PROJECT WORK FOR ADDITIONAL

MATHEMATHICS 2010

CURRICULUM DEVELOPMENT DIVISION

MINISTRY OF EDUCATION MALAYSIA

PROJECT WORK 4

STATISTICS

NAME : ABDUL MAJID BIN ABD AZIZ

CLASS : 5 SCIENCE 1

IC NUMBER : 930206-08-5689

SUBJECT TEACHER : SIR RAFHAN BIN AHMAD

SEKOLAH MENENGAH RAJA DR NAZRIN SHAH

KG GAJAH

INTRODUCTION

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,

and I gladly choose to do this individually.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

APPRECIATION

Alhamdullilah,thank you to Allah for giving the will to me to complete

this Additional Mathematics project.Secondly, I would like to thank the

principle of Sekolah Menengah Raja Dr. Nazrin Shah, Mr. Hj. Omar Bin

Bakar A.M.P,P.P.T for giving me the permission to do my this Additional

Mathematics Project Work. I also like to thank my Additional

Mathematics teacher, Sir Rafhan for the guide and giving useful and

important information for me to complete this project work.

Besides that, I would like to thank my parents for their support and

encouragement. Lastly, a special thanks to all my friends for their help

and cooperation in searching for information and completing this

project work.

A BRIEF HISTORY OF STATISTICS 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 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.

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 statistics departments. Statistics is also

taught in departments as diverse as psychology, education, and public

health.

PART 1 The prices of goods sold in shops vary from one shop to

another.Shoppers tend to buy goods which are not only

reasonably priced 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 different shops.

QUESTION

a) Collect pictures,newspaper cuttings or photos on items that you

have chosen.Design a collage to illustrate the chosen items

Answer: FOODS :

DETERGENTS :

STATIONARY :

(b) Record the items and their prices systematically as in Table

1.Since items maybe differently packed,be sure to use consistent

measurements for each item selected so that comparison can be

done easily and accurately.

Answer:

CATEGORY ITEM PRICE(RM)

KOOP SARJANA

KOOP IUI

KIOSK PUM

FOOD 1.SELF-RAISING FLOUR (1000 g)

4.00 3.70 3.60

2.SUGAR (1000g) 2.00 1.90 1.80

3.BUTTER (250g) 4.70 4.50 4.30

4.EGGS (GRADE A) 1 DOZEN

5.90 5.50 5.00

TOTAL PRICE 16.60 15.60 14.70

DETERGENT 1.SOAP (3 BARS) 3.20 3.00 2.80

2.LIQUID DISHWASHER (1000ml)

4.29 3.90 3.20

3.CLOTHES DETERGENT (3KG)

18.90 17.00 16.50

4.TOILET CLEANER (500ml)

5.50 5.50 5.50

TOTAL PRICE 31.89 29.40 28.00

STATIONERY 1.SHARPENER 1.50 1.30 1.00

2.PENCIL (2B) 1 DOZEN 5.00 4.80 4.50

3.PEN 1.30 1.20 1.00

4.ERASER 1.30 1.20 1.10

TOTAL PRICE 9.10 8.50 7.60

GRAND TOTAL 57.59 53.50 50.30

(c) Create at least two suitable graphical representations (the

use of ICT is encouraged) to compare and contrast the price of

the items chosen.

Answer:

1)FOODS

0

1

2

3

4

5

6

KOOP SARJANA

SELF-RAISING FLOUR (1000 g)

SUGAR (1000g)

BUTTER (250g)

EGGS (GRADE A) 1 DOZEN

0

1

2

3

4

5

6

KOOP IUI

SELF-RAISING FLOUR (1000 g)

SUGAR (1000g)

BUTTER (250g)

EGGS (GRADE A) 1 DOZEN

0

1

2

3

4

5

KIOSK PUM

SELF-RAISING FLOUR (1000 g)

SUGAR (1000g)

BUTTER (250g)

EGGS (GRADE A) 1 DOZEN

2)DETERGENT

0

5

10

15

20

KOOP SARJANA

SOAP (3 BARS)

LIQUID DISHWASHER (1000ml)

