SMK Sultan Abu BakarJalan Beserah, 25300, Kuantan, Pahang.

Additional Mathematics Project WorkK2/2014

Application of Index Number and Statistics

Name : Nur Syahirah binti Shafek Hamlan

I/C :

SPM ID:

Class : 5SC2

__________________ Teacher’s signature

CONTENT

OBJECTIVE 1

FOREWORD 2

INTRODUCTION 3-4

PART ONE 5-12

PART TWO 13-16

PART THREE 17-18

FURTHER EXPLORATION 19-21

CONCLUSION 22

REFLECTION 23

OBJECTIVE

I was given a task from the Pahang State Education Department to carry out an

Additional Mathematics project work this year. I was asked to complete this task and submit

an individually written report. Upon completion of the Additional Mathematics Project Work,

I want to gain valuable experiences and able to:

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

routine problems.

Experience a classroom environment which is challenging, interesting, meaningful

and hence improving our thinking skills.

Experience a classroom environment where knowledge and skills are applied in

meaningful ways in solving real-life problems.

Experience a classroom environment where expressing ones mathematical thinking,

reasoning and communication are highly encouraged and expected.

Experience a classroom environment that stimulates and enhances effective ways of

learning.

Prepare for the demands of my 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 myself not only to be an independent learner but also collaborate, cooperate and

share knowledge in an engaging and healthy environment.

Use technology especially ICT appropriately and effectively.

Train myself to appreciate the intrinsic values of mathematics and to become more

creative and innovative.

Realise the importance and the beauty of mathematics.

FOREWARD

Praise to god for giving me the strength and health to do this project work. Not to forget

my parents for providing everything for me such as money to buy anything that are related to

this project work, computer, internet, books, time, energy and also their valuable advices to

me. They also supported me and encouraged me to complete this task that I will never

procrastinate in doing it. I also would like to thank my teacher, Puan Rabaiah binti Sujak for

guiding me throughout this work. Not to mention, my friends who are also doing this project

with me. They shared ideas and gave suggestions to me which are very helpful to me in order

to complete this task. I had some difficulties in doing this task but thanks to them, I had this

task done on time.

INTRODUCTIONHISTORY 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 price, quantity, value and 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.

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.

BACKGROUNDThe prices of goods sold in shops vary from one shop to another. Shoppers tend to buy goods

which are reasonably priced, giving value for their money. We, as consumers, should learn to

spend wisely and keep our daily expenses to the minimum.

PART ONEI was required to carry out a survey on three different items based on the following

categories: food, detergent and stationary. The survey should be done in three different shops.

(a) Pictures, newspapers cuttings or photos on items that I have chosen are collected.

A collage to illustrate the chosen items was designed.

(b) The items and their prices are recorded systematically as in Table 1(a), 1(b) and 1(c). Since items may be differently packed, consistent measurements for each item are made sure so that comparison can be done easily and accurately.

Category ItemPrice (RM)

Shop A:Giant

Shop B:Tesco

Shop C:Mydin

Food

1. Fresh milk (1 litre) 6.90 7.49 6.50

2. Margarine(250g) 5.80 5.00 5.30

3. Wheat flour (1kg) 1.30 2.50 1.50

Total price (RM) 14.00 14.99 13.33Table 1(a)

Category ItemPrice (RM)

Shop P:Giant

Shop Q:Tesco

Shop R:Mydin

Detergent

1. Bar soap (3 bars) 4.50 4.30 3.99

2. Liquid detergent(1 litre) 7.99 8.00 7.80

3. Powdered detergent

(5kg)15.99 16.20 15.50

Total price (RM) 28.48 28.50 27.29Table 1(b)

Category ItemPrice (RM)

Shop X:Giant

Shop Y:Tesco

Shop Z:Mydin

Stationeries

1. Pencil (1 dozen) 9.90 8.20 7.90

2. Colouring pencils

(12 pencils/box)9.50 8.00 8.50

3. Whiteboard marker (1 item)

4.80 4.50 4.30

Total price (RM) 24.20 20.70 20.70Table 1(c)

(c) Some suitable graphical representations (encouraged by the use of ICT) were created to compare and contrast the prices of each category.

