Additional Mathematics Project Work 2
2013
Content Page No.
Contents
Page No.
1.
Acknowledgement Acknowledgement
2
2.
Objectives
3
3.
Introduction
4-6
4.
Part 1
7-13
5.
Part 2
14-17
6.
Part 3
18-27
7.
Part 4
28-30
8.
Reflection
31
9.
Reference
32
Page 1
Additional Mathematics Project Work 2
2013
Acknowledgement First and foremost, I would like to thank the principle of Sekolah Menengah Kebangsaan Sultan Sulaiman Shah, Puan Hajah Siti Zaleha binti Harun for giving me the permission to do this Additional Mathematics Project Work.
Secondly, I would also like to show my gratitude towards the endless guidance and support given by my Addi Additi tion onaal Mat Mathem hematics sub subjject ect tea teache cher, Puan Puan Ras Rashida hidah h throug oughout hout the the proc proces esss of do doing ing this pr proje oject work. She taught me and my friends patiently on ways to accomplish this project work.
Apart from that, I would like to acknowledge and range my earnest appreciation to my parents for giving me supports in every ways, such as money, to buy anything that are related to this project work and their advises, which is the most needed for this project.
Lastly, a special thank you to all my friends for aiding me in solving some of the calculations. They were helpful to share information and ideas in order to get this project work done. And especially to God, who made all things possible.
Thank you, everyone.
Page 2
Additional Mathematics Project Work 2
2013
Acknowledgement First and foremost, I would like to thank the principle of Sekolah Menengah Kebangsaan Sultan Sulaiman Shah, Puan Hajah Siti Zaleha binti Harun for giving me the permission to do this Additional Mathematics Project Work.
Secondly, I would also like to show my gratitude towards the endless guidance and support given by my Addi Additi tion onaal Mat Mathem hematics sub subjject ect tea teache cher, Puan Puan Ras Rashida hidah h throug oughout hout the the proc proces esss of do doing ing this pr proje oject work. She taught me and my friends patiently on ways to accomplish this project work.
Apart from that, I would like to acknowledge and range my earnest appreciation to my parents for giving me supports in every ways, such as money, to buy anything that are related to this project work and their advises, which is the most needed for this project.
Lastly, a special thank you to all my friends for aiding me in solving some of the calculations. They were helpful to share information and ideas in order to get this project work done. And especially to God, who made all things possible.
Thank you, everyone.
Page 2
Additional Mathematics Project Work 2
Objectives
The aims of carrying out this project work are:
To apply and adapt a variety of problem-solving strategies to solve problems
To improve thinking skills
To promote effective mathematical communication
To develop mathematical knowledge through problem-solving in a way that increases students’ interest & confidence
To use language of mathematics to express mathematical ideas precisely
To provide learning environment that stimulates and enhances effective learning.
To develop positive attitude towards mathematics
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Additional Mathematics Project Work 2
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Introduction History of early price indices
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 brough t by Spain from the New World from the effect due tocurrency 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 leve ls in England had risen 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 behalf of the Oxford students and published his findings anonymously in a volume entitled Chronicon Preciosum.
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Index number theory
Price index formulas can be evaluated based on their relation to economic concepts (like cost of living) 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
giving prices for a base period and a reference period while
and
and
are vectors give quantities for
[5]
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 (each quantity of an item is multiplied by the same factor of either
, for the first period, or
, for the later period) then the index value will
be one. 2. Proportionality test:
If each price in the original period increases b y 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.
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Additional Mathematics Project Work 2
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5. Symmetric treatment of time (or, in parity measures, symmetric treatment of place):
Reversing the order of the time periods should pro duce 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:
A price index for lower later prices should be low er 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 price index for periods
and
,
,
, the price index for periods
and
times the
should be equivalent to the price index for periods
and
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PART 1 Page 7
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PRICE INDEX
A price index ( plural : “price indices” or “price indexes”) is a normalized average (typically a weighted average) of prices for a given class of goods or services in a 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 indexes 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
GDP deflator
WEIGHTAGE
The assignment of a quota (as of members of a legislature) to a particular segment of the population as a special favor or concession in a proportion above that allowable on a strictly numerical basis. It is a value given to each item to show the relative importance of each item in a certain condition.
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COMPOSITE INDEX A grouping of equities, indexes or other factors combined in a standardized way, providing a useful statistical measure of overall market or sector performance over time. Also known simply as a "composite". Usually, a composite index has a large number of factors which are averaged together to form a product representative of an overall market or sector. These indexes are useful tools for
measuring and tracking price level changes to an entire stock market or sector. Therefore, they provide a useful benchmark against which to measure an investor's portfolio. The goal of a well diversified portfolio is usually to outperform the main composite indexes.
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WEIGHTAGE REPRESENTATIONS
I.
probability weights - The most common type of weights are probability weights. These weights represent the probability that a case (or subject) was selected into the sample from a population. These weights are calculated by taking the inverse of the sampling fraction. For example, if you have a population of 10 widgets and you select 3 into your sample, your sampling fraction would be 3/10 and your pweight would be 10/3 = 3.33. You frequently find this type of weight in surve y data.
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II.