CLOTHES DETERGENT (3KG)

TOILET CLEANER (500ml)

0

2

4

6

8

10

12

14

16

18

KOOP IUI

SOAP (3 BARS)

LIQUID DISHWASHER (1000ml)

CLOTHES DETERGENT (3KG)

TOILET CLEANER (500ml)

0

2

4

6

8

10

12

14

16

18

KIOSK PUM

SOAP (3 BARS)

LIQUID DISHWASHER (1000ml)

CLOTHES DETERGENT (3KG)

TOILET CLEANER (500ml)

3)STATIONERY

0

1

2

3

4

5

KOOP SARJANA

SHARPENER

PENCIL (2B) 1 DOZEN

PEN

ERASER

0

1

2

3

4

5

KOOP IUI

SHARPENER

PENCIL (2B) 1 DOZEN

PEN

ERASER

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

KIOSK PUM

SHARPENER

PENCIL (2B) 1 DOZEN

PEN

ERASER

(d) Based on the graphical representation that you have

constructed in Part 1(c), interpret,discuss and draw

conclusions.Comments on your findings.

Answer:

Based on the graphical representation that I have constructed

in Part 1(c), it is shown that there are large and small

differences among the pries of items in each category between

the shops.In the food category, the smallest price difference

are of those 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 KOOP SARJANA, while the lowest price items

come frm the KIOSK PUM.the graph 1(d) will show the

conclusion of the difference among the shops based upon the

shops grand total.

Graph 1(d)

46

48

50

52

54

56

58

GRAND TOTAL

KOOP SARJANA

KOOP IUI

KIOSK PUM

(e) Identify an item that has a large price difference among the

shops. Calculate the mean and standard deviation of that

particular item. Hence, suggest and discuss possible reasons for

the price difference.

Answer:

Liquid Dishwasher

Mean = 18.9+17+16.5 3 = 17.47

Standard Deviation

= √(∑x²)/N – (x)²

= √18.9²+17²+16.5²

3 = 0.97

The large price difference of clothes detergent among the shops

maybe because of the standard of the shop.A high standard shop

or supermarket, the items sold intend to be much more

expensive than a regular shop or supermarket.Also, the price

difference of the items may also due to the quality of the item

present.A better quality means a higher price.

PART 2

Every year SMK Indah organises a carnival to raise funds for the

school. This year the school plans to install air conditioners in the

school library. Last year, during the carnival, your class made and

sold butter cakes. Because of the popularity of the butter

cakes,your class has decided to carry out the same project for

this year’s carnival.

QUESTION

(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:

The Giant Supermarket.This is because the total price of the

ingredients from this shop is the lowest from the three shops.

(b) Complete Table 2 with the prices of the items found in the

Shop/supermarket that you have chosen

Answer:

INGREDIENT QUANTITY PER CAKE

PRICE IN THE YEAR

2009(RM)

PRICE IN THE YEAR

2010(RM) Self-Raising Flour 250g 0.90 0.90

Sugar 200g 0.35 0.36

Butter 250g 3.30 4.30

Eggs (Grade A) 5 eggs (300g) 1.25 2.10

Table 2

(i) Calculate the price index for each of the ingredients in Table

2 for the year 2010 based on the year 2009

Answer :

INGREDIENT QUANTITY

PER CAKE PRICE IN

THE YEAR 2009(RM)

PRICE IN THE YEAR 2010(RM)

PRICE INDEX FOR THE YEAR2010 BASED ON THE YEAR 2009(I)

Self-Raising Flour

250g 0.90 0.90 100

Sugar 200g 0.35 0.36 102.86

Butter 250g 3.30 4.30 130.30

Eggs (Grade A)

5 eggs

(300g)

1.25 2.10 168

1.Self-Raising Flour I = 0.9

0.9 x 100 = 100 2.Sugar I = 0.36

0.35 x 100 = 102.86 3.Butter I = 4.3

3.3 x 100 = 130.30 4.Eggs (Grade A) I = 2.1

1.25 x 100 = 168

(ii) Calculate the composite index for making a butter cake in

the year 2010 based on the year 2009.Discuss how you

obtained your answers.