1) Bar charts

Giant Tesco Mydin0

1

2

3

4

5

6

7

8

Fresh MilkMargarineWheat flour

Shops

Price

(RM

)

Food

Giant Tesco Mydin0

2

4

6

8

10

12

14

16

18

Bar soapLiquid detergentPowdered detergent

Shops

Price

(RM

)

Detergent

Giant Tesco Mydin0

2

4

6

8

10

12

PencilColouring pencilsWhiteboard marker

Shops

Price

(RM

)

Stationeries

2) Pie charts

Giant33%

Tesco35%

Mydin31%

Total price for food

Giant34%

Tesco34%

Mydin32%

Total price for detergent

Giant37%

Tesco32%

Mydin32%

Total price for stationeries

(d) Based on the graphical representations that I have constructed in Part 1(c), it shows that there are large and small differences among the prices of items in each category between the shops. The bar charts show that products from Mydin are much cheaper than products from Tesco and Giant. From the pie charts, we can conclude that shop which is selling the products with higher price is Giant. As a conclusion, shop which is selling their products with lower prices among all of the shops is Mydin, followed by Tesco and the shop which is selling their products with higher prices among all of the three shops is Giant.

(e) An item that has a large price difference among the shops was identified. Possible reasons for the price difference are suggested.

Pencil (one dozen):

Giant = RM 9.90

Tesco = RM 8.20

Mydin = RM 7.90

Using the formula:

Mean, x=∑ xN

, where ∑ x is sum of all data and N is is number of data.

9.90+8.20+7.903

=RM 8.67

Standard deviation ,σ=√∑ x2

N−x2=√(9.90+8.20+7.90)2

3−(8.67)2=12.254

The large price differences of pencils among the shops are 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. The price difference of the items may also due to the quality of the item present. A better quality means a higher price.

PART TWOEvery year, my school organizes a carnival to raise funds for school improvement projects. This year, my school is planning to install CCTV in order to enhance its security. Last year, during the carnival, my class made and sold nasi lemak. Due to the popularity of this local food, my class has decided to carry out the same project for this year’s carnival.

(a) Table 2 shows the prices of the given items for the year 2014.

Items(For 10 packets of

Nasi Lemak)Quantity Price in the year

2013 (RM)Price in the year

2014 (RM)

I

For the flavoured rice:Rice 1 kg 2.80 3.00Santan 150 g 1.50 1.90Onion 10 g 0.15 0.30Ginger 10 g 0.15 0.20Salt 2 g 0.05 0.08

II

For the sambal:Red onion 10 g 0.05 0.12Garlic 5 g 0.04 0.08Dried chillies 100 g 1.00 1.25Anchovies 200 g 2.00 2.70Cooking oil 100 g 0.25 0.32

III

Miscellaneous: Fried groundnuts 200 g 0.50 0.84Fried anchovies 150 g 1.50 1.80Cucumber 200 g 0.20 0.315 Eggs 300 g 1.50 1.80

Banana leaves (1 bundle) 300 g 1.00 1.20

Table 2

(b) The price index for each of the items in Table 2 for the year 2014 based on the year 2013 was calculated by using the formula below:

Price Index , I=P1

P0×100

Where, P0 = Price of the item at the base time.

P1 = Price of the item at the specific time.

