AdditionalMathematics Project Work 4 Lukmanulhakim awaluddin 930423125069 S.m.k agama kota kinabalu lukman ℜ Acknowledgement.................................................. ℜ Objectives............................................................... ℜ Introduction ........................................................... ℜ Part 1...................................................................... ℜ Part 2...................................................................... ℜ Part 3...................................................................... ℜ Further Explorations............................................... ℜ Reflections............................................................ ℜ Conclusion.............................................................. Acknowledgement 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 I’m 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. Introduction 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 comparing production, 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 numbers 1. Establishes trends Index 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 making It 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 rupee Usually 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 data Index 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 year’s 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 series can be eliminated by the usage of an appropriate price index and thus making the series real. Types of index numbers Three are three types of principal indices. They are: 1. Price Index The 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 Index A 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 Index The 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 = ˝1 ˝u × 1uu I = index number Q 1 = quantity at specific time Q o = quantity at base time • The composite index is the weighted mean for all the items in a certain situation. Ī = ¿w I ¿w Ī = Composite index W = weightage I = index number Part 1 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 Category Item Price Giant Servay khidmat Food 1.self-raising flour 2.70 3.70 3.30 2.sugar 1.80 1.60 1.35 3.butter 3.60 2.90 3.00 4.Eggs(grade A) 3.60 2.90 3.00 Total price 11.70 12.00 12.15 Detergent 1.Washing powder 19.00 21.00 20.50 2.dish washer 4.00 3.20 2.10 3.liquid bleach 6.00 5.50 4.90 4.tile cleaner 10.20 9.80 9.50 Total price 39.20 39.50 38.00 Stationary 1.pencil(shaker) 8.90 9.20 8.20 2.highlighter 3.50 2.90 3.80 3.permenent marker 3.50 2.90 3.80 4.card indexing 14.70 15.00 16.00 Total price 30.60 30.50 32.00 GRAND TOTAL 81.50 82.00 82.15 0 2 4 6 8 10 12 14 giant 0 5 10 15 20 25 giant 0 2 4 6 8 10 12 14 16 giant servay khidmat Food Self Raising Flour Sugar Butter Eggs servay khidmat Detergent washing powder dish washer liquid bleach tile cleaner servay khidmat Stationery pencil highlighter permenant marker card indexing Self Raising Flour Sugar Butter Eggs washing powder dish washer liquid bleach tile cleaner pencil highlighter permenant marker card indexing 0 5 10 15 20 25 30 35 40 45 food 0 5 10 15 20 25 30 35 40 food detergent stationary detergent stationary giant servay khidmat giant servay khidmat Question (D) Based on all the graph in question 1(C) , we can conclude that giant hypermarket offers the lowest price for their customers. Then followed by servayl and Khidmat. This is because the supplier of the giant gives the special price for it as it buy by bulk. servay offer the normal price for their customer as it does not get special price from the supplier. While, khidmat have to sold the items at the higher price because the shop buy the items by bulk from Giant. 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. Giant 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. Giant is a bigmall and it provides many items that we need in our life. Eventhough Giant 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. Servay and khidmat cannot offer the prices as giant because they are situated outside the urban area like giant . So the population of the customer will not be as many as customer in giant. These shops get the supply for their goods from giant. Even they buy by bulk with giant but their prices still will be higher than giant. 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. graph below will show the conclusion of the difference among the shops based upon the shops grand total. 81.1 81.2 81.3 81.4 81.5 81.6 81.7 81.8 81.9 82 82.1 82.2 giant n, 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. will show the conclusion of the difference among the shops based upon servay khidmat grand total n, 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 will show the conclusion of the difference among the shops based upon 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 ˲ = ¿x N = 19+21+20.5 3 = 2u.2u Calculate the standard deviation o = Vo 2 Or o = ¹ ¿x 2 N − X ) 2 = ¹ 19 2 +21 2 +(20.5) 2 3 − ( 21 6 ) 2 = 0.8498 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. giant 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 servay and Khidmat 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 servay and khidmat cannot buy by bulk the stationery items with their supplier. Part2 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 year’s carnival. Question (a) From the data in Part 1, I would go to Giant to purchase the ingredients for the butter cakes. This is because giant 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, giant is located not far from my school. So it is easier to my friends and I to go there. Ingredient Quantity per cake Price in 2009 (Rm) Price in 2010 (Rm) Price index 2010 based 2009 Self-raising flour 250g 0.90 0.675 75 Sugar 200g 0.35 0.36 102.86 Butter 250g 3.30 3.60 109.10 Eggs(grade A) 5 (300g) 1.20 1.80 144 (i) Calculate Price Index I = ˜1 ˜u × 1uu Self raising-four = 1.00 0.90 × 100 = 111.11 Sugar = 0.36 0.35 × 100 =102.86 Butter = S.Su S.Su × 1uu =106.06 Eggs (Grade A) = 1.37 1.25 × 100 =109.60 (ii) Composite index Ī = ¿w I ¿w = (5×111.11) + (4×102.86) + (5×106.06) + (6×109.60) 5+4+5+6 =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 15.00 On 2010, suitable price is : ˲ 1S × 1uu = 1u7.74% ˲ × 1uu = 1u7.74 × 1S ˲ = 1616.1u 1uu ˲ = 16.2u 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 library By 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 m 3 One unit of air conditioner with 3HP is for 81 m 3 For 6S1.9um 3 = 651.90 81 = 8.048 This 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 1800 Cost for a cake = RM 6.23 Selling price = RM 16.20 Profit =RM 16.20- RM6.23 = RM 9.97 Number of cakes to buy 1 unit of air conditioner = 18uu 9.97 = 18u.S4 = 181 cakes As a committee member for the carnival, I am required to prepare an estimated budget to organise this year’s 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. Expenditure Ammount in 2009 (RM) Amount in 2010 (RM) Index Weightage Food 1200 1.15 x 1200 =1380 115 12 Games 500 1 x 500 =500 100 5 Transportation 1300 1.15 x 1300 =345 115 3 Decoration 200 1 x 200 =200 100 2 Prizes 600 1 x 600 =600 100 6 Tonts 800 1.15 x800 =920 115 8 miscellaneous 400 1.3 x400 =520 130 4 Composite Index Ī = ¿w I ¿w = 115(12)+ 100(5)+ 115(3)+100(2)+ 100(6)+ 115(8)+ 130(4) (12+5+3+2+6+8+4) = 446S 4u = 111.62S 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. (a) 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 I 2011 2009 × 1uu = I 2010 2009 × I 2011 2010 I 2011 2009 = 111.62S × 111.62S × 1 1uu I 2011 2009 = 124.6u Further Explorations 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 Welshman 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 risen six to eightfold over the preceding century. [1] While Vaughan can be considered a forerunner of price index research, his analysis did not actually involve calculating an index. [1] 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. [2] Formal calculation Further information: List of price index formulas Given a set C of goods and services, the total market value of transactions in C in some period t would be where represents the prevailing price of c in period t represents the quantity of c sold in period t If, across two periods t 0 and t n , 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 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 t 0 and t n while quantities stay the same: P will double. Now consider what happens if all the quantities double between t 0 and t n while all the prices stay the same: P will double. In either case the change in P is identical. As such, P 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 formulas used to calculate price indices are the Paasche index (after the German economist Hermann Paasche [ˈpaːʃɛ]) and the Laspeyres index (after the German economistEtienne Laspeyres [lasˈpejres]). The Paasche index is computed as while the Laspeyres index is computed as where P is the change in price level, t 0 is the base period (usually the first year), and t n 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 (period 0) 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 she 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 base year prices but current 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 systematically overstates inflation, while the Paasche index understates 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 c then, ceteris paribus, quantities of that good should go down. Fisher index and Marshall-Edgeworth index A third index, the Marshall-Edgeworth index (named for economists Alfred Marshall and Francis Ysidro Edgeworth), tries to overcome these problems of under- and overstatement by using the arithmethic means of the quantities: A fourth, the Fisher index (after the American economist Irving Fisher), is calculated as the geometric mean of P P and P L : Fisher's index is also known as the “ideal” price index. However, there is no guarantee with either the Marshall-Edgeworth index or the Fisher index that the overstatement and understatement will thus exactly one 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 (Paasche, Laspeyres, Fisher, or Marshall-Edgeworth) against reality. Normalizing index numbers Price indices are represented as index numbers, number values that indicate relative change but not absolute values (i.e. one price index value can be compared to another or a base, but the number alone has no meaning). 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 2001: original index value was $2.60; $2.60/$2.50 = 104%, so our new index value is 104 2002: original index value was $2.70; $2.70/$2.50 = 108%, so our new index value is 108 2003: original index value was $2.80; $2.80/$2.50 = 112%, so our new index value is 112 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 (in this case, year 2000). Relative ease of calculating the Laspeyres index As can be seen from the definitions above, if one already has price and quantity data (or, alternatively, price and expenditure 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 (e.g., the Paasche index) for a new period requires both new price data and new quantity data (or, alternatively, both new price data and new expenditure 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. [3] Calculating indices from expenditure data Sometimes, especially for aggregate data, expenditure data is more readily available than quantity data. [4] 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 (by definition) we have and therefore 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 alternative is to take the base period for each time period to be the immediately preceding time period. This can be done with any of the above indices, but here's an example with the Laspeyres index, where t n is the period for which we anchors