Add Math Project 2012 Js

Name : ALIBABA JCHOW IC Number : 999999-07-5594

Class : 5 Science 99Teacher’s Name : En.TAYLOR SWICH

ALIBABA 5 S 11 ADDITIONAL MATHEMATICS PROJECT 2012 1 / 41

No Title Page1 Aim and Objective 22 Introduction 33 Appreciation 44 History of Indices 5-165 Activity 1: Collecting Information 17-276 Activity 2: Analysing the Data 28-307 Activity 3: Interpreting the Result 318 Activity 4: Further Exploration 32-349 Activity 5: Conclusion 3510 Reflection 36


To investigate the rising cost of living in a family over a three years period.

At the end of the project, I will able to

Evaluate the total monthly expenses for each expense item in a family.

Calculate the rate of increase in total monthly expenses for family in three years.

State the priority of the family’s expenditure. Calculate the composite index of the family expenditure in

2011 based on year 2009. Forecast or predict the index values of each item if the cost

of living increases at the rate for the next three years Recommend ways and means to cut the cost of living by

using numerical argument.


The increase in prices of goods has always been a topic of discussion in today’s society because the price increases causes an increment in the cost of living of a family. A hike


in petrol price, causing food company to charge extra transportation fee, is a major factor associated with the exorbitant price of food today. The government has implemented various measures to control price increases, such as providing subsidies and list a few essential items as controlled goods in order to reduce the burden on the people. Nevertheless, consumers are also responsible for planning their family expenditure. A study could be done to have a better understanding on how a family spend their money. This study is suitable as the project work of Additional Mathematics for SPM candidates 2012.

Thank you to God for giving the will to do my Additional Mathematic Project. Secondly, I would like to thank the principle of Sekolah Menengah Tanjung Rambutan, Mr. ALI BABA for giving the permission to do my Additional Mathematic Project Work. I also like to thank my Additional Mathematic teacher, Mr. Micheal Jackson for her guidance 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 price index is a normalized average of prices for a given class of goods or services in given region, during a given interval of time. It is a statistic designed to help to compare how these prices, taken as a whole, differ between time periods or geographical locations.

Price indices have several potential uses. For particularly broad indices, the index can be said to measure the economy’s price level or a cost of living. More narrow price indices can help producers with business plans and pricing. Sometimes, they can be useful in helping to guide investment.


Some notable price indices include consumer price index, producer price index and GDP deflator.

No clear consensus has emerged on who created the first price index. The earliest reported research in this area came from Welsh poet Henry Rice Vaughan who examined price level change in his 1675 book A Discourse of Coin and Coinage. Vaughan wanted to separate the inflationary impact of the influx of precious metals brought by Spain from the New World from the effect due to currency debasement. Vaughan compared labor statutes from his own time to similar statutes dating back to Edward III. These statutes set wages for certain tasks and provided a good record of the change in wage levels. Vaughan reasoned that the market for basic labor did not fluctuate much with time and that a basic laborers salary would probably buy the same amount of goods in different time periods, so that a laborer's salary acted as a basket of goods. Vaughan's analysis indicated that price levels in England had rose six to eightfold over the preceding century.

While Vaughan can be considered a forerunner of price index research, his analysis did not actually involve calculating an index. In 1707 Englishman William Fleetwood created perhaps the first true price index. An Oxford student asked Fleetwood to help show how prices had changed. The student stood to lose his fellowship since a fifteenth century stipulation barred students with annual incomes over five pounds from receiving a fellowship. Fleetwood, who already had an interest in price change, had collected a large amount of price data going back hundreds of years. Fleetwood proposed an index consisting of averaged price relatives and used his methods to show that the value of five pounds had changed greatly over the course of 260 years. He argued on


http://en.wikipedia.org/wiki/William_Fleetwood

http://en.wikipedia.org/wiki/Edward_III

http://en.wikipedia.org/wiki/Statute_of_Labourers_of_1351

http://en.wikipedia.org/wiki/Debasement

http://en.wikipedia.org/wiki/New_World

http://en.wikipedia.org/wiki/Spain

http://socserv2.socsci.mcmaster.ca/~econ/ugcm/3ll3/vaughan/coin

http://socserv2.socsci.mcmaster.ca/~econ/ugcm/3ll3/vaughan/coin

http://en.wikipedia.org/wiki/Henry_Vaughan

http://en.wikipedia.org/wiki/Wales

behalf of the Oxford students and published his findings anonymously in a volume entitled Chronicon Preciosum.

