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Logarithms This series of slides demonstrates the usage of logarithms. Logs have a variety of applications in economics and finance. To obtain the spreadsheet that contains the examples click on the following: ftp://rohan.sdsu.edu/faculty/vandenberg/Logs.xls ms. Logs have a lick on the following: How do logarithms work? ROW/COL B C D E 4 5 6 Number 1 Number 2 Result of Multiplying 7 Number 10 10 100 8 Power of the number 1 1 2 9 10 Notice we get the same result by multiplying the numbers as we do by adding the power of the numbers. 11 The power of the number is also its exponent. By default we do not write the power if it is 1. 12 13 Works for any numbers Result of Multiplying 14 Number 100 10 1000 15 Power of the number 2 1 3 16 17 We also that we can write: 1,000,000 18 as #NAME? 19 or =10^6 (remember an ^ means raised to the power) 20 21 Thus 22 Number 1,000,000 1,000,000 1,000,000,000,000 23 Power 6 6 12 24 25 In fact the spread sheet might use: 1E+12 to represent the number. 26 As you can see the E+12 means move the decimal point 12 space to the right and you get the answer. 27 28 It would not be hard to figure the exponent for any number which is an even power of 10 29 30 So then we can see that 100,000 31 is the same as #NAME? 32 or =10^5 33 34 Rather than count the number of 10's need to get the number a spreadsheet provides a function to do this 35 36 Log of number (or the number 37 Number of 10s) Formula 38 10 1 #NAME? 39 100 2 #NAME? 40 1000 3 #NAME? 41 10000 4 #NAME? 42 100000 5 #NAME? 43 1000000 6 #NAME? 44 45 The log function is particularly helpful if we wish to find the power of 10 when the number is not exactly a 46 We can write 12,345 47 as =10^4.09149109426795 48 or the log of 12,345 = 4.09149109426795 49 50 Works for any numbers 51 Number 1 Number 2 Result of Multiplying 52 Number 12,345 678 8,369,910 53 Power of the number 4.091491094 2.831229694 6.922720788 54 55 56 In days before electronic calculators and computers logs were commonly used because addition is easier 57 Tables of logarithms were available as were slide rules which were base on logs. 58 59 To reverse the process, to convert a logarithm into a number, you need to do the opposite. 60 Again for even powers of 10 that is not difficult. For uneven powers of 10 use the spreadsheet power symb 61 62 Log of a number is: 4.091491094 63 Exponent form: =10 ^ 4.09149109426795 64 The number is 12,345 #NAME? 65 66 Therefore 10^(log(Number)) = Number 67 Example Number 68 12,345 12,345 #NAME? F G H I Mathematically we can say: =10*10 =1+1 adding the power of the numbers. te the power if it is 1. Mathematically we can say: =100*10 =2+1 an ^ means raised to the power) =1000000*1000000 =6+6 nt the number. e right and you get the answer. even power of 10 sheet provides a function to do this 0 when the number is not exactly a power of 10 Mathematically we can say: =12345*678 =4.09149109426795+2.83122969386706 nly used because addition is easier than multiplication to do the opposite. 10 use the spreadsheet power symbol (^) to do it. Log Scales ROW/COL B C D E F G 4 Frequently data is hard to understand without using logs 5 Here is some Forecasted data for the next few years. Look at the graphs below 6 7 Alternative log of way to 8 Year Forecast number express the # #NAME? 9 2000 10 1 10 ^1 10 2001 100 2 10 ^2 11 2002 1,000 3 10 ^3 12 2003 10,000 4 10 ^4 13 2004 100,000 5 10 ^5 14 2005 1,000,000 6 10 ^6 15 2006 10,000,000 7 10 ^7 16 2007 100,000,000 8 10 ^8 17 2008 1,000,000,000 9 10 ^9 18 2009 10,000,000,000 10 10 ^10 19 2010 100,000,000,000 11 10 ^11 20 2011 1,000,000,000,000 12 10 ^12 21 22 23 24 Here is one picture of the data 25 Notice that it is hard to see what is going on. 26 27 28 Original Data 29 30 31 1,200,000 Millions 32 33 1,000,000 34 35 36 800,000 37 38 600,000 Values 39 40 41 400,000 42 43 200,000 44 45 0 46 47 1995 2000 2005 2010 2015 48 -200,000 49 50 51 Time 52 53 54 55 Here is a picture using the log scale 56 Notice that you can see the series is going straight up. 57 58 59 60 61 Using the log scale of the Y axis 62 63 64 65 66 1,000,000,000,000 10 ^12 