VIEWS: 37 PAGES: 15 CATEGORY: Childrens Literature POSTED ON: 11/29/2009
Unit II Prices Introduction to Block A Unit I introduced a number of ways of seeing Mathematically : looking with numbers , with relationships , with generality , with symbols. Many situations give rise to differences or changes which need to be identified and quantified . Much of statistics develops initially from a need to find ways of comparing things , either individual measurements or more commonly whole batches of numerical data . The mathematical ways of seeing used most often here will be seeing with numbers and seeing with graphs and diagram 1) Are we getting better off ? The main aim of this section is to introduce some ideas about making valid comparisons and focus on ways of extracting information from tables and graphs . Activities 1,2,3 compare the price of a loaf of bread in 1594 , as a percentage of the daily wage , with the price of a loaf of breed in 1994 . 1 Unit 2 Cost of a loaf of bread in 1594 ( ad a percentage of daily wage ) = ½ X 100% = 4.2 % ___ 12 ( Activity 4 ) : cost of Bread in 1994 ( As percentage of d. wage ) 56 X 100 = 3600 1.6% Compare this to the price of Bread in 1594 ( as percentage of daily wage ) ( Activity 9 ) : use the data in table 4 ( page 14 ) calculate the percentage increase in the average bread price between Feb. 1992 and Feb 1993 . Compare this to the increase from 1981 to 1982 ( in both cases the increase in price was 1 penny but the percentage increase in different ) ( Activity 10 ) : using percentages to make comparisons Table 5 , page 17 , gives the average U.K male earnings ( weekly ). for all industries and services from 1980 to 1993 . Percentage increase in the average weekly salary between 1980 and 1993 = 274 – 111.7 X 100 = 145.6 % 111.7 Percentage increase in the price of bread in the same period ( Activity 9 ) = 55 - 32 X 100 = 71.9 % 32 The percentage increase in the average weekly salary is more than the percentage increase in the price of bread . Are we getting better off ? 2 Unit 2 2) A typical shopping basket : The main aim of this section is to discuss how best to measure average price increases. We will try to identify a “ typical basket of goods and use these goods to try to analyze price changes over time , since studying the price of only are item like bread will help us to know whether people are getting better off or not . Table 7 : What happens when we change the units for one of the items ( potatoes are in 50 kg bags instead of 1 kilo loose ) Average ( mean ) price rise is £ 12.30 = £ 2.46 = 246 pence 5 ( Altering the units in which the potatoes have been measured made a dramatic difference to the overall average ) Table 8 : Average percentage price increase is 444 = 88.8% 5 Activity 12 : Calculating on average price measure using percentages solves the problem of which units to use but it gives each item of the basket the same (( weight )) or emphasis which may not be realistic . The weight chosen by the government (in this unit ) are the amounts of money spent on each item by a typical household over a typical week . 3 Unit 2 3) A stastical interlude – averages : The main aim of this section is to discuss several ways of finding averges and to introduce you to the satistical facilities of your calculator . 3 .1 The mean of a batch of numbers is the sum of all the number in the batch divided by the batch size . The median is essentically the middle value of the batch ( when the values are pleased in size order ) Example 1 : Finding the median when the batch size is an odd number. Sort the numbers into ascending order . The single middle value is the median of the batch . Example 2 : Finding the median when the Batch Size is an even number sort the numbers into ascending order The mean of the two middle numbers is the median of the batch . 3 .2 Summary of Various Averges : Mean : ∑ x n ; add the x values in the batch and divide by the batch size n. X × f ∑ x f ; for data given with frequencies ; add all the products n and divide the result by the sum of frequencies . 4 Unit 2 weighted ∑ x w add all the products X × W and divided the result by the mean ∑ w sum of the weights . Median : Sort the values in the batch into ascending order ( if necessary ) . if the batch size is odd then the median is the middle value if the batch size is even , then the median is the mean of the two midde values . See examples 3,4,5 , read the discussion about the difference between a mean and weighted mean page 31 ) . 