Given a set of goods and services, the total market value of transactions in in some period would be

where

represents the prevailing price of in period represents the quantity of sold in period

If, across two periods and , the same quantities of each good or service were sold, but under different prices, then

and

would be a reasonable measure of the price of the set in one period relative to that in the other, and would provide an index measuring relative prices overall, weighted by quantities sold.

Of course, for any practical purpose, quantities purchased are rarely if ever identical across any two periods. As such, this is not a very practical index formula.

One might be tempted to modify the formula slightly to


http://en.wikipedia.org/wiki/Index_(mathematics)

http://en.wikipedia.org/wiki/Measurement

This new index, however, doesn't do anything to distinguish growth or reduction in quantities sold from price changes. To see that this is so, consider what happens if all the prices double between and while quantities stay the same: will double. Now consider what happens if all the quantities double between and while all the prices stay the same: will double. In either case the change in is identical. As such, is as much a quantity index as it is a price index.

Various indices have been constructed in an attempt to compensate for this difficulty.

Paasche and Laspeyres price indices

The two most basic formulae used to calculate price indices are the Paasche index and the Laspeyres index

The Paasche index is computed as

while the Laspeyres index is computed as

where is the relative index of the price levels in two periods, is the base period, and the period for which the index is

computed.

Note that the only difference in the formulas is that the former uses period n quantities, whereas the latter uses base period quantities.

When applied to bundles of individual consumers, a Laspeyres index of 1 would state that an agent in the current period can afford to buy the same bundle as he consumed in


the previous period, given that income has not changed; a Paasche index of 1 would state that an agent could have consumed the same bundle in the base period as he is consuming in the current period, given that income has not changed.

Hence, one may think of the Laspeyres index as one where the numeraire is the bundle of goods using current year prices but base year quantities. Similarly, the Paasche index can be thought of as a price index taking the bundle of goods using current prices and current quantities as the numeraire.

The Laspeyres index tends to overstate inflation, while the Paasche index tends to understate it, because the indices do not account for the fact that consumers typically react to price changes by changing the quantities that they buy. For example, if prices go up for good then, ceteris paribus, quantities of that good should go down.

Fisher index and Marshall–Edgeworth index

A third index, the Marshall–Edgeworth index, tries to overcome these problems of under- and overstatement by using the arithmetic means of the quantities:

A fourth, the Fisher index, is calculated as the geometric mean of and :

Fisher's index is also known as the “ideal” price index.


http://en.wikipedia.org/wiki/Geometric_mean

http://en.wikipedia.org/wiki/Geometric_mean

http://en.wikipedia.org/wiki/Marshall%E2%80%93Edgeworth_index

http://en.wikipedia.org/wiki/Ceteris_paribus

http://en.wikipedia.org/wiki/Numeraire

However, there is no guarantee with either the Marshall–Edgeworth index or the Fisher index that the overstatement and understatement will exactly cancel the other.

While these indices were introduced to provide overall measurement of relative prices, there is ultimately no way of measuring the imperfections of any of these indices against reality.

Practical measurement considerations

Normalizing index numbers

Price indices are represented as index numbers, number values that indicate relative change but not absolute values. Price indices generally select a base year and make that index value equal to 100. You then express every other year as a percentage of that base year. In our example above, let's take 2000 as our base year. The value of our index will be 100. The price

2000: original index value was $2.50; $2.50/$2.50 = 100%, so our new index value is 100




When an index has been normalized in this manner, the meaning of the number 108, for instance, is that the total cost for the basket of goods is 4% more in 2001, 8% more in 2002 and 12% more in 2003 than in the base year.

Relative ease of calculating the Laspeyres index


http://en.wikipedia.org/wiki/Index_(economics)

http://en.wikipedia.org/wiki/Measurement

As can be seen from the definitions above, if one already has price and quantity data for the base period, then calculating the Laspeyres index for a new period requires only new price data. In contrast, calculating many other indices for a new period requires both new price data and new quantity data for each new period. Collecting only new price data is often easier than collecting both new price data and new quantity data, so calculating the Laspeyres index for a new period tends to require less time and effort than calculating these other indices for a new period.

Calculating indices from expenditure data

Sometimes, especially for aggregate data, expenditure data is more readily available than quantity data. For these cases, we can formulate the indices in terms of relative prices and base year expenditures, rather than quantities.