2013
frequency weights - Frequency weights are whole (i.e., integer) numbers that tell the program how many cases each case represents. It is a kind of short cut: if you have five rows of data that are identical, you can use a frequency weight with a value of 5 and spare yourself having to input the same row five times.
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Additional Mathematics Project Work 2
III.
2013
importance weights - Importance weights are just what you think the y should be - they are weights that indicate how "important" a case is. There is no standard way of calculating this type of weight.
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IV.
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analytic weights - Analytic weights are perhaps the least common type of weight. This type of weight is used when the cases are actually an average. If the averages are based on different numbers of observations (for example, some averages are based on three observations and others are based on 30 observations), some cases (averages) are measured with more precision than others, and you want the more precisely measured cases to have a greater weight than the less precisely measured cases. The more measurements used in the average, the more precise the average will be. The weights are proportional to the inverse of the variance.
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PART 2 Page 14
Additional Mathematics Project Work 2 I.
My family's monthly expenditure for the year 2013.
Item Food Accomodation (Rental / Loan) Transportation (Petrol/ Loan / Bus fare etc) Clothing Education Recreation Utilities (Water / Electricity / Telephone) Medication Miscellaneous TOTAL
II.
2013
Average Monthly Expenditure for the year 2013 (to the nearest RM) 1500 2400
Percentage of monthly expenses (to the nearest %) 25 40
600
10
90 600 120 450
1.5 10 2 7.5
120 120 6000
2 2 100
Average monthly expenses for the year 2012 to calculate price index.
Item
Food Accomodation (Rental / Loan) Transportation (Petrol / Loan / Bus fare etc) Clothing Education Recreation Utilities (Water / Electricity / Telephone) Medication Miscellaneous TOTAL
Average monthly expenses for the year 2012 as the base year (RM) 1200 2400
Average monthly expenses for the year 2013 (RM) 1500 2400
500
600
70 400 100 350
90 600 120 450
100 120 5240
120 120 6000
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Additional Mathematics Project Work 2
III.
Prince indices for the year 2013 based on the year 2012 and its weightage.
Item Food Accomodation (Rental / Loan) Transportation (Petrol / Loan / Bus fare etc) Clothing Education Recreation Utilities (Water / Electricity / Telephone) Medication Miscellaneous TOTAL
IV.
2013
Price indices for the year 2013 based on the year 2012 I = 1500 x 100 = 125 1200
Weightage
I = 2400 x 100 = 100 2400 I = 600 x 100 = 120 500 I = 90 x 100 = 129 70 I = 600 x 100 = 150 400 I = 120 x 100 = 120 100 I = 450 x 100 = 129 350 I = 120 x 100 = 120 100 I = 120 x 100 = 100 120 1093
40
25
10 1.5 10 2 7.5 2 2 100
COMPOSITE INDEX
̅ ∑ ∑
̅ )()() ()()()()()()( = 1166 / 100 =RM 116.66
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Additional Mathematics Project Work 2 V.
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CONCLUSION
My family’s expenditure for the year 2013 is higher compa red to the year 2012 as the living cost had gone up by 16.66%. Food’s expenses increased by 25%, transportation’s, recreation’s, and medication’s expenses gone up by 20%, clothing’s and utilities’ cost increased by 29%, and education’s expenses gone up by 50%. However, the expenses for accommodation and miscellaneous are constant.
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PART 3 Page 18
Additional Mathematics Project Work 2
I.
THE PRICES OF TELEVISIONS BY CASH PAYMENT
Brand
Price (RM)
Size of Television (inches)
Shop A
Shop B
Shop C
24 32 40 24 32 40
419.00 539.95 785.00 415.00 538.00 734.00
415.90 519.00 790.00 420.00 539.00 755.00
414.00 539.00 786.00 417.00 535.00 741.00
Samsung
Panasonic
i.
2013
MEAN
∑
x
x
n
SAMSUNG
1. 24 inches =
= RM 416.30
2. 32 inches =
= RM 532.65
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Mean Price (RM) 416.30 532.65 787.00 417.30 537.30 743.30
Standard Deviation (RM)
2.06 9.66 2.16 5.70 6.20 11.20
Additional Mathematics Project Work 2
3. 40 inches
= RM 787
PANASONIC
1. 24 inches =
= RM 417.30
2. 32 inches =
= RM 537.30 3. 40 inches
= RM 743.30
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Additional Mathematics Project Work 2
ii.
STANDARD DEVIATION
∑ √ () SAMSUNG
1. 24 inches
√ () = RM 2.06 2. 32 inches
√ () = RM 9.66 3. 40 inches
√ () = RM 2.16
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Additional Mathematics Project Work 2
PANASONIC
1. 24 inches
√ ( ) = RM 5.70 2. 32 inches
√ () = RM 6.20 3. 40 inches
√ ( ) = RM 11.20
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Additional Mathematics Project Work 2
II.
THE PRICES OF TELEVISIONS BY INSTALLMENT
Brand
Price (RM)
Size of Television (inches)
Shop A
Shop B
Shop C
24 32 40 24 32 40
838.00 1079.90 1570.00 830.00 1076.00 1460.00
831.80 1038.00 1580.00 840.00 1079.00 1510.00
828.00 1078.00 1572.00 835.00 1070.00 1482.00
Samsung
Panasonic
i.
