time. For example, if the index of industrial production is 215.1 in 1992-93 (base

year 1980-81) it means that the industrial production in that year was up by 2.15times compared to 1980-81. But it does not, however, mean that the net increase

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 of the 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.

2. Helps in policy making

It is widely known that the dearness allowances paid to the employees is linked

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

it is the cost of living index, which forms the basis of many a wages agreement

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

policy making.


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3. Determines purchasing power of the rupee

Usually index numbers are used to determine the purchasing power of the rupee.

Suppose the consumers price index for urban non-manual employees increased

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

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

paise in 1984 its purchasing power is 49.5 paise in 1992.

4. Deflates time series data

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

example, nominal income(income at current prices) can be transformed into realincome(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 of increase in the volume of production as one would have it but because

of increase 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 series real.

Types of index numbers

Three are three types of principal indices. They are:

1. Price Index

The most frequently used form of index numbers is the price index. A priceindex compares charges in price of edible oils. If an attempt is being made to

compare the prices of edible oils this year to the prices of edible oils last year, it

involves, firstly, a comparison of two price situations over time and secondly, the

heterogeneity of the edible oils given the various varieties 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


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Whole Price Index (WPI). Consumer Price Index (CPI) are some of the 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 oils in 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 indexused in this country and elsewhere is the index of industrial production (HP). The

index of industrial production measures the increase or decrease in the level of

industrial 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 suchmeasures changes in net monetary worth. Though the value index enables

comparison of value of a commodity in a year to the value of that commodity in a

base 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

Index numberIs a measure used to show the change of a certain quantity for astated period of time by choosing a specific time as the base year. In general an

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

expressed as a percentage.

H = 1

0100

I = index number

Q1 = quantity at specific time

Qo = quantity at base time

The composite index is the weighted mean for all the items in a certain situation.

= = Composite index

W = weightage

H = index number


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Part 1


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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

different shops. Informations below shows the results from my research.

Question (a)

Picture

Stationery

Food

Detergent


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Question (b)

DataCategory 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 12.15Detergent 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

1214

giant

0

5

10

15

20

25

giant

0

2

4

6

8

10

12

14

16

giant

servay khidmat

Food

S

S

B

E

servay khidmat

Detergent

li

t

servay khidmat

Stationery

pen

hig

per

car

lf Raising Flour

gar

utter

ggs

ashing powder

ish washer

quid bleach

ile cleaner

cil

lighter

menant marker

d indexing


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0

5

10

15

20

25

30

35

40

45

food

0

5

10

15

20

25

30

35

40

food

detergent stationary

detergent stationary

giant

servay

khidmat

giant

servay

khidmat


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Question (D)

Based on all the graph in question 1(C) , we can conclude that giant hypermarketoffers the lowest price for their 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 offer the normal price for their customer as it does not get special price from the

supplier. While, khidmat have to sold the items at the higher price because the shop buy

the items by bulk from Giant.

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

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

the shop, and the rent for the 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 of the 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 of the goods sold. As example the shop with status mall will offer the lowest pricethan the shop with status mini market. The size of the shop also will influenced the

price. When the size of the shop is bigger its mean it can sell many different items in 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 items that we need in our life. Eventhough Giant have to pay rent for the place,

but it not gives too much effects to the price of goods sold as it 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 of the customer will not be as many

as customer in giant. These shops get the supply for their 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 for their customers. They just sold

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

that they have to pay will influenced the price of the goods sold.


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As a conclusion, the

a shop. So, we must be a

graph below will show the

the shops grand total.

81.1

81.2

81.3

81.4

81.5

81.6

81.7

81.8

81.9

82

82.1

82.2

giant

re are many factors that affect the price

smart customer to ensure we can get t

conclusion of the difference among th

servay khidmat

grand total

f the goods solds in

e lowest price. The

shops based upon

grand total


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Question (e)

The item that has large price different among the shops is marker. Mydin Mallsold it at RM 3.00, Si Comel sold it at RM3.90 while Embat Shop sold it at RM 3.60.