Items(For 10 packets of

Nasi Lemak)Quantity Price in the year

2013 (RM)Price in the year

2014 (RM)

Price Index for the year

2014 based on the year 2013

(I)

I

For the flavoured rice:Rice 1 kg 2.80 3.00 107.14Santan 150 g 1.50 1.90 126.67Onion 10 g 0.15 0.30 200.00Ginger 10 g 0.15 0.20 133.33Salt 2 g 0.05 0.08 160.00

II

For the sambal:Red onion 10 g 0.05 0.12 240.00Garlic 5 g 0.04 0.08 200.00Dried chillies 100 g 1.00 1.25 125.00Anchovies 200 g 2.00 2.70 135.00Cooking oil 100 g 0.25 0.32 `128.00

III

Miscellaneous:Fried groundnuts 200 g 0.50 0.84 168.00Fried anchovies 150 g 1.50 1.80 120.00Cucumber 200 g 0.20 0.31 155.005 Eggs 300 g 1.50 1.80 120.00

Banana leaves (1 bundle) 300 g 1.00 1.20 120.00

(c) The composite index for the cost of a packet of nasi lemak in the year 2014 based on the year 2013 was calculated by using the formula below:

Composite Index Number , I =∑ I iW i

∑W i

Where, I i = Index number

W i = Weightage

Items(For 10 packets of

Nasi Lemak)Quantity

Price Index for the year 2014 based on the year 2013 (I)

Weightage (W) I iW i

I

For the flavoured rice:Rice 1 kg 107.14 1.000 107.14Santan 150 g 126.67 0.150 19.00Onion 10 g 200.00 0.010 2.00Ginger 10 g 133.33 0.010 1.33Salt 2 g 160.00 0.002 0.32

II

For the sambal:Red onion 10 g 240.00 0.010 2.40Garlic 5 g 200.00 0.005 1.00Dried chillies 100 g 125.00 0.100 12.50Anchovies 200 g 135.00 0.200 27.00Cooking oil 100 g `128.00 0.100 12.80

III

Miscellaneous: Fried groundnuts 200 g 168.00 0.200 33.60Fried anchovies 150 g 120.00 0.150 18.00Cucumber 200 g 155.00 0.200 31.005 Eggs 300 g 120.00 0.300 36.00

Banana leaves (1 bundle) 300 g 120.00 0.300 36.00

Total 2.737 340.09

∑ I i W i

∑ W i

= 340.092.737

= 124.256(d) In the year 2013, the nasi lemak was sold at RM2.50 each. A suitable selling price for a packet of the nasi lemak in the year 2014 was suggested.

Price of a packet of nasi lemak in the year 2013 = RM2.50

Price of a packet of nasi lemak in the year 2014 =

Using the formula = P1

P0×100

x

2.50×100=124.256

¿3.10

The suitable selling price for a packet of nasi lemak in the year 2014 is RM3.10. The increasing in price is suitable because of the rise of price of the ingredients.

PART THREEIn order to upgrade the security, your school has decided to install closed-circuit TV (CCTV).

(a) A suitable number of cameras to be installed and their placement were suggested.

I suggested two cameras to be installed at my school. The cameras will be installed near the office entrance and near the principal’s cark park. This action will be done to avoid any robbery or crime case happening at my school.

(b) An installation for a CCTV costs RM 800.

The total cost for two installations of CCTVs are RM 1600.

(c) My class intends to sponsor 10% of the cost. The number of packets of nasi lemak must my class sell is calculated.

10% × RM 1600 = RM 160

Price of a packet of nasi lemak = RM 3.10 + RM 0.40 (profit)

= RM 3.50

Packets of nasi lemak must be sold = 1603.50 = 45.7

Approximately 46 packets of nasi lemak must be sold to sponsor 10% of the cost.

(d) I think my class can achieve the 10% target because many visitors from Kuantan will come and visit our school during the carnival. This will increase the amount of our nasi lemak sold. Last year, we had sold more than 100 packets of nasi lemak. So we don’t have to

Block A

worry if no one’s going to buy our nasi lemak. From the selling, we can sponsor 10% for the cost of installation of the CCTVs.

I don’t think that my class should carry out a different project since we can get a lot of profit from selling this food. Moreover, nasi lemak is one of the most popular foods here in Malaysia so I bet everyone loves nasi lemak.