2013
MEAN
∑
x
x
n
SAMSUNG
1. 24 inches =
= RM 832.60 2. 32 inches =
= RM 1065.00
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Mean Price (RM) 832.60 1065.00 1574.00 835.00 1075.00 1484.00
Standard Deviation (RM)
4.10 31.80 4.30 4.10 3.70 20.50
Additional Mathematics Project Work 2
3. 40 inches
= RM 1574.00
PANASONIC
1. 24 inches =
= RM 835.00
2. 32 inches =
= RM 1075.00 3. 40 inches
= RM 1484.00
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Additional Mathematics Project Work 2
iii.
STANDARD DEVIATION
∑ √ () SAMSUNG
1. 24 inches
√ () = RM 4.10 2. 32 inches
√ () = RM 31.80 3. 40 inches
√ () = RM 4.30
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Additional Mathematics Project Work 2
PANASONIC
1. 24 inches
√ () = RM 4.10 2. 32 inches
√ () = RM 3.70 3. 40 inches
√ () = RM 20.50
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Additional Mathematics Project Work 2
III.
2013
Determine the brand and size of the television that you have decided to buy. Give your reasons.
I have decided to buy the Panasonic television which is 24 inches in size. This is because Panasonic is the world's fourth-largest television manufacturer and it produces high quality of electrical equipments which can last longer compared to Samsung. Other than that, a small size of television is enough for m y family as my family is a small family. Then, we also cannot afford for an expensive television as our monthly expenses exceeded RM 5000, so a cheap television which only cost RM 415.00 by cash payment will be more appropriate and economical.
VI.
The Ministry of Domestic Trade and Consumer Affairs wishes to present the Fair Price Shop Award for one of the above shops. If you are one of the panels for this award, determine the shop that deserves the award. Do you consider the value of the mean and the value of standard deviation in making your decision? Give your justifications.
The shop that deserves the award will be Shop A. This is because the prices of the televisions in Shop A are nearer to the mean values and the costs are more reasonable and cheaper compared to other shops. Then, the standard deviation of Shop A is smaller than Shop B and C. Therefore, the prices of televisions in Shop A are more consistent. Thus, Shop A should be chosen to receive the Fair Price Shop Award by the Ministry of Domestic Trade and Consumer Affairs.
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PART 4 Page 28
Additional Mathematics Project Work 2
(a )
2013
Your family has a fixed monthly income. In order to buy the television, your family needs to make some adjustment on the various types of expenditure. [You can choose to pay by cash or by in stal lm ent ] Show the average monthly expenditure that you have modified in a table.
MONTHLY INCOME : RM 6000
Item Food Accomodation (Rental / Loan) Transportation (Petrol/ Loan / Bus fare etc) Clothing Education Recreation Utilities (Water / Electricity / Telephone) Medication Miscellaneous TOTAL EXTRA
Average Monthly Expenditure for the year 2013 (to the nearest RM) 1500 2400
Modified Average Monthly Expenditure for the year 2013 (to the nearest RM) 1400 2400
600
550
90 600 120 450
80 600 80 440
120 120 6000 0
120 110 5780 220
The adjustment should be carried on for two months in order to save enough money to buy the television by cash because buying television by installment will cost the double value of the original price. So, saving RM 220 for two months will save RM 440 which is enough to buy the television of my choice. Although it takes time, it is more economical as we can save our money.
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Additional Mathematics Project Work 2
(b)
2013
Assuming you have just started working with a monthly salary of RM2 500.You intend to save 10% of your salary every month. Plan your monthly expenditure as in Table 1 above and add other items such as savings and contributions to your parents.
Item Food Accomodation (Rental / Loan) Transportation (Petrol/ Loan / Bus fare etc) Clothing Education Recreation Utilities (Water / Electricity / Telephone) Medication Miscellaneous Savings Contributions to parents TOTAL
Average Monthly Expenditure for the year 2013 (to the nearest RM) 600.00 400.00
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Percentage of monthly expenses (to the nearest %) 24 16
300.00
12
100.00 200.00 100.00 250.00
4 8 4 10
100.00 100.00 250.00 100.00 2500.00
4 4 10 4 100
Additional Mathematics Project Work 2
REFLECTION ♥ ADDITIONAL MATHEMATICS ♥
When I Know You For The First Time, I Feel Curious And Obsessed To Know All About You, And I Even Stalk You, Day And Night, You Are So Clingy And Addictive, You Make Me Turns To You 24-Hours A Day, Telling Me Little By Little About The Knower Of All Know, I Feel So Grateful, That He Has Given Me The Chance To Get To Know You Additional Mathematics, You Start From, Addition, Subtraction, Multiplication And Division, Then You Expand To, Squares, Cubes, Square Roots, And Cube Roots, And Now Your Are In, Logarithms, Differentiation And Integration, You Have No Full Stop, I Will Fill You In My Life, I Will Be With You Forever, I LOVE YOU ADDITIONAL MATHEMATICS. ♥
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