Add Math

PROJECT WORK FOR ADDITIONAL MATHEMATICS 2010

Statistics in Our Life

NAME : ANTHONIUS ANAK GRENSON

SUBJECT TEACHER : SIR SAHARUDDIN

I/C NUMBER : 930113-13-5157

SCHOOL : SEKOLAH MENENGAH SAINS SABAH

Contents

No Contents Page1 Acknowledgement2 Introduction3 Part 14 Part 25 Part 36 Conclusion7 Further Exploration8 References

Acknowledgement

I would like to thank my Additional Mathematics’ teacher, Sir Ssaharuddin as he gives us important guidance and commitment during this project work.

I also would like to thank to all my friends for helping me and always supporting me to complete this project work.

For their strong support. I would like to express my gratitude to my beloved parents.

IntroductionThis project is carried out by every student who taking Additional Mathematic in their SPM examination. This project carries such aims:-

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

We are expected to submit the project work within three weeks from the first day the task is being administered to us. Failure to submit the written report will result in us not receiving certificate.

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 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.

How about statistic 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, economy, 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 1The 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.

a) Collect pictures, newspaper cuttings or photos on items that you have chosen. Design a collage to illustrate the chosen items

Answer:

Food

Self-raising flour

Sugar

Eggs (Grade A)

Butter

Detergent

Stationery

Question

(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)Gui Brothers G-Mart Superstore

Food1.Self-Raising Flour(1000g)

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

Detergent1.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

Stationery1.Sharpener 1.50 1.30 1.002.Pencil(2B-1 dozen)

5.00 4.80 4.50

3.Pen 1.30 1.20 1.004.Eraser 1.30 1.20 1.10

Total Price 9.10 8.50 7.60 Grand Total 57.59 53.50 50.30

Question

(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)

2)

3)

Question

(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 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 differences 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 item doesn’t have any obvious price difference. The graph also show that most of the items that are high priced comes from the Gui Brothers 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.

Graph 1(d)

Question

(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

= √(∑х²)/N – ( х L )²

= √ 18.9²+17²+16.5²

3 - (17.47)²

=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 2Every 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 Superstore. This is because the total price of the ingredients from this shop is the lowest from the three shops.

Question

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

Answer:

Ingredient QuantityPer cake

Price in the year 2009(RM)

Price in the year 2010(RM)

Self-raising flour 250g 0.90 0.90Sugar 200g 0.35 0.36Butter 250g 3.30 4.30Eggs(Grade A) 5 eggs (300g) 1.25 2.10

Question

(i) Calculate the price index for each of the ingredients in Table 2 for the year 2010 based on the year 2009

Answer:

Ingredient QuantityPer cake

Price in the year 2009(RM)

Price in the year

2010(RM)

Price index for the year 2010 based on

the year 2009 (Ι)Self-raising flour

250g 0.90 0.90 100

Sugar 200g 0.35 0.36 102.86Butter 250g 3.30 4.30 130.30Eggs(Grade A)

5 eggs (300g)

1.25 2.10 168

1. Self-raising flour

Ι=0.90.9

×100=100

2. Sugar

Ι=0.360.35

× 100=102.86

3. Butter

Ι=4.33.3

× 100=130.30

4. Eggs(Grade A)

Ι=2.1

1.25×100=168

Question

(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), Weight

Totalweight

Ingredients Weightage (W)Self-raising

flour250

1000=14

Sugar 200

1000=15

Butter 250

1000=14

Eggs(Grade A) 300

1000=3

10

Composite index

¿ 1

4 (100 )+¿1

5 (102.86 )+¿

1

4 (130.30 )+¿3

10 (168 )1

¿

¿¿

=128.54

Question

(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=ϰ15

×100=128.54 %

ϰ × 100=128.54 × 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.

Question

(c)(i) Find out from reliable sources 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.

Question

(ii) Work in groups 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=3.6×9.17×20.12

=664.20m³

1 unit of air conditioner is for 27m ³

For 664.20m³=664.20

27

=24.6

That means our school library needs 25 unit of air conditioner.

Question

(iii) If your class intends to sponsor one air conditioner for the school library, how many butter cakes must your class 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 =7.66

Selling price =RM19.30

Profit =19.30-7.66

=RM11.64

Number of cakes to buy 1 unit of air conditioner =

700

11.64 = 60.13 = 60 cakes

Part 3As 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 1380.00Games 500.00 500.00Transportation 300.00 345.00Decorations 200.00 200.00Prizes 600.00 600.00Tents 800.00 920.00Miscellaneous 400.00 520.00

Table 3

Question

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.

Solution.

Expenditure Amount in 2009(RM)

Amount in 2010(RM)

Price Index, I

I=P1P 0

×100 %

Weightage,W

Food 1200.00 1380.00 115 12Games 500.00 500.00 100 5Transportation 300.00 345.00 115 3Decorations 200.00 200.00 100 2Prizes 600.00 600.00 100 6Tents 800.00 920.00 115 8Miscellaneous 400.00 520.00 130 4

Composite Index

Ī = ∑ IiWi∑W

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

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

=4465

40 =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.

Question.

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.

Solution.

Composite index for the year 2009 to the year 2010

=111.625

Composite index for the year 2010 to the year 2011

=111.625

Ī 20112009

× 100=Ī 20102009

× Ī 20112010

Ī 20112009

=111.625×111.625×1

100

Ī 20112009

=¿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.

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

Unhealthy

A level that may have harmful impacts on patients and members of sensitive groups (children, aged or weak people), and also cause the general public unpleasant feelings.

251-350

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: 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 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 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.