PROJECT

ADDITIONAL MATHEMATIC

2010

INDEX NUMBER

NAMA : MUHAMMAD ASHRAF BIN SHAHARUDIN

KELAS : 5 SCIENCE 1

TEACHER : PN. ONG YOKE BEE

OBJECTIVES

We students taking Additional Mathematics are required to carry out a project work while we are in Form 5.This year the Curriculum Development Division, Ministry of Education has prepared four tasks for us.We are to choose and complete only ONE task based on our area of interest.This project can be done in groups or individually,but each of us are expected to submit an individually written report.Upon completion of the Additional Mathematics Project Work,we are to gain valuable experiences and able to:

Apply and adapt a variety of problem solving strategies to solve routine and non-routine problems;

Experience classroom environments which are challenging, interesting and meaningful and hence improve their thinking skills.

Experience classroom environments where knowledge and skills are applied in meaningful ways in solving real-life problems

Experience classroom environments where expressing ones mathematical thinking,reasoning and communication are highly encouraged and expected

Experience classroom environments that stimulates and enhances effective learning. Acquire effective mathematical communication through oral and writing,and to use the

language of mathematics to express mathematical ideas correctly and precisely Enhance acquisition of mathematical knowledge and skills through problem-solving in

ways that increase interest and confidence Prepare ourselves for the demand of our future undertakings and in workplace Realise that mathematics is an important and powerful tool in solving real-life problems

and hence develop positive attitude towards mathematics. Train ourselves not only to be independent learners but also to collaborate, to

cooperate, and to share knowledge in an engaging and healthy environment Use technology especially the ICT appropriately and effectively Train ourselves to appreciate the intrinsic values of mathematics and to become more

creative and innovative Realize the importance and the beauty of mathematics

FOREWARD

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

project work. Not forgotten my parents for providing everything, such as money, to buy anything

that are related to this project work and their advise, which is the most needed for this project.

Internet, books, computers, and etc. 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 my teacher, MrsPn.

Ong Yoke Bee for guiding me and my friends throughout this project. We had some difficulties

in doing this task, but she taught us patiently until we knew what to do. She tried and tried to

teach us until we understand what we supposed to do with the projectwork.Last but not least, my

friends who were doing this project with me and sharing our ideas. They were helpful that when

we combined and discussed together, we had this task done.

HISTORY OF INDEX NUMBER

          Index numbers are meant to study the change in the effects of such factors which cannot be

measured directly. According to Bowley, “Index numbers are used to measure the changes in some

quantity which we cannot observe directly”. For example, changes in business activity in a country

are not capable of direct measurement but it is possible to study relative changes in business activity

by studying the variations in the values of some such factors which affect business activity, and

which are capable of direct measurement.

            Index numbers are commonly used statistical device for measuring the combined fluctuations

in a group related variables. If we wish to compare the price level of consumer items today with that

prevalent ten years ago, we are not interested in comparing the prices of only one item, but in

comparing some sort of average price levels. We may wish to compare the present agricultural

production or industrial production with that at the time of independence. Here again, we have to

consider all items of production and each item may have undergone a different fractional increase (or

even a decrease). How do we obtain a composite measure? This composite measure is provided by

index numbers which may be defined as a device for combining the variations that have come in

group of related variables over a period of time, with a view to obtain a figure that represents the

‘net’ result of the change in the constitute variables.

    Index numbers may be classified in terms of the variables that they are intended to measure. In

business, different groups of variables in the measurement of which index number techniques are

commonly used are (i) price, (ii) quantity, (iii) value and (iv) business activity. Thus, we have

index of wholesale prices, index of consumer prices, index of industrial output, index of value of

exports and index of business activity, etc. Here we shall be mainly interested in index numbers

of prices showing changes with respect to time, although methods described can be applied to

other cases. In general, the present level of prices is compared with the level of prices in the past.

The present period is called the current period and some period in the past is called the base

period.

Index Numbers:

            Index numbers are statistical measures designed to show changes in a variable or group

of related variables with respect to time, geographic location or other characteristics such as

income, profession, etc. A collection of index numbers for different years, locations, etc., is

sometimes called an index series.

Simple Index Number:

            A simple index number is a number that measures a relative change in a single variable

with respect to a base.

Composite Index Number:

            A composite index number is a number that measures an average relative changes in a

group of relative variables with respect to a base.

Add Maths : Projek Work 4/2010

Part 1(a)

The price of goods sold in shop vary from one shop to another. Shoppers tend to buy goods which are not only reasonably price but also give value for their money.