Calculate the mean

=

=#$#$".'

%

=20.20 Calculate the standard deviation

= $Or

=

I)$

= # $#($".')% ($# )$

= 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 or

stationery 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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Part2


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Every year my school organises a carnival to raise funds for the 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 out the same project for this years carnival.

Question (a)

From the data in Part 1, I would go to Giant to purchase the ingredients for the butter

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

want to buy. So my class will able to sold the butter cakes at the low price and get some

profits form the sale. Futhermore, giant is located not far from my school. So it is 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 0.36102.86

Butter 250g 3.30 3.60

109.10

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

144


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(i) Calculate Price Index

H = 10 100 Self raising-four

= 1.000.90 100

= 111.11 Sugar

=0

.36

0.35 100=102.86

Butter

= 3.503.30 100=106.06

Eggs (Grade A)

=1

.37

1.25 100=109.60

(ii) Composite index

=

= ('###.##) ("$.) ('#".") (#".")'&' =107.74


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To calculate composite index firstly use 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 :

15 100=107.74 100= 107.74 15

= 1616.10100 =16.20

Thus, the suitable price for the butter cake for the year 2010 is RM 16.20. The

increase in price is also suitable because of the rise in the price of the 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 of volume. If we buy an

air conditioner with 3HP, it is suitable for a room around 81m.

(ii) Estimate the volume of school library

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

Height=3.6mWidth=9.0m

Length=20.12m

Volume of the room=3.6 x 9.0 x 20.12

=651.90m%One unit of air conditioner with 3HP is for 81 m%For651.90m% = '#."#

= 8.048This means our school library needs 8 unit of air conditioner.

(iii) My class intends to sponsor one air conditioner for the school library. The

calculation below is to find how many butter cakes we must sell in order to

buy the air conditioner.

1 unit of 3 HP air conditioner = RM 1800Cost for a cake = RM 6.23Selling price = RM 16.20Profit =RM 16.20- RM6.23

= RM 9.97

Number of cakes to buy 1 unit of air conditioner =

18009.97 = 180.54 = 181 cakes


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As a committee member for the carnival, I am required to prepare an estimatedbudget to organise this years carnival. I has taken into consideration the increases

in expenditur from the previous year due to inflationThe price of food, transportation

and tents has increased by 15%. The cost of games, prizes and decorations remains

the same,whereas the cost of miscellaneous items has increase 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

=

= ##'(#$) #""(') ##'(%)#""($) #""() ##'() #%"(&)(#$'%$&) = 446540 =111.625

The total price for the year 2010 increase by 111.625%. This is because some price

in the year 2009 increased in the year 2010.

(a) The change in the composite index for the estimate budget for the carnivalfrom the year 2009 to the year 2010 is the same as the change from the

year

2010 to the year 2011. Below are the calculation to determine the

composite index of the budget for the year 2011 based on the year 2009.

Composite index for the year 2009 to the year 2010

=111.625Composite index for the year 2010 to the year 2011

=111.625

$"##$"" 100 = $"#"$"" $"##$"#"

$"##

$"" =111.625 111.625

1

100

=124.60


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Further Explorations

History of early price indices

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

earliest reported research in this area came from Welshman Rice Vaughan

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

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

precious metals brought bySpain from the New Worldfrom the effect due

to currency debasement. Vaughan compared labor statutes from his own timeto similar statutes dating back to Edward III. These statutes set wages for

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

Vaughan reasoned that the market for basic labor did not fluctuate much with

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

goods in different time periods, so that a laborer's salary acted as a basket of

goods. Vaughan's analysis indicated that price levels in England had risen six

to eightfold over the preceding century.[1]

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

Englishman William Fleetwoodcreated perhaps the first true price index. An

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

student stood 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 interest in 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 methods to showthat the value of five pounds had changed greatly over the course of 260

years. He argued on behalf of the 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 market value of transactionsin Cin some periodtwould be

where

represents the prevailing price ofc in periodt

represents the quantity ofc sold in periodt

If, across two periods t0

andtn, the same quantities of each good

or service were sold, but under different prices, then

and

would be a reasonable measure of the price of the set in one period relative to

that in the other, and would provide an indexmeasuring relative prices overall,

weighted by quantities sold.