the value of the series: Each term answers the question "by what factor have prices increased between period When you multiply these all together, you get the increased since period t 0 . Nonetheless, note that, when chain indices are in use, the numbers cannot be said to be "in period prices. th 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 Calculating indices from expenditure data Sometimes, especially for aggregate data, expenditure data is more readily available than quantity formulate the indices in terms of relative prices and base year expenditures, rather than quantities. Here is a reformulation for the Laspeyres index: be the total expenditure on good c in the base period, then (by definition) we and therefore also . We can substitute these values into our Laspeyres formula as follows: A similar transformation can be made for any index. chained calculations So far, in our discussion, we have always had our price indices relative to some fixed alternative is to take the base period for each time period to be the immediately preceding time period. This can be done with any of the above indices, but here's an example with the Laspeyres is the period for which we wish to calculate the index and t 0 is a reference period that answers the question "by what factor have prices increased between period t n − 1 When you multiply these all together, you get the answer to the question "by what factor have prices Nonetheless, note that, when chain indices are in use, the numbers cannot be said to be "in period th 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 Sometimes, especially for aggregate data, expenditure data is more readily available than quantity formulate the indices in terms of relative prices and base year be the total expenditure on good c in the base period, then (by definition) we . We can substitute these values 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 any of the above indices, but here's an example with the Laspeyres is a reference period that and period t n ". answer to the question "by what factor have prices Nonetheless, note that, when chain indices are in use, the numbers cannot be said to be "in period t 0 " Index number theory Price index formulas can be evaluated in terms properties per se. 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 P 0 and P n are vectors period while and 1. Identity test: The identity test basically means that if prices remain the quantities remain in the same proportion to each other (each quantity of an item is multiplied by the same factor of either or β, for the later period) 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 per 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 (or, in parity measures, symmetric treatment of place): 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 perio going from the earlier period to the more recent. Index number theory Price index formulas can be evaluated in terms of their mathematical . 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 are vectors giving prices for a base period and a reference give quantities for these periods. [5] The identity test basically means that if prices remain the quantities remain in the same proportion to each other (each quantity of an item is multiplied by the same factor of either α, for the first period, , for the later period) then the index value will be one. Proportionality test: e in the original period increases by a factor α then the index should increase by the factor α. 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. Commensurability test: The index should not be affected by the choice of units used to measure es. Symmetric treatment of time (or, in parity measures, symmetric treatment of place): 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. of their mathematical . Several different tests of such properties have been proposed in index number theory literature. W.E. Diewert summarized past , giving prices for a base period and a reference The identity test basically means that if prices remain the same and quantities remain in the same proportion to each other (each quantity of , for the first period, , for the later period) then the index value will be one. e in the original period increases by a factor α then the index The price index should not change if the prices in both periods are iods are increased by another factor. In other words, the magnitude of the values of The index should not be affected by the choice of units used to measure Symmetric treatment of time (or, in parity measures, symmetric Reversing the order of the time periods should produce a reciprocal index value. If the index is calculated from the most recent time period d, it should be the reciprocal of the index found 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 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 t m , t n , t r , the price index for periods t m and t n times the price index for periods t n and t r should be equivalent to the price index for periods t m and t r . Quality change Price indices often capture changes in price and quantities for goods and services, but they often fail to account for improvements (or often deteriorations) 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. [6] The problem discussed above can be represented as attempting to bridge the gap between the price for the old item in time t, P(M) t , with the price of the new item in the later time period,P() t + 1 . [7] The overlap method uses prices collected for both items in both time periods, t and t+1. The price relative P(N) t + 1 /P(N) t 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. P(N) t + 1 /P(M) t 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. [8] 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 (physical, geographic, economic, etc.) to M and N. 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