You are required to carry out a survey on four different items based on the following categories i.e. food,detergent and stationery. The survey should be done in three differnt shop.

(a) Collect picture,newspaper cuttings or photos on item that you have chosen. Design a collage to illustrate the chosen item.

Part 1(b):

Record the item and their prices systematically as such in Table 1. Since item may be differently packed, be sure to use consistent measurements for each item selected so that comparison can be done easily and accurately

The data in the table below is only serve as a REFERENCE to you:

Category Item Tesco Giant The StoreFood Self-rising flour 2.70 3.70 3.30

Sugar 1.80 1.60 1.35Butter 3.60 3.80 4.50Eggs (Gred A) 3.60 2.90 3.00Total Price 11.70 12.00 12.15

Detergent Washing Powder 19.00 21.00 20.50Dish Washer 4.00 3.20 3.10Liquid Bleach 6.00 5.50 4.90Tile Cleaner 10.20 9.80 9.50Total Price 39.20 39.50 38.00

Stationery Pencil(Shaker) 8.90 9.20 8.20Highlighter 3.50 3.40 3.10Permanent Marker

3.50 2.90 3.80

Card Indexing 14.70 15.00 16.00Grand Total 81.50 82.00 82.15

Part 1(c):

Create at least two suitable graphical representations (the use of ICT is encouraged) to compare contrast the price of the item chosen.

Food Detergent Stationery0

5

10

15

20

25

30

35

40

45

TescoGiantThe Store

Food Detergent Stationery0

5

10

15

20

25

30

35

40

45

TescoGiantThe Store

Grand Total

TescoGiantThe Store

Part 1 (d):

Based on the graphical representations that you have constructed in part 1(c), interpret, discuss and draw consclusion. Comment on your findings.

Based on the graphical representations that I have constructed in Part 1(c), it is shown that there are large and small differences among the prices of items in each category between the shops.In the food category, the smallest price difference are of those of sugar, while the highest is the price of eggs. Besides food, detergent also shows a large price difference between its items. Among them is the price of liquid dishwasher and clothes detergent. On the other hand,stationery items doesn’t have any obvious price difference. The graph also show that most of the items that are high priced comes from the Boulevard Supermarket, while the lowest price items come from the Discount Store.The graph 1(d) will show the conclusion of the difference among the shops based upon the shops grand total.

Food Detergent Stationery0

5

10

15

20

25

30

35

40

45

TescoGiantThe Store

Part 1(e):

The item that has a large price difference = washing powder

Mean , X = ∑ x

N =

19+21+20.53

= 1216

= 20.20 (2dp)

Standard deviation = √∑ x2

N−(mean)2

= √ 192−212+20.52

3−( 121

6)

2

=√ 1222.253

−( 1216

)2

=0.8498Reason of difference in price :

Bulk buying from the retailer gives the lower cost resulted in lower relling price. Hypermarket own brands are much cheaper than the expansive brand-don’t involve in

advertiving resulted in lower – selling price Sales and discount offer by the supermarket.

Part 2(a)Every year SMK Indah organises a carnival to raise funds for the school. This year the school plans to install air conditioners in school libary. Last year, during the carnival, your class made and sold butter cakes. Because of the popularity of butter cakes, your class has decided to carry out the same project for this yera carnival.

(a)Suggest a shop from Part 1 which you would go to purchase the ingredients for the butter cakes. State and discuss your reasons for purchasing from the shop you suggested.

Answer : Shop A will be the choice.offer the cheapest pricer.

Part 2(b) :

I. Complete Table 2 with the prices of items found in the shop/supermarket that you have chosen.

Ingredient Quantity per cake Price in 2009(Rm)

Price in 2010(RM)

Price index 2010 based 2009

P20102009

=

p2010p2009

×100

Self-raising flour 250g 0.90 0.675 0.6750.9

×100=75

Sugar 200g 0.35 0.36 0.300.35

×100=102.86

Butter 250g 3.30 3.60 0.363.2

×100=109.10

Eggs (Grade A) 5eggs(300g) 1.25 1.80 1.81.25

×100=144

II. Compoite index = ∑ pi .wi

∑ wi

= 75 (250 )+102.86 (200 )+109.10 (250 )+144 (300)

250+200+250+300

= 109797

1000= 109.797

III. 109.797 = Price of butter cake∈2010Price of butter cake∈2009

×100

= P2010

15×100

Thus, p2010=109.797×15100

=16.469−16.50 (2d . p )

Part 2(c) :i. Find out from reliable sources how to determine suitable capacity of air-conditioner to be

installed based on volume/size of a room.ii. Work in group to estimate the volume of yout school libary. Explain how you arrive at your

answer. Hence, determine the number of air conditioners with the appropriate capacity required for your libary.

iii. If your class intends to sponsor one air conditioner for the school libary, how many butter cakes must your class sell in order to buy the air conditioner?