Of course, for any practical purpose, quantities purchased are rarely if ever

identical across any two periods. As such, this is 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 quantities sold from price changes. To see that this is so, consider

what happens if all the prices double between t0andtn while quantities staythe same:Pwill double. Now consider what happens if allthe quantities double between t0 andtn while all the prices stay thesame:Pwill double. In either case the change inPis identical. As such,Pisas much a quantity index as it is a price index.

Various indices have been constructed in an attempt to compensate for thisdifficulty.


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Paasche and Laspeyres price indices

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

index(after the German economistHermann Paasche[pa]) and

the Laspeyres index(after the German economistEtienneLaspeyres[laspejres]).

The Paasche index is computed as

while the Laspeyres index is computed as

wherePis the change in price level, t0 is the base period (usually the firstyear), andtn the period for which the index is computed.

Note that the only difference in the formulas is that the former uses period n

quantities, whereas the latter uses base period (period 0) quantities.

When applied to bundles of individual consumers, a Laspeyres index of 1

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

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

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

consumed the same bundle in the base period as she is consuming in the

current period, given that income has not changed.

Hence, one may think of the Laspeyres index as one where the numeraire is

the bundle of goods using base year prices but current quantities. Similarly,

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

goods using current prices and current quantities as the numeraire.

The Laspeyres index systematically overstates inflation, while the Paascheindex understates it, because the indices do not account for the fact that

consumers typically react to price changes by changing the quantities that

they buy. For example, if prices go up for goodc then, ceteris paribus,quantities of that good should go down.

Fisher index and Marshall-Edgeworth index

A third index, the Marshall-Edgeworth index(named for economistsAlfred

MarshallandFrancis Ysidro Edgeworth), tries to overcome these problems of

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


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A fourth, the Fisher index(after the American economistIrving Fisher), is

calculated as the geometric mean ofPPandPL:

Fisher's index is also known as the ideal price index.

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

the Fisher index that the overstatement and understatement will thus exactly

one cancel the other.

While these indices were introduced to provide overallmeasurementof

relative prices, there is ultimately no way of measuring the imperfections ofany of these indices (Paasche, Laspeyres, Fisher, or Marshall-Edgeworth)

against reality.

Normalizing index numbers

Price indices are represented as index numbers, number values that indicate relative change but not

absolute values (i.e. one price index value can be compared to another or a base, but the numberalone has no meaning). Price indices generally select a base year and make that index value equal to

100. You then express every other year as a percentage of that base year. In our example above,

let's take 2000 as our base year. The value of our index will be 100. The price

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




When an index has been normalized in this manner, the meaning of the number 108, for instance, is

that the total cost for the basket of goods is 4% more in 2001, 8% more in 2002 and 12% more in

2003 than in the base year (in this case, year 2000).

Relative ease of calculating the Laspeyres index

As can be seen from the definitions above, if one already has price and quantity data (or,

alternatively, price and expenditure data) for the base period, then calculating the Laspeyres index for

a new period requires only new price data. In contrast, calculating many other indices (e.g., the

Paasche index) for a new period requires both new price data and new quantity data (or, alternatively,

both new price data and new expenditure data) for each new period. Collecting only new price data is


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often easier than collecting both

index for a new period tends to r

new period.[3]

Calculating indices from e

Sometimes, especially for aggre

data.[4]For these cases, we can

expenditures, rather than quantit

Here is a reformulation for the L

Let be the total expenditu

have an

into our Laspeyres formula as fol

A similar transformation ca

Chained vs non-chaine

So far, in our discussion, we hav

alternative is to take the base pe

period. This can be done with an

index, where tn is the period for

anchors the value of the series:

Each term

answers the question "by what f

When you multiply these all toge

increased since periodt0.