Here is a reformulation for the Laspeyres index:

Let be the total expenditure on good c in the base period,

then we have and therefore also . We can substitute these values into our Laspeyres formula as follows:

A similar transformation can be made for any index.

Chained vs non-chained calculations

So far, in our discussion, we have always had our price indices relative to some fixed base period. An alternative is to take the base period for each time period to be the immediately preceding time period. This can be done with ALIBABA 5 S 11 ADDITIONAL MATHEMATICS PROJECT 2012 12 / 41

any of the above indices. Here's an example with the Laspeyres index, where is the period for which we wish to calculate the index and is a reference period that anchors the value of the series:

Each term

answers the question "by what factor have prices increased between period and period ". When you multiply these all together, you get the answer to the question "by what factor have prices increased since period ".

Nonetheless, note that, when chain indices are in use, the numbers cannot be said to be "in period " prices.

Index number theory

Price index formulas can be evaluated based on their relation to economic concepts or on their mathematical properties. Several different tests of such properties have been proposed in index number theory literature. W.E. Diewert summarized past research in a list of nine such tests for a price index

, where and are vectors giving prices for a base period and a reference period while and give quantities for these periods.

1. Identity test:


The identity test basically means that if prices remain the same and quantities remain in the same proportion to each other then the index value will be one.

2. Proportionality test:

If each price in the original period increases by a factor α then the index should increase by the factor α.

3. Invariance to changes in scale test:

The price index should not change if the prices in both periods are increased by a factor and the quantities in both periods are increased by another factor. In other words, the magnitude of the values of quantities and prices should not affect the price index.

4. Commensurability test:

The index should not be affected by the choice of units used to measure prices and quantities.

5. Symmetric treatment of time:

Reversing the order of the time periods should produce a reciprocal index value. If the index is calculated from the most recent time period to the earlier time period, it should be the reciprocal of the index found going from the earlier period to the more recent.

6. Symmetric treatment of commodities:

All commodities should have a symmetric effect on the index. Different permutations of the same set of vectors should not change the index.

7. Monotonicity test:


http://en.wikipedia.org/wiki/Permutations

A price index for lower later prices should be lower than a price index with higher later period prices.

8. Mean value test:

The overall price relative implied by the price index should be between the smallest and largest price relatives for all commodities.

9. Circularity test:

Given three ordered periods , , , the price index for periods and times the price index for periods and should be equivalent to the price index for periods and

.

Quality change

Price indices often capture changes in price and quantities for goods and services, but they often fail to account for variation in the quality of goods and services. Statistical agencies generally use matched-model price indices, where one model of a particular good is priced at the same store at regular time intervals. The matched-model method becomes problematic when statistical agencies try to use this method on goods and services with rapid turnover in quality features. For instance, computers rapidly improve and a specific model may quickly become obsolete. Statisticians constructing matched-model price indices must decide how to compare the price of the obsolete item originally used in the index with the new and improved item that replaces it. Statistical agencies use several different methods to make such price comparisons.


The problem discussed above can be represented as attempting to bridge the gap between the price for the old item at time t, , with the price of the new item at the later time period, .

The overlap method uses prices collected for both items in both time periods, t and t+1. The price relative / is used.

The direct comparison method assumes that the difference in the price of the two items is not due to quality change, so the entire price difference is used in the index. / is used as the price relative.

The link-to-show-no-change assumes the opposite of the direct comparison method; it assumes that the entire difference between the two items is due to the change in quality. The price relative based on link-to-show-no-change is 1.

The deletion method simply leaves the price relative for the changing item out of the price index. This is equivalent to using the average of other price relatives in the index as the price relative for the changing item. Similarly, class mean imputation uses the average price relative for items with similar characteristics to M and N.


Appendix

Respondent’s detail

Name of respondent : TAN BUCK KOW

Occupation : Technician

Item expenditure

Estimated monthly expenditure in the year 2009 and 2011 of each item.


No Item of expenditure

Year 2009 (RM)

Year 2011 (RM)

1 Food 900 1,000

2 House Rental 500 650

3 Transportation 550 600

4 Utilities 1,200 1,350

5 Education 500 600

Total 3,650 4,200

Name of interviewer : Tan Swee Min Class :

…………………...... ...…5S11...

