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PROJECT WORK FOR ADDITIONAL

MATHEMATICS 2010

Work 4

Name :Syafiq Phua Bin Fadhlee PhuaClass : 5 Science Bestari

Index number : SH11OO

I/C number : 930526 13 6035

Subject teacher : CIKGU RAMLAH BINTI BAKAR

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Acknowledgement................................................

..

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

....................................................

Introduction

...........................................................

Part

1......................................................................

Part

2......................................................................

Part

3......................................................................

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Then I would like to thank to my teacher, Cikgu Ramlah Binti Bakar 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.

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

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effective learning.

vii. To develop positive attitude towards mathematics.

A 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. The relation between statistics and probability

theory developed rather late, however. In the 19th century, statistics

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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 reasoning

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.

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.

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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,

econometrics, 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

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of mathematics but rather as a distinct, albeit allied, field.

Many universities maintain separate mathematics and

statistics departments. Statistics is also taught in

departments as diverse as psychology, education, and

public health.

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

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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.

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.

2. 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 notcapable 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 2.15 times 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 timeswhile other sectors may have increased

their production onlymarginally.

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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.

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

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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 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 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.

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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 price

index 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

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 index

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used 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 such

measures 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

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

stated 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.

I = index number

Q1 = quantity at specific time

Qo = quantity at base time

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

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=

= Composite index

W = weightage

= index number

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

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Food

Detergent

Question (b)

Data

Category Item

Price (RM)

Mydin Mall Embat Shop Si Come

Food

1.Self raising flour(500g) 2.00 2.50 2.10

2. Sugar(1000g) 1.80 1.90 1.85

3. Butter (250g) 3.50 3.60 3.60

4. Eggs (Grade A) 8.20 8.50 8.40

Total price 15.50 16.50 15.95

Detergent

1. Dishwash 1.80 2.00 1.95

2. Softener 3.90 4.20 4.00

3. Floor cleaner 10.90 11.00 11.30

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4. PowderD/gent 9.50 9.50 9.60

Total price 26.10 26.70 26.85

Stationery

1. Eraser 0.80 0.80 0.70

2. Marker 3.00 3.60 3.90

3. Pencil colour 5.50 5.40 5.30

4. Scissor 2.30 2.30 2.40

Total price 11.60 12.10 12.30

Grand total

53.20 55.30 55.10

Question (c)

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Question (D)

0

1

2

3

4

56

7

8

9

Mydin Mall Embat Shop Si Comel

Food

Self Raising Flour

Sugar

Butter

Eggs

0

2

4

6

8

10

12


Detergent

Dishwash

Softener

Floor Cleaner

Powder D/gent

0

1

2

3

4

5

6


Stationery

Eraser

Marker

Pencil Colour

Scissor

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Based on all the graph in question 1(C) , we can conclude that Mydin Mall offers

the lowest price for their customers. Then followed by Si Comel and Embat Shop. This

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

Si comel offer the normal price for their customer as it does not get special price from

the supplier. While, Embat Shop have to sold the items at the higher price because the

shop buy the items by bulk from Mydin Mall.

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.

Mydin Mall 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 price

than 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. Mydin Mall is a bigmall and it

provides many items that we need in our life. Eventhough Mydin Mall 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.

Si Comel and Embat Shop cannot offer the prices as Mydin Mall because they

are situated outside the urban area like Mydin Mall. So the population of the customer

will not be as many as customer in Mydin Mall. These shops get the supply for their

goods from Mydin Mall. Even they buy by bulk with Mydin Mall but their prices still will

be higher than Mydin Mall. 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

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they not have too much customers, so the rent that they have to pay will influenced the

price of the goods sold.

As a conclusion, there are many factors that affect the price of the goods solds in

a shop. So, we must be a smart customer to ensure we can get the lowest price. The

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

the shops grand total.

50

51

52

53

54

55

56

57

58

Grand Total

Mydin Mall

Si Comel

Embat Shop

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Question (e)

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

0.3742

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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 Mydin Mall to purchase the ingredients for the

butter cakes. This is because Mydin Mall 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, Mydin Mall is located not far from my

school. So it is easier to my friends and I to go there.