4) Price indices Activity : 4 .1 Weighted mean of the percentage price increase from 1980 to 1994 : ∑ x w ∑ w = 87% ( table 9 , P. 34 ) Activity 19 : Calculating the mean using the ten week period expenditures as weights ( table 10 ) gives the same answer as before . 4 .2 Price ratios : Activity 20 Percentage Price rise = 39 – 30 x 100 = 30% 100 Ratio = 39 = 1.3 30 Price ratio = 1 + Percentage Price rise 100 Perceutage price rise = ( ratio – 1 ) × 100 % 5 Unit 2 Activity 21 : Weighted mean of price ratios for 1994 relative to 1980 = 1.87 This is equivalent to a price increase of 87% . conclusion it does not matter whether we use percentage price increases or price ratios to find the “ average ” percentage price increase . 4 .3 Price indices : January bread prices Year 1987 44 1988 46 1989 48 1990 50 1991 53 1992 54 1993 55 1994 51 Price (p) Calculating price ratio relative to January 1987 price ratio for Jan. of any year = Average price in Jan. of that year Average price in Jan. of 1987 Price Ratio Relative to Jan. 1987 Year Price Ratio 1987 1.000 1988 1.045 1989 1.091 1990 1.136 1991 1.205 1992 1.227 1993 1.250 1994 1.159 A Price index for bread I the base year method Year Index 1987 100 1988 104.5 1989 109.1 1990 113.6 1991 120.5 1992 122.7 1993 125.0 1994 115.9 base year has value of 100 100 × price ratio relative to base year 6 Unit 2 One year price ratio Year Price ratio 1987 1.000 1988 1.045 1989 1.043 1990 1.042 × 1991 1.060 1992 1.019 1993 1.019 1994 1.927 Value of price index = value of price index In given year in previous year price ratio in given year relative to previous year Values of price index for bread ( the previous year method ) Year Index 1987 100 1988 104.5 1989 109.0 1990 113.6 1991 120.4 1992 122.7 1993 125.0 1994 115.9 Table 12 Year – on – year price ratios for the shopping basket Year Price Ratio 1987 1.000 1988 1.043 1989 1.050 1990 1.065 1991 1.070 1992 1.038 1993 1.005 1994 1.016 The price index for the shopping basket ( Activity 23 ) Year Index Activity 24 : 1987 100 1988 104.3 1989 109.5 1990 116.6 1991 124.8 1992 129.5 1993 130.2 1994 132.3 Price Ratio fro 1994 relative to 1990 (a) = value of the index in 1994 = 132.2 = 1.134 value of the index in 1990 Percentage rise in the price = 13.4 % ( b ) Price ratio for 1992 relative to 1989 = 129.5 = 1.183 109.5 Percentage rise in the price 18.3 % 116.6 7 Unit 2 Summary A price index The starting value of the index (that is , in the base year ) is 100 . The new value of index is calculated using the formula: Value of price index = value of price index x price ratios for the year in one year in previous year relative to the previous year 5) The Retail Price index : : The retail price index is calculated in essentially the same way as the price index for the shopping basket of five items described in subsection 4.3 ; However it is calculated once a month instead of once a year , and it is based on a very large basket and the weights assigned to the basket are updated once a year to reflect changes in spending patterns . Each month the price ratio for the basket is calculated relative to the previous January . Then the value of index is obtained by multiplying the value of the index for the previous January by the price ratio , For example RPI for may 1994 = RPI for Jan 1994 × Price Ratio for May 94 Relative to January 94 The items in the basket are grouped into five fundamental groups each group is divided into forteen more detailed subgroups , each subgroup may be split into ninty five sections. Finally, the sections are comprised of the six hundred or so individual items on which the RPI is calculated. 8 Unit 2 Weighted means are used to calculate in turn . Section price rations Subgroup price ratios Group price ratios The all item price ratios ( Frame 3 ) shows how to calculate the RPI for April 1994 value of the RPI in Jan 1994 = 141.3 . All item price ratio ( from the Table ) = ∑ x w ∑ w = 1.020128 value of the RPI for April 1994 = value of the RPI for Jan. 1994 × all item Ratio = ( 141.3 ) × ( 1.020128 ) = 144.1 Frame 5 shows how to calculate the RPI fro March . All items price ratio for March 1994 relative to Jan 1994 = 1.008896 Value of RPI = In March 1994 Value of the RPI in Jan 94 × All item ratio for march 94 relative to Jan. 94 = 141.3 × 1.008896 = 142.5288861 = 142.5 Now to calculate the RPI fro Jun 1994 : All item price ratio fro Jun 94 relative to Jan. 94 = 1.023864 Value of RPI in Jun 94 = value of RPI in Jan. 94 × all item price ratio = 141.3 × 1.023864 = 144.7 9 Unit 2 6) Using the RPI : : We have three uses : ( I ) To calculate the annual rate of inflation The annual rate of inflation = percentage increase in the RPI So the annual rate of inflation = The value of RPI in later year - 1 -1 The value of RPI in later year The value of RPI in the earlier year The value of RPI in the earlier year × 100% Example :The date : Date RPI Dec. 1992 139.2 Dec. 1993 141.9 Jan. 1993 137.9 Jan. 1994 141.3 1) Find the annual rate of inflation in Dec. 1993 relative to the year 1992 . Solution :- The annual rate of inflation in 1993 = The value of RPI in 1993 - 1 The value of RPI in 1992 × 100% = 141.9 - 1 139.2 × 100% = 1.9 % 2) Find the annual rate of inflation in Jan. 1994 The annual rate of inflation in Jan. 1994 = 141.3 - 1 134.9 × 100 % = 2.5% 3) Find the annual rate of inflation in July 1994 , if the value of the RPI July 1994 was 144.0 ; its value in July 1993 was 140.7 10 Unit 2 Solution : The annual rate of inflation in July 1994 = The value of RPI in 1994 - 1 The value of RPI in 1993 1.023 – 1 × 100% × 100% = 144.0 - 1 140.7 × 100% = 2.3 % II ) RPI - linked pensions .( The second use ) Pension at later date = pension at earlier date × 1) RPI at later date RPI at earlier date A pension was £90 per week in December 1992 . Find the pension in December 1993 Solution : Pension in December 1993 = Pension in December 1992 X RPI in December 1993 RPI in December 1992 = £ 90 X 141.9 139.2 ≈ £ 69.68 2) An index – linked pension was £ 104 per week in July 1993 what should it be in July 1994 ? Solution : Pension in July 1994 = Pension in July 1993 X RPI in July 1994 RPI in July 1993 = £ 104 X 144.0 140.4 ≈ £ 106.44 11 Unit 2 3) A pension was £68 per week in January 1993 . In January 1994 it should be : Pension in January 1993 X RPI in January 1993 = £ 68 X 141.3 ≈ £ 69.68 137.9 RPI in January 1994 Solution :- Pension in July 1994 = pension in July 1993 X RPI in July 1994 RPI in July 1993 = 104 X 144.0 = £ 106.44 140.4 III) Third use is in the purchasing power of the pound . The purchasing power ( in pence ) of it the pound at a given date compared with an earlier date is : Value of RPI at the earlier date Value of RPI at the later date × 100 P Examples : 1. Find the purchasing power ( in pence ) of the pound in December 1993 Compared with a year earlier ( 1992 ) Solution : The value of RPI in 1992 The value of RPI in 1993 × 100 P = 144.0 140.7 × 100 P = 98 P = The purchasing power of the pound in December 12 Unit 2 2. Find the purchasing power of the pound in July 1994 compared with July 1993 . Solution : The purchasing power of the pound in July 1994 137.9 144.0 × 100 P = 96 P 3. Find the purchasing power of the pound in Jan, 1994 compared With December 1992 . Solution : The purchasing power of the pound in July 1994 139.2 141.3 × 100 P = 99 P 7) Some mathematical themes : The main aim of this section is to review some of the mathematical skills and idea you have been using , and for you to reflect on some of their more general features and applications . 7.1 Relative and absolute comparisons : Example : Refer to table ( 15) Page 67 : As you can see there are many more births in the U.K ( 800000 is far more than 50000 ) , but using the birth rates of the two countries gives the following results : 13 Unit 2 a) The birth rate for the Republic of Ireland in 1992 was 500000 X 1000 = 14 ( rounded to the nearest whole no. ) 3500000 b) The birth rate for the UK was 800,0000 X 1000 = 14 57700,000 ( rounded to the nearest whole no. ) Thus the relative comparison is more meaningful than absolute comparison. There are two other examples pages 68,69 reflecting the same idea . 7.2 Ratio and proportion : If two ratios are equal then we will have a propotion that is if a,b,c and d are real numbers , and if 1. 2. a = c then ad = bc provided that b,d are # O. b d a = c then b = d b d a c Example : Solve the proportion 2 = 6 x 3 then 6 x = 6 → x = 1 Now for a more practical example of proportion . Activity 39 : More ice cream = The ingredients for six servings of hazelnut ice cream are given in the table page 73 . We need to complete the table for eight servings . 14 Unit 2 So if we need 225g of toasted hazelnut for 6 servings , then 225 for 6 x for 8 6 8 g i . e 225 = x x = 8 × 225 6 Note also that two shapes which are in proportion to each other have the same shape but different size , they are said to be similar. See the figure. Notice that these two triangles are similar . i.e their corresponding angles are congruent and their corresponding sides are proportional . Finally, this unit has looked at a variety of ways of comparing prices , and the construction of a price index . Important statistical ideas that contributed to this included mean , weighted mean and median , as well as the general notion of an index. 15 Unit 2