Respondent’s signature Date

Appendix


Name of respondent : CHEN KIM HOCK

Occupation : Salesman

Item expenditure



Year 2009 (RM)

Year 2011 (RM)


1 Food 750 900



4 Utilities 1,500 1,600

5 Education 700 850

Total 3,800 4,450


…………………...... ...…5S11...


Appendix


Name of respondent : TAN SOON YAN


Item expenditure


http://aspsms.jitsin.edu.my/sms/asp_sms_bs_guardian.php?act=enq&gid=7974



Year 2009 (RM)

Year 2011 (RM)

1 Food 1,000 1,200



4 Utilities 1,000 1,150

5 Education 1,000 1,150

Total 3,850 4,650


…………………...... ...…5S11...

Respondent’s signature

Date

Appendix


Name of respondent : CHON YEE YEE

Occupation : Store keeper ALIBABA 5 S 11 ADDITIONAL MATHEMATICS PROJECT 2012 20 / 41

Item expenditure



Year 2009 (RM)

Year 2011 (RM)

1 Food 800 900

2 House Rental - -


4 Utilities 800 950

5 Education 600 700

Total 2,550 3,050


…………………...... ...…5S11...


Appendix

Respondent’s detailALIBABA 5 S 11 ADDITIONAL MATHEMATICS PROJECT 2012 21 / 41

Name of respondent : TEH CHU YAN

Occupation : Pharmacist

Item expenditure



Year 2009 (RM)

Year 2011 (RM)

1 Food 1,100 1,200



4 Utilities 950 1,150

5 Education 550 700

Total 3,500 4,250


…………………...... ...…5S11...



Appendix


Name of respondent : KHAW JING XUAN

Occupation : Supervisor

Item expenditure



Year 2009 (RM)

Year 2011 (RM)

1 Food 900 1,100



4 Utilities 350 400

5 Education 400 550

Total 2,430 3,100


…………………...... ...…5S11...

Respondent’s signature DateALIBABA 5 S 11 ADDITIONAL MATHEMATICS PROJECT 2012 23 / 41

Appendix


Name of respondent : LIM YU SIN


Item expenditure



Year 2009 (RM)

Year 2011 (RM)

1 Food 1,000 1,150



4 Utilities 1,200 1,350

5 Education 350 500

Total 3,450 4,150



…………………...... ...…5S11...


Appendix


Name of respondent : WILL TANG

Occupation : Supervisor

Item expenditure



Year 2009 (RM)

Year 2011 (RM)

1 Food 900 1,000



4 Utilities 1,000 1,200

5 Education 300 450

Total 2,800 3,600



…………………...... ...…5S11...


Appendix


Name of respondent : NG JING CAO


Item expenditure



Year 2009 (RM)

Year 2011 (RM)

1 Food 900 1,150

2 House Rental - -



5 Education 500 650

Total 2,700 3,450



…………………...... ...…5S11...


Appendix


Name of respondent : KHOO PEH XIN


Item expenditure



Year 2009 (RM)

Year 2011 (RM)

1 Food 700 350





5 Education 450 600

Total 2,700 3,400


…………………...... ...…5S11...




Expenditure of the year 2009

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 Mean

1 Food 900 750 1000 800 1100 900 1000 900 900 700 895

2 House Rental 500 350 400 - 300 300 500 300 - 400 305

3 Transportation 550 500 1150 350 600 480 400 300 400 300 433

4 Utilities 1200 1500 1000 800 950 350 1200 1000 900 850 975

5 Education 500 700 1000 600 550 400 350 300 500 450 535

Total 3650 3800 3850 2550 3500 2430 3450 2800 2700 2700 3143

NoItem of expenditure

Expenditure of the year 2011

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 Mean

1 Food 1000 900 1200 900 1200 1100 1150 1000 1150 350 1045

2 House Rental 650 450 550 - 450 450 650 450 - 500 415

3 Transportation 600 650 600 500 750 600 500 500 550 450 570


4 Utilities 1350 1600 1150 950 1150 400 1350 1200 1100 1000 1125

5 Education 600 850 1150 700 700 550 500 450 650 600 675

Total 4200 4450 4650 3050 4250 3100 4150 3600 3450 3400 3830


NoItem of Expenditure Average 2009

(RM)Average 2011

(RM)

1 Food 895 1045



4 Utilities 975 1125

5 Education 535 675

Total 3143 3830


Percentage of increase in expenditure of each item in 2011 based on 2009


Calculation Percentage(%)

