ADDITIONAL MATHEMATICS PROJECT WORK

Work 4

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

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

Introduction ...........................................................

Part 1......................................................................

Part 2......................................................................

Part 3......................................................................

Further explorations...............................................

Reflections............................................................

Conclusion..............................................................

First of all, I would like to say Alhamdulillah, for giving me the strength and health to do this project work and finish it on time.

Not forgotten to my parents for providing everything, such as money, to buy anything that are related to this project work, their advise, which is the most needed for this project and facilities such as internet, books, computers and all that. They also supported me and encouraged me to complete this task so that I will not procrastinate in doing it.

Then I would like to thank to my teacher, Mdm Fazilah for guiding me throughout this project. Even I had some difficulties in doing this task, but she taught me patiently until we knew what to do. She tried and tried to teach me until I understand what Im supposed to do with the project work.

Besides that, my friends who always supporting me. Even this project is individually but we are cooperated doing this project especially in disscussion and sharing ideas to ensure our task will finish completely.

Last but not least, any party which involved either directly or indirect in completing this project work. Thank you everyone.

The aims of carrying out this project work are:

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

ii. To improve thinking skills.

iii. To promote effective mathematical communication.

iv. To develop mathematical knowledge through problem solving in a way that increases students interest and confidence.

v. To use the language of mathematics to express mathematical ideas precisely.

vi. To provide learning environment that stimulates and enhances effective learning.

vii. To develop positive attitude towards mathematics.

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 increasingly used probability theory, whose initial results were found in the17th and 18th centuries, particularly in the analysis of games of chance (gambling). By 1800, astronomy used probability models and statistical theories, particularly the method of least squares, which was invented by Legendre and Gauss. Early probability theory and statistics was systematized and extended by Laplace; following Laplace, probability and statistics have been in continual development.

In the 19th century, social scientists used statistical r easoning and probability models to advance the new sciences of experimental psychology and sociology; physical scientists used statistical reasoning and probability models to advance the new sciences of thermodynamics and statistical mechanics.

The development of statistical reasoning was closely associated with the development of inductive logic and the scientific method. Statistics is not a field of mathematics but an autonomous mathematical science , like computer science or operations research. Unlike mathematics, statistics had its origins in public administration and maintains a special concern with demography and economics. Being concerned with the scientific method and inductive logic, statistical theory has close association with the philosophy of science ; with its emphasis on learning from data and making best predictions, statistics has great overlap with the decision science and microeconomics. With its concerns with data, statistics has overlap with information science and computer science .

Statistics Today

During the 20th century, the creation of precise instruments for agricultural research, public health concerns (epidemiology, biostatistics, etc.),industrial quality control, and economic and social purposes (unemployment rate, econometry, etc.) necessitated substantial advances in statistical practices.

Today the use of statistics has broadened far beyond its origins. Individuals and organizations use statistics to understand data and make informed decisions throughout the natural and social sciences, medicine, business, and other areas. Statistics is generally regarded not as a subfield of mathematics but rather as a distinct, albeit allied, field. Many universities maintain separate mathematics and stati stics departments. Statistics is also taught in departments as diverse as psychology, education, and public health.

Index Number

Index numbers are today one of the most widely used statistical indicators. Generally used to indicate the state of the economy, index numbers are aptly called barometers of economic activity. Index numbers are used in comparingproduction, sales or changes exports or imports over a certain period of time. The role-played by index numbers in Indian trade and industry is impossible to ignore. It is a very well known fact that the wage contracts of workers in our country are tied to the cost of living index numbers.

By definition, an index number is a statistical measure designed to show changes in a variable or a group or related variables with respect to time, geographic location or other characteristics such as income, profession, etc.

Characteristics of an Index Numbers

1. These are expressed as a percentage: Index number is calculated as a ratio of the current value to a base value and expressed as a percentage. It must be clearly understood that the index number for the base year is always 100. An index number is commonly referred to as an index.

2. Index numbers are specialized averages: An index number is an average with a difference. An index number is used for purposes of comparison in cases where the series being compared could be expressed in different units i.e. a manufactured products index (a part of the whole sale price index) is constructed using items like Dairy Products, Sugar, Edible Oils, Tea and Coffee, etc. These items naturally are expressed in different units like sugar in kgs, milk in liters, etc.The index number is obtained as a result of an average of all these items, which are expressed in different units. On the other hand, average is a single figure representing a group expressed in the same units.

3. Index numbers measures changes that are not directly measurable: An index number is used for measuring the magnitude of changes in such phenomenon, which are not capable of direct measurement. Index numbers essentially capture the changes in the group of related variables over a period 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 sectors might have increased their production more than 2.15 times while other sectors may have increased their production only marginally.

Uses of index numbers1. Establishes trendsIndex 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 makingIt 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 rupeeUsually index numbers are used to determine the purchasing power of the rupee. Suppose the consumers price index for urban non-manual employees increased from 100 in 1984 to 202 in 1992, the real purchasing power of the rupee can be found out as follows: 100/202=0.495 It indicates that if rupee was worth 100 paise in 1984 its purchasing power is 49.5 paise in 1992.