Question (b)

Ingredient Quantity per cakePrice in the year 2009

(RM)Price in the year 201

(RM)

Self-raising flour 250 g 0.90 1.00

Sugar 200 g 0.35 0.36

Butter 250 g 3.30 3.50

Eggs (Grade A) 5 eggs (300 g) 1.25 1.37

(i) Calculate Price Index

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Self raising-four

100090 100 11111Sugar

0360

35

100=102.86

Butter

=106.06

Eggs (Grade A)

1371

25

100=109.60

(ii) Composite index

=

=107.74

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.

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(iii)

On 2009, RM 15.00On 2010, suitable price is :

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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(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 =

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

budget 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 Amount in 2009 (RM) Amount in 2010 (RM)

Food 1200.00 1380.00

Games 500.00 500.00

Transportation 300.00 345.00

Decorations 200.00 200.00

Prizes 600.00 600.00

Tents 800.00 920.00

Miscellaneous 400.00 520.00

Table 3

(b) Calculation to find the composite index

ExpenditureAmount in2009 (RM)

Amount in2010 (RM)

Price Index,

Weightage,

W

Food 1200.00 1380.00 115 12

Games 500.00 500.00 100 5

Transportation 300.00 345.00 115 3Decorations 200.00 200.00 100 2

Prizes 600.00 600.00 100 6

Tents 800.00 920.00 115 8

Miscellaneous 400.00 520.00 130 4

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Composite Index

=

=

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.

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

from 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

=111.625

Composite index for the year 2010 to the year 2011

=111.625

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AIR POLLUTION INDEX

Air pollution is the introduction of chemicals, particulate matter, or biological

materials that cause harm or discomfort to humans or other living organisms, or

damages the natural environment into the atmosphere. The atmosphere is a complex

dynamic natural gaseous system that is essential to support life on planet Earth.

Stratospheric ozone depletion due to air pollution has long been recognized as a threat

to human health as well as to the Earth's ecosystems.

The Air Quality Index (AQI) (also known as the Air Pollution Index (API) or

Pollutant Standard Index (PSI) is a number used by government agencies to

characterize the quality of the air at a given location. As the AQI increases, an

increasingly large percentage of the population is likely to experience increasingly

severe adverse health effects. To compute the AQI requires an air pollutant

concentration from a monitor or model. The function used to convert from air pollutant

concentration to AQI varies by pollutant, and is different in different countries. Air

quality index values are divided into ranges, and each range is assigned a descriptor

and a color code. Standardized public health advisories are associated with each AQI

range. An agency might also encourage members of the public to take public

transportation or work from home when AQI levels are high.

Limitations of the AQI

Most air contaminants do not have an associated AQI. Many countries monitor

ground-level ozone, particulates, sulphur dioxide, carbon monoxide and nitrogen

dioxide and calculate air quality indices for these pollutants.

Causes of Poor Air QualityThe AQI can worsen (go up) due to lack of dilution of air emissions by fresh air.

Stagnant air, often caused by an anticyclone or temperature inversion, or other lack of

winds lets air pollution remain in a local area.

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Indices by location

South Korea

The Ministry of Environment of South Korea uses the Comprehensice Air-quality

Index (CAI) to describe the ambient air quality based on health risk of air

pollution. The index aims to help the public easily understand air quality level and

protect the health of people from air pollution. - The CAI has values of 0 through

500, which are divided into six categories. The higher the CAI value, the greater

the level of air pollution. - Of values of the five air pollutants, the highest is the

CAI value.

CAI Description Health Implications

0-50 GoodA level that will not impact patients suffering

from diseases related to air pollution.

51-

100Moderate

A level which may have a meager impact on

patients in case of chronic exposure.

101-

150

Unhealthy for

sensitive

groups

A level that may have harmful impacts on

patients and members of sensitive groups.

151-

250Unhealthy

A level that may have harmful impacts on

patients and members of sensitive groups

(children, aged or weak people), and also

cause the general public unpleasant

feelings.

251-

350Very unhealthy

A level which may have a serious impact on

patients and members of sensitive groups in

case of acute exposure.

351-

500Hazardous

A level which may need to take emergency

measures for patients and members of

sensitive groups and have harmful impacts

on the general public.