1 Food 1045-895 x 100 895

16.76

2 House Rental 415-305 x 100 305

36.07

3 Transportation 570-433 x 100 433

31.64

4 Utilities 1125-975 x 100 975

15.38

5 Education 675-535 x 100 535

26.17


Expenditure index for each item


Calculation Expenditure index in

2011 based 2009

1 Food 1045 x 100 895

117

2 House Rental 415 x 100 305

136

3 Transportation 570 x 100 433

132

4 Utilities 1125 x 100 975

115

5 Education 675 x 100 535

126

Total 19675 x 100 3143

626



Weightage of each item in percentage

(%)

Weightage of each item in degree ()

1 Food 26.80 28.06 x 360 100

= 96.48

2 House Rental 24.90 24.90 x 360 100

= 89.64

3 Transportation 12.50 12.50 x 360 100

= 45.00

4 Utilities 24.50 24.50 x 360 100

= 88.20

5 Education 11.30 11.30 x 360 100

= 40.68


Expenditure composite index

Expenditure index x Weightage

Total weightage

117(26.8) + 136(24.9) + 132(12.5) + 115(24.5) + 126(11.3)=124.13

100

No. Item of Expenditure Year 2011

1 Food Increased

2 House Rental Increased

3 Transportation Increased

4 Utilities Increased

5 Education Increased

The conclusion of the expenditure composite index in total from year 2009 to year 2011 has increased.


In the year 2013, the costs of living are expected to increase by 20% from year 2011. The expenditure index of each item in year 2013 based on year 2009 are calculated as shown in the table below:


Percentage of each item in 2011

based on 20009 (%)

Percentage of each

item in 2013

based on 2009 (%)

Expenditure index in 2011 based

on 2009

Calculation

1 Food 100 120 117117 x 120

100= 140.40

2House Rental

100 120 136 136 x 120 100 = 163.20

3Transportati

on100 120 132 132 x 120 100 = 158.40


4 Utilities 100 120 115 115 x 120 100 = 138.00

5 Education 100 120 126 126 x 120 100 = 151.20

Method I

I =2013

2009

140.4(26.8)+163.2(24.9)+158.4(12.5)+138(24.5)+151.2(11.3)= 149

100

Method II

II =2013

2009

Percentage of increase in year 2013 based on 2011 is composite index in year 2011 based on 2009

120 x 626 = 751


100

The impact of 20% increase in expenditure index in year 2013 can be reduced by controlling expenses for certain items. My suggestions on how to reduce the percentage increase of expenditure for some item and its reason are shown below:

No. Item of Expenditure

Method to Reduce the expenditure

Percentage

1. FoodCook instead of buying

Eat at food stall instead of restaurant

2%

2. House Rental Look for cheaper rental or rental flat or smaller house.

2%

3. TransportationCar polling or use public transportation

Plan your traveling

3%


4. UtilitiesSwitch off lights when not in use.

Smaller house less consumption

2%

5. EducationBorrow from library, friends and photocopy books.

Reduce tuition fees, refer to school teacher

1%

Total 10%

The impact of my suggestion on the composite index of expenditure decrease in each item.

Three important items in a family expenditure are food, utilities and finally education.

The cost of living for a family in the year 2011 is RM 3,400 when compared to year 2009 RM 2,700 has increased.

There will be a major impact on a family if the cost of living increases by 20% in 2013 without increase in income. The


family may have to borrow money to spend for food, utilities and education. The family may not able to safe money for future use. They have to depend on government for medicine and other facilities available.Daily life will be a struggle and lost of happiness at home. If my suggestions are able to reduce the cost of living, it will decrease the amount that been calculated in the composite index.

Under my estimation on the percentage of reduction in the cost of living in year 2013 if my suggestion is accepted are between 5-10% reduction.

After completing this Additional Mathematic project, I have learned the usage and importance of index number in the use of our daily life through problem solving.This project has encouraged me to be able to work with friends as a group. Through this, I have learned the values of team work in a group. I also learned how to gather information from the internet, analyzing data, interpreting data, collecting and enhancing my thinking abilities and skills and promote the effective mathematical () and other students. Besides that, I have also obtained some () values through the completion this project. This project has thought me to be


responsible and ().As an evidence, I have finished the project given to me, before the deadline that was set by my teacher.I proposed that this project should be carried out every year as it brings a lot of benefits to the students such as it tests the students ability and application of Additional Mathematic in our daily life.


Documents

Add Math Project 2012 Js