4. Deflates time series dataIndex 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 seriescan be eliminated by the usage of an appropriate price index and thus making the series real.

Types of index numbersThree are three types of principal indices. They are:

1. Price IndexThe 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 IndexA 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 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 IndexThe 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.

Calculating index number

Index number Is 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 numberQ1 = quantity at specific timeQo = quantity at base time

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

=

= Composite index W = weightage= index number

The prices of good sold in shops are vary from one shop to another. Shoppers tend to buy goods which are not only reasonably priced but also give value for their money. I had carried out a survey on four different items based on the following categories which is food, detergent and stationery. The survey was done in three different shops. Informations below shows the results from my research.

Question (a) Picture

Stationery

Food

Detergent

Question (b) Data

CategoryItemPrice (RM)

Mydin MallEmbat ShopSi Comel

Food

1.Self raising flour(500g)2.002.502.10

2. Sugar(1000g)1.801.901.85

3. Butter (250g)3.503.603.60

4. Eggs (Grade A)8.208.508.40

Total price15.5016.5015.95

Detergent

1. Dishwash1.802.001.95

2. Softener 3.904.204.00

3. Floor cleaner10.9011.0011.30

4. Powder D/gent9.509.509.60

Total price26.1026.7026.85

Stationery

1. Eraser0.800.800.70

2. Marker3.003.603.90

3. Pencil colour5.505.405.30

4. Scissor2.302.302.40

Total price11.6012.1012.30

Grand total

53.2055.3055.10

Question (c)

Question (D)

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 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.

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

The difference of the price of the marker in these three shops is maybe due to the price given by the supplier to the shops. Mydin Mall can sold it at lowest prices because the demand of the buyers for the the item is high so it can buy by bulk with the supplier. So the shop can get the special price. The demand of the item in Embat Shop and Si Comel are low. This is because the customers are more interested to buy the stationery items in mall or stationery shops as there are more options to choose. So Embat Shop and Si Comel cannot buy by bulk the stationery items with their supplier.

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)

IngredientQuantity per cakePrice in the year 2009 (RM)Price in the year 2010 (RM)

Self-raising flour250 g0.901.00

Sugar200 g0.350.36

Butter 250 g3.303.50

Eggs (Grade A)5 eggs (300 g)1.251.37

(i) Calculate Price Index

Self raising-four

Sugar =102.86

Butter

=106.06

Eggs (Grade A)

=109.60

(ii) (iii) 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.

(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.

Question (c)

(i) To determine suitable capacity of air conditioner to be installed based on volume/ size of a room

For common usage, air conditioner is rated according to horse power (1HP), which is approximately 700W to 1000W of electrical power. It is suitable for a room size 1000ft which is around 27m of volume. If we buy an air conditioner with 3HP, it is suitable for a room around 81m.

(ii) Estimate the volume of school libraryBy using a measuring tape, the dimension for the library is: Height=3.6m Width=9.0m Length=20.12m Volume of the room=3.6 x 9.0 x 20.12 =651.90 One unit of air conditioner with 3HP is for 81 For 8.048This means our school library needs 8 unit of air conditioner.

(iii) My class intends to sponsor one air conditioner for the school library. The calculation below is to find how many butter cakes we must sell in order to buy the air conditioner.

1 unit of 3 HP air conditioner = RM 1800Cost for a cake = RM 6.23Selling price = RM 16.20Profit =RM 16.20- RM6.23 = RM 9.97

Number of cakes to buy 1 unit of air conditioner =

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 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%.

(a) Table 3 has been completed based on the above information.

ExpenditureAmount in 2009 (RM)Amount in 2010 (RM)

Food1200.001380.00

Games500.00500.00

Transportation300.00345.00

Decorations200.00200.00

Prizes600.00600.00

Tents800.00920.00

Miscellaneous400.00520.00

Table 3

(b) Calculation to find the composite index

ExpenditureAmount in 2009 (RM)Amount in 2010 (RM)Price Index,

Weightage,W

Food1200.001380.0011512

Games500.00500.001005

Transportation300.00345.001153

Decorations200.00200.001002

Prizes600.00600.001006

Tents800.00920.001158

Miscellaneous400.00520.001304

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

AIR POLLUTION INDEXAir 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.

CAIDescriptionHealth Implications

0-50GoodA level that will not impact patients suffering from diseases related to air pollution.

51-100ModerateA level which may have a meager impact on patients in case of chronic exposure.

101-150Unhealthy for sensitive groupsA level that may have harmful impacts on patients and members of sensitive groups.

151-250UnhealthyA 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 unhealthyA level which may have a serious impact on patients and members of sensitive groups in case of acute exposure.

351-500HazardousA 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 ppmv but PM10 particulate matter is reported in g/m.

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: 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.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.

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 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 countrys economical state sometimes depend on it.

After spending countless hours,days and night to finish this project and also sacrificing my time for chatting and movies 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 play Time..Sacrificing my Chatting...Sacrificing my Facebook...Sacrificing my internet...Sacrifing 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...I LOVE U ADDITIONAL MATHEMATICS..

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 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.