67 100,000,000,000 10 ^11 68 10,000,000,000 10 ^10 log values are graphed 69 70 1,000,000,000 10 ^9 71 100,000,000 10 ^8 72 10,000,000 10 ^7 73 74 1,000,000 10 ^6 75 100,000 10 ^5 76 10,000 10 ^4 77 1,000 10 ^3 78 79 100 10 ^2 80 10 10 ^1 81 1 82 83 1995 2000 2005 2010 2015 84 Time 85 86 87 88 Using the logs of the data rather than a log axis 89 You see the same effects 90 91 92 93 Transformed data 94 95 96 14 97 98 12 10 ^12 the log of the values 99 10 ^11 100 10 10 ^10 101 10 ^9 Using the log of the values 102 10 ^9 103 8 10 ^8 104 10 ^7 105 6 10 ^6 106 107 10 ^5 108 4 10 ^4 109 10 ^3 110 2 10 ^2 111 10 ^1 112 113 0 114 1998 2000 2002 2004 2006 2008 2010 2012 115 Time H I 2015 This result was achieved by right clicking the axis and then choosing to check this log box. 10 ^12 2012 The famous log scale. If you live in California you should be familiar with a famous log scale. It is called the Richter Scale, it measures an earthquake's power. So an earthquake with a Richter reading of 4 has a certain power. If another earthquake with a Richter reading of 5 occurs, how much more powerful is it. Since Richter readings are in log form then a 5 is 10 times more powerful than a 4 So the difference between a 3 and a 7 magnitude is substantial. A 7 is 10,000 more powerful than a 3. For more information try: http://neic.usgs.gov/neis/general/handouts/richter.html owerful than a 4 Other bases for logs. ROW/COL B C D E F G H I 4 5 6 There is nothing magic about base 10 (or common logarithms). You could build logs on any base you want. 7 In fact the log function has an option to set the base to any you may want to use. 8 9 Number log to the base 2 of the number 10 2 1 =2 ^ 1 11 4 2 =2 ^ 2 12 8 3 =2 ^ 3 13 16 4 =2 ^ 4 14 32 5 =2 ^ 5 15 64 6 =2 ^ 6 =log(number, base) 16 17 Other bases 18 Base 19 Number 2 5 10 16 20 2 1.0000 0.4307 0.3010 0.2500 #NAME? 21 4 2.0000 0.8614 0.6021 0.5000 22 5 2.3219 1.0000 0.6990 0.5805 23 10 3.3219 1.4307 1.0000 0.8305 24 16 4.0000 1.7227 1.2041 1.0000 25 20 4.3219 1.8614 1.3010 1.0805 26 #NAME? 27 The blue numbers are the logs of the number for different bases 28 Notice that when the base is equal to the number the log is always 1 29 30 Natural logarithms 31 32 Look at the following expansion: 33 X (1+1/X)^X 34 1 2.00000 As X expands the 35 2 2.25000 value of expression 36 3 2.37037 increases, but at a 37 4 2.44141 slower and slower 38 5 2.48832 rate and reaches a 39 6 2.52163 limit. This limit is a 40 7 2.54650 number which like 41 8 2.56578 pi, never rounds off 42 9 2.58117 even. This number 43 10 2.59374 is called "e." It has a 44 1010 2.71694 lot of useful 45 2010 2.71761 46 3010 2.71783 mathematical 47 4010 2.71794 properties. Because 48 5010 2.71801 of this it is used 49 6010 2.71806 frequently. As an 50 7010 2.71809 example the 51 8010 2.71811 derivative of e^x is e^x. Very convenient. mathematical properties. Because of this it is used frequently. As an example the derivative of e^x is 52 9010 2.71813 e^x. Very 53 10010 2.71815 convenient. 54 11010 2.71816 55 12010 2.71817 56 very large 2.7182818284591 <==Value of "e" used by Excel 57 1000 #NAME? 58 Natural logarithms use "e" as their base. 59 So the log to the base e is the value you need to raise e to get the number 60 61 Spreadsheets use a special set of functions to manipulate the natural logs. 62 63 To get the natural logarithm use =ln(number) 64 To get the anti-logarithm (to reverse the process) use =exp(mumber) 65 66 67 Number ln(number) Result 68 1 0.000000 = 2.71828182845905 ^ 0 = 1 69 2 0.693147 = 2.71828182845905 ^ 0.693147180559945 = 2 70 5 1.609438 = 2.71828182845905 ^ 1.6094379124341 = 5 71 7 1.945910 = 2.71828182845905 ^ 1.94591014905531 = 7 72 10 2.302585 = 2.71828182845905 ^ 2.30258509299405 = 10 73 25 3.218876 = 2.71828182845905 ^ 3.2188758248682 = 25 74 75 So again exp(ln(number)=number 76 77 Number 78 1000 1000 #NAME? 79 80 81 82 83 gs on any base you want. Natural logs are frequently used in finance. ROW/COL B C D E F G H 5 6 7 The compound value of a $ is: 8 9 10 11 12 V = P *(1+ i/n)^(n*t) 13 14 where: 15 16 V is the ending value 17 P is the present value you start with 18 I is the rate of interest 19 n is the frequency of compounding 20 t is the length of time 21 22 Notice that part of the formula looks a lot like the expansion to get the value of "e" 23 If we let P=1 and t=1 it is exactly the same formula. 