Answer:

To calculate the composite index,weightage is needed

(W), 𝑊𝑒𝑖𝑔 ℎ𝑡𝑇𝑜𝑡𝑎𝑙 𝑊𝑒𝑖𝑔 ℎ𝑡

INGREDIENT WEIGHTAGE (W)

Self-Raising Flour 250

1000 = 14

Sugar 2001000 =

15

Butter 2501000 =

14

Eggs (Grade A) 3301000 =

310

Composite Index =

14 (100)+

15 (102.86)+

14 (130.30)+

310(168)

1 = 128.54

(iii) In the year 2009,the butter cake was sold at RM15.00

each.Suggest a suitable selling price for the butter cake in

the year 2010.Give reasons for your answer.

Answer:

On 2009,RM 15.00

On 2010, price = 𝜘

15 x 100 = 128.54%

𝜘 x 100 = 128.54 x 15

𝜘 = 1928.1

100

𝜘 = 19.30

Thus,the suitable price for the butter cake for the year 2010

is RM19.30.The increase in price is also suitable because of

the rise in the price of the ingredients.

(c)

(i) Find out from reliable source how to determine

suitable

Capacity of air conditioner to be installed based on

the

volume/size of a room.

Answer:

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 1000ft³

which is around 27m³ of volume.

(ii) Work in group to estimate the volume of your school

Library.Explain how you arrive at your answer.Hence,

determine the number of air conditioners with the

appropriate capacity required for your library.

Answer:

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

Height = 3.6m

Width = 9.17m

Length = 20.12m

Volume of the room = 3.6 x 9.17 x 20.12

=664.20m³

1 unit of air conditioner is for 27m³

For 664.20m³ = 664.20

27

= 24.6

That means our school library needs 25 unit of air conditioner

(iii) If your class intends to sponsor one air conditioner for

the

School library, how many butter cakes must you sell in

order to buy the air conditioner

Answer:

1 unit of 1HP air conditioner =RM700

Cost for a cake = 0.9 + 0.36 + 4.3 + 2.1

= RM 7.66

Selling price = RM19.30

Profit = 19.30 – 7.66

= RM 11.64

Number of cakes = 700

11.64

to buy 1 unit of = 60.13 air conditioner = 60 cakes

PART 3 As a committee member for the carnival, you are required to prepare an estimated budget to organize this year’s carnival. The committee has to take into the consideration the increase in expenditure from the previous year due to inflation. The 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%. QUESTION (a) Complete Table 3 based on the information given above

Answer:

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) Calculate the composite index for the estimated budget of the

carnival in the year 2010 based on the year 2009. Comment on

your answer.

Answer:

EXPENDITURE AMOUN

T IN

2009

(RM)

AMOUN

T IN

2010

(RM)

PRICE

INDEX

, I

I = 𝑃1𝑃0 X

100%

WEIGHTAG

E, W

FOOD 1200.00 1380.00 115 12

GAMES 500.00 500.00 100 5

TRANSPORTATIO

N

300.00 345.00 115 3

DECORATIONS 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

Composite Index

I = ∑𝐼𝑖𝑊𝑖

∑𝑊

= 115 12 +100 5 +115 3 +100 2 +100 6 +115 8 +130(4)

12+5+3+2+6+8+4

=

446540

= 111.625

The total price for the year 2010 increase by 11.625%.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. Determine

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

the year 2009.

Answer:

Composite index for the year 2009 to the year 2010

= 111.625

Composite index for the year 2010 to the year 2011

= 111.625

I2011

2009 X 100 = I2010

2009 X I2011

2009

I2011

2009 = 111.625 X111.625 X

1

100

I2011

2009 = 124.60

FURTHER EXPLORATION

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.

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 g/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 tree major U.S. stock indices: the NASDAQ

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

nation²and 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 quality of air index and to detect any pollutants in our country’s air.This is to ensure the air is clean and safe for us ti 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 unterpretations 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 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 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 nessecities.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 Adter spending countless hours,days and night to finish this project and also sacrificing my time video games and mangas 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... Sacrifing my Anime... Sacrificing my Manga... 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...