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Stock Market Index

A comparison of three major U.S. stock indices: the NASDAQ Composite, Dow Jones Industrial Average,and S&P 500. All three have the same height at March 2007. Notice the large dot-com spike on the

NASDAQ, a result of the large number of tech. companies on that index.

A stock market index is a method of measuring a section of the stock

market. Many indices are cited by news or financial services firms and are used

as benchmarks, to measure the performance of portfolios such as mutual funds.

Types of indices

Stock market indices may be classed in many ways. A 'world' or 'global'

stock market index includes (typically large) companies without regard for where

they are domiciled or traded. Two examples are MSCI World and S&P Global

100.

A national index represents the performance of the stock market of a

given nationand by proxy, reflects investor sentiment on the state of its

economy. The most regularly quoted market indices are national indices

composed of the stocks of large companies listed on a nation's largest stock

exchanges, such as the American S&P 500, the Japanese Nikkei 225, and the

British FTSE 100.

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The concept may be extended well beyond an exchange. The Wilshire

5000 Index, the original total market index, represents the stocks of nearly every

publicly traded company in the United States, including all U.S. stocks traded on

the New York Stock Exchange (but not ADRs or Limited Partnerships), NASDAQ

and American Stock Exchange. Russell Investment Group added to the family of

indices by launching the Russell Global Index.

More specialised indices exist tracking the performance of specific sectors

of the market. Some examples include the Wilshire US REIT which tracks more

than 80 American Real Estate Investment Trusts and the Morgan Stanley

Biotech Index which consists of 36 American firms in the biotechnology industry.

Other indices may track companies of a certain size, a certain type of

management, or even more specialized criteria one index published by Linux

Weekly News tracks stocks of companies that sell products and services based

on the Linux operating environment.

Index versions

Some indices, such as the S&P 500, have multiple versions. These

versions can differ based on how the index components are weighted and on

how dividends are accounted for. For example, there are three versions of the

S&P 500 index: price return, which only considers the price of the components,

total return, which accounts for dividend reinvestment, and net total return, which

accounts for dividend reinvestment after the deduction of a withholding tax. As

another example, the Wilshire 4500 and Wilshire 5000 indices have five versions

each: full capitalization total return, full capitalization price, float-adjusted total

return, float-adjusted price, and equal weight. The difference between the full

capitalization, float-adjusted, and equal weight versions is in how index

components are weighted.

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USES AND IMPORTANCE OF AIR POLLUTION INDEX

AND STOCK MARKET INDEX

As everyone can see, the air pollution index is use by the government to

measure the quality of air index and to detect any pollutants in our countrys air.

This is to ensure the air is clean and safe for us to inhale. Besides that, an early

warning can be given to us if the air pollution is too high for us to get out of our

homes. This warning is given based upon readings and interpretations of the air

index.

As for the stock market index, it is mainly for the business entrepreneurs.

This type of index is used to determine the outcome of a stock market and also

the conclusion of a stock market. The stock market index is important because a

countrys economical state sometimes depend on it.

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After spending countless hours, days and night to finish this project and also sacrificing

my time for chatting and movies in this midyear holiday, there are several things that I can

say...

Additional Mathematics...

From the day I born...

From the day I was able to holding pencil...

From the day I start learning...

And...

From the day I heard your name...

I always thought that you will be my greatest obstacle and rival in excelling in my life...

But after countless of hours...Countless of days...

Countless of nights...

After sacrificing my precious

time just for you...

Sacrificing my play Time..

Sacrificing my Chatting...

Sacrificing my Facebook...

Sacrificing my internet...

Sacrificing my Anime...

Sacrificing my Movies...

I realized something really

important in you...

I really love you...

You are my real friend...

You my partner...

You are my soulmate...

ILOVE U ADDITIONALMATHEMATICS

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CONCLUSION

After doing research, answering questions, drawing graphs and

some problem solving, I saw that the usage of statistics is important

in daily life. It is not just widely used in markets but also in interpreting

the condition of the surrounding like the air or the water. Especially in

conducting an air-pollution survey. In conclusion, statistics is a daily

life necessity. Without it, surveys cant be conducted, the stock

market cant be interpret and many more. So, we should be thankful

of the people who contribute in the idea of statistics.

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