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.

1. Air Quality Index

What is Air Quality Index?

An air quality index (AQI) is a number used by government agencies to communicate to the public how polluted the air is currently or how polluted it is forecast to become. As the AQI increases, an increasingly large percentage of the population is likely to experience increasingly severe adverse health effects. Different countries have their own air quality indices which are not all consistent. Different countries also use different names for their indices such as Air Quality Health Index, Air Pollution Index and Pollutant Standards Index.

Definition and usage

Air quality is defined as a measure of the condition of air relative to the requirements of one or more biotic species or to any human need or purpose. 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 colour code. Standardized public health advisories are associated with each AQI range.

The AQI can go up (meaning worse air quality) due to a lack of dilution of air pollutants. Stagnant air, often caused by an anticyclone, temperature inversion, or low wind speeds lets air pollution remain in a local area, leading to high concentrations of pollutants and hazy conditions. An agency might encourage members of the public to take public transportation or work from home when AQI levels are high.

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.

Air Pollution Index in Malaysia

API Status Health Effect Health Advice0 - 50 Good Low pollution

without any bad effect on health.

No restriction for outdoor activities to the public. Maintain healthy lifestyle.

51 - 100 Moderate Moderate pollution that does not pose any bad effect on health.

No restriction for outdoor activities to the public. Maintain healthy lifestyle.

101 - 200 Unhealthy Worsen the health condition of high risk people who is the people with the heart and lung compilations.

Limited outdoor activities fir the high risk people. Public need to reduce the extreme outdoor activities.

201 - 300 Very unhealthy Worsen the health condition and low tolerance if physical exercises to people and heart and lung compilations. Affect public health.

Old and high risk people are advised to stay indoors and reduce physical activities. People with health compilations are advised to see doctor.

> 300 Hazardous Hazardous to high risk people and public health.

Old and high risk people are prohibited for outdoor activities. Public are advised to prevent from outdoor activities.

> 500 Emergency Hazardous to high risk people and public health.

Public are advised to follow orders from National Security Council and follow the announcement in mass media.

2. Body Mass Index (BMI)

The body mass index (BMI), or Quetelet index, is a measure of relative weight based on an individual's mass and height. It is defined as the individual's body mass divided by the square of their height – with the value universally being given in units of kg/m2.

BMI= mass(kg)[height (m)]2

BMI can also be determined using a table or from a chart which displays BMI as a function of mass and height using contour lines, or colours for different BMI categories, and may use two different units of measurement.

The BMI is used in a wide variety of contexts as a simple method to access how much an individual's body weight departs from what is normal or desirable for a person of his or her height. There is however often vigorous debate, particularly regarding at which value of the BMI scale the threshold for overweight and obese should be set, but also about a range of perceived limitations and problems with the BMI.

'BMI' provides a simple numeric measure of a person's thickness or thinness, allowing health professionals to discuss overweight and underweight problems more objectively with their patients.

CONCLUSIONAfter doing this research, I have learnt that index numbers are daily life necessities. Without it, surveys cannot be conducted. Mathematics are not just numbers. They are a very important tool to solve real life problems and enhance our thinking skills. We should be thankful to the people who contribute in the idea of index number.

REFLECTIONWhile I was conducting the project, I have learnt and practice a lot of moral values.

We should be patient when facing hard times when doing a task and try to find ways to solve problems that we are facing by seeking help from friends or teachers. We should also be independent when doing a task and try to learn something new from it and appreciate the intrinsic values of mathematics and to become a more creative and innovative person.

“Pure mathematics is, in its way, the poetry of logical ideas.” – Albert Einstein

“One of the most amazing things about mathematics is the people who do maths aren’t usually interested in application, because mathematics itself is truly a beautiful art form. It’s structures and patterns, and that’s what we love, and that’s what we get off on.” – Danica McKellar.

“For the things of this world cannot be made known without knowledge of mathematics.”- Roger Bacon