Nonetheless, note that, when ch

prices.

new price data and new quantity data, so calcula

quire less time and effort than calculating these

penditure data

ate data, expenditure data is more readily availa

ormulate the indices in terms of relative prices a

ies.

speyres index:

re on good c in the base period, then (by definiti

d therefore also . We can substi

lows:

be made for any index.

calculations

e always had our price indices relative to some fi

riod for each time period to be the immediately p

y of the above indices, but here's an example wit

hich we wish to calculate the index andt0 is a r

ctor have prices increased between periodtn 1

ther, you get the answer to the question "by what

in indices are in use, the numbers cannot be sai

ting the Laspeyres

other indices for a

ble than quantity

d base year

n) we

ute these values

xed base period. An

eceding time

h the Laspeyres

ference period that

and periodtn".

factor have prices

d to be "in periodt0"


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Index number th

Price index formulas can

properties per se. Severa

proposed in index numbresearch in a list of nine

whereP0 andPn are vec

period while and

1. Identity test:

The identity test ba

quantities remain ian item is multiplie

or, for the later p

2. Proportionality tes

If each price in the

should increase by

3. Invariance to chan

The price index sh

increased by a fact

by another factor. I

quantities and pric

4. Commensurability

The index should

prices and quantiti5. Symmetric treatm

treatment of place)

Reversing the orde

index value. If the i

to the earlier time

going from the earl

ory

be evaluated in terms of their mathe

l different tests of such properties ha

r theory literature. W.E. Diewert sumuch tests for a price index

tors giving prices for a base period a

give quantities for these periods.[5]

sically means that if prices remain th

the same proportion to each other (by the same factor of either, for th

riod) then the index value will be on

:

original period increases by a factor

the factor .

ges in scale test:

uld not change if the prices in both p

or and the quantities in both periods

n other words, the magnitude of the v

s should not affect the price index.

test:

ot be affected by the choice of units

s.nt of time (or, in parity measures, sy

:

r of the time periods should produce

ndex is calculated from the most rece

eriod, it should be the reciprocal of th

ier period to the more recent.

atical

e been

arized past,

d a reference

same and

ach quantity ofe first period,

.

then the index

riods are

re increased

alues of

sed to measure

metric

reciprocal

nt time period

e index found


34/36

6. Symmetric treatment of commodities:

All commodities should have a symmetric effect on the index.

Differentpermutations of the same set of vectors should not change the

index.7. Monotonicity test:

A price index for lower later prices should be lower than a price index

with higher later period prices.

8. Mean value test:

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

the smallest and largest price relatives for all commodities.

9. Circularity test:

Given three ordered periods tm, tn, tr, the price index for

periods tm andtn times the price index for periods tn andtr should be

equivalent to the price index for periods tm andtr.


35/36

Quality change

Price indices often capture changes in price and quantities for goods and services, but they often fail

to account for improvements (or often deteriorations) in the quality of goods and services. Statisticalagencies generally use matched-model price indices, where one model of a particular good is priced

at the same store at regular time intervals. The matched-model method becomes problematic when

statistical agencies try to use this method on goods and services with rapid turnover in quality

features. For instance, computers rapidly improve and a specific model may quickly become obsolete.

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

obsolete item originally used in the index with the new and improved item that replaces it. Statistical

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

The problem discussed above can be represented as attempting to bridge the gap between the pricefor the old item in time t,P(M)t, with the price of the new item in the later time period,P()t + 1.

[7]

The overlap method uses prices collected for both items in both time periods, t and t+1. The price

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

The direct comparison method assumes that the difference in the price of the two items is not due

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

price relative.

The link-to-show-no-change assumes the opposite of the direct comparison method; it assumes

that the entire difference between the two items is due to the change in quality. The price relative

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

The deletion method simply leaves the price relative for the changing item out of the price index.

This is equivalent to using the average of other price relatives in the index as the price relative for

the changing item. Similarly, class mean imputation uses the average price relative for items with

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


36/36

After spending countless

also sacrificing my time

holiday,there are severa

Additional Mathematics...

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OVE ADDITIONAL MATHE

project and

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ATICS..

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