Reliable sources from the internet from the technician (electrical/home suppliance)

Volume = Area of the libary × Height of the libary determine the shape of the libary from plan of the libary determine how to find the area according to yhe shape of the libary. determine the height. determine the average number of student enter the libary at yhe peak hour check with the data from (a), estimate the number of air conditioner needed

The price of the air conditionerTheunit priceof the butter cake

=thenumber of butter cakethat are required ¿ sell

Part 3As a commite member for the carnivel, you are required to prepared an estimate budget to organise this year’s carnival. The commite has take into consideration the increase in expenditure from the privious year due to inflation. The price of food, trasportation and tents has increased by 15%. The cost of games, prizes and decorations remains the same, where as the cost of miscellaneous items has increased by 30%.

a) Complete Table 3 based on the information given above. Expenditure Amount

in 2009(RM)

Amount in 2010 (RM)

PriceIndex

Weight

Food 1200.00 1.15 × 1200 = 1380 115 12Games 500.00 1 × 500 = 500 100 5

Transportation 300.00 1.15 × 300 = 345 115 9Decorations 200.00 1 × 200 = 200 100 2

Prizes 600.00 1 × 600 = 600 100 6Tents 800.00 1.15 × 800 = 920 115 8Miscellaneous 400.00 1.3 × 400 = 520 130 4

b) Calculate the composite index for the estimated budget of the carnival in the year 2010 based on the year 2009. Comment on your answer.

composite index= εIWiεWi

¿ 446540

¿111.625

c) The change in the composite index for the estimated 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. Determine the composite index of the budget for the year 2011 based on the year 2009.

Given that 20112010

∧2010

2009=111.625

Thus, 20102009

=20112010

×20102009

= (111.625 ) (111.625 )

100

= 124.60 (2 d.p)

FURTHER EXPLONATION

Index numbers are being used in many different daily situations, for example air pollution index, stock market index, gold index and property index.

Obtain information from the internet or other reliable sources on the importance of two different types of index number of your choice. Elaborate the use and the importance of these index numbers in daily life.

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 Quality

The 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.

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.

51-100

ModerateA 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.

151-250

UnhealthyA 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.

251-350

Very unhealthyA level which may have a serious impact on patients and members of sensitive groups in case of acute exposure.

351-500

HazardousA level which may need to take emergency measures for patients and members of sensitive groups and have harmful impacts on the general public.

Malaysia

The air quality in Malaysia is reported as the API or Air Pollution Index. Four of the index's

pollutant components (i.e., carbon monoxide, ozone, nitrogen dioxide and sulfur dioxide) are

reported in PM10 particulate matter is reported in μg/m³.

Unlike the American AQI, the index number can exceed 500. Above 500, a state of emergency is

declared in the reporting area. Usually, this means that non-essential government services are

suspended, and all ports in the affected area closed. There may also be a prohibition on private

sector commercial and industrial activities in the reporting area excluding the food sector.

Stock Market Index

A comparison of three major U.S. stock indices: theNASDAQ Composite, Dow Jones Industrial

Average, andS&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 nation—and 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.

The concept may be extended well beyond an exchange. The Dow Jones Total Stock Market

Index, as its name implies, 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) and most traded on the 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.

The Morgan Stanley Biotech Index, for example, 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.[1] 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.

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 air quality index and to detect any pollutants in our country’s 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 to high for us to get out of our homes. This warning is given based upon readings and interpretations of the air pollution 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 country’s economical state sometimes depend on it.

CONCLUSION

After doing research, answering questions, drawing graphs and some problem solving, I saw that the usage of index number 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 essecities. Without it, surveys can’t be conducted, the stock market can’t be interpret and many more. So ,we should be thankful of the people who contribute in the idea of statistics.

REFLECTION

After spending countless hours, days and night to finish this project and also sacrificing my time video games and magazine in this mid year 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 Computer Games...Sacrificing my Video Games...Sacrificing my Facebook ...Sacrificing my Internet...Sacrificing my Anime...Sacrificing my magazine...I realized something really important in you...

I really love you...You are my real friend...You my partner...You are my soulmate...I LOVE U ADDITIONAL MATHEMATICS...