24 25 26 27 28 V = 1 *(1+ i/n)^(n*1) 29 30 If n =1 then we get: (which is frequently the form of the compound amount formula) 31 32 33 34 35 V = P *(1+ i/1)^(1*t) 36 Impact of compounding 37 Using an interest rate of: 5.0000% 38 Assume the time period is 1 year and the amount is $1 39 40 Frequency of compounding n V Annual Rate of return 41 Annual 1 $1.05000 5.0000% 42 Semi-Annual 2 $1.05063 5.0625% 43 Quarterly 4 $1.05095 5.0945% 44 Monthly 12 $1.05116 5.1162% 45 Daily 365 $1.05127 5.1267% 46 Hourly 8760 $1.05127 5.1271% 47 Minutely 525600 $1.05127 5.1271% 48 Secondly 31536000 $1.05127 5.1271% 49 Infinite =EXP(i) $1.05127 5.1271% 50 51 Notice that as the frequency of compounding increase our rate of return increase, but at a decreasing rate. 52 Until the difference between daily, secondly and infinite is nil. 53 54 55 So if you allow infinite compounding the future value of a dollar formula becomes: 56 57 58 59 60 V = P*exp(i*t) 61 62 Remembering that ln and e^x are opposite transactions we can determine an equivalence between rates of i 63 64 65 You would indifferent between the following rates compounded once a year, or the rates compounded conti Compounded Compounded Compound amount of $1 66 once a year continuously at the end of a year ln(1+Annual Annual Rate of Rate of Once a 67 return Return) year Continuously 68 5.0000% 4.8790% $1.0500 $1.0500 69 5.0625% 4.9385% $1.0506 $1.0506 70 5.0945% 4.9690% $1.0509 $1.0509 71 5.1162% 4.9896% $1.0512 $1.0512 72 5.1267% 4.9997% $1.0513 $1.0513 73 5.1271% 5.0000% $1.0513 $1.0513 74 75 You can always find a rate that will cause you to be indifferent between an annual rate and continuously com 76 Continuously 77 Annual Rate Compounded 78 5.0000% 4.8790% #NAME? 79 80 Or you find a continuously compounded rate that is equal to an annual rate by using exp(i)-1 81 Continuously 82 Compounded Annual Rate 83 5.5000% 5.6541% #NAME? 84 85 Try it by changing the green numbers 86 87 88 In many cases theoretical consideration require that we use continuously compounded rates in finance. 89 In general when using the CAPM and Black/Scholes you should use continuously compounded rates. 90 I ln(Annual Rate of Return) 4.8790% 4.9385% 4.9690% 4.9896% 4.9997% 5.0000% 5.0000% 5.0000% 5.0000% e, but at a decreasing rate. uivalence between rates of interest and compounding the rates compounded continuously. al rate and continuously compounded one by taking the ln(1+i) sing exp(i)-1 ounded rates in finance. ly compounded rates. The fundamental equation of finance: ROW/COL B C D E F G H I 5 6 7 The compound amount equation is: 8 9 10 11 V = P *(1+ i)^t 12 13 This equation can also be stated as follows. 14 15 16 17 ln(V )= ln(P)+t*ln(1+ i) 18 19 20 I= 10.00% 21 P= $100.00 22 t= 10 23 24 ln(v)= 5.558272 #NAME? 25 26 V= $259.37 #NAME? 27 28 Using the FV function $259.37 #NAME? 29 30 An alternative would be: 31 32 33 34 35 36 ln($1.00 )=t*ln(1+ i) 37 38 39 I= 10.00% 40 P= $1.00 41 t= 10 42 43 ln(1)= 0.9531018 #NAME? 44 FV Factor 2.5937425 #NAME? 45 46 For $100 47 V= $259.37 #NAME? 48 49 50 Another alternative for annual compounding would be: 51 52 53 54 55 56 V=P*exp(t*ln(1+ i)) 57 58 I= 10.00% 59 P= $100.00 60 t= 10 61 62 V= $259.37 #NAME? 63 64 For continuous compounding the formula would be: 65 66 67 68 69 V=P*exp(t*i) 70 71 I (continuously) = 10.00% 72 P= $100.00 73 t= 10 74 75 V= $271.83 #NAME? 76 77 78 Note the difference of $12.45 #NAME? 79 is caused by the frequency of compounding 80 81 82 83 84 Stock Returns ROW/COL B C D E F G 3 4 As you know many things in this world are normally distributed. This is very convenient for statistical analy 5 6 Unfortunately stock returns are not normally distributed. You can described the reason in a variety of ways 7 8 However, there is a fair amount of evidence that stock returns are approximately normally distributed if you 9 To do this is fairly simple. 10 11 12 Below is the data for IBM from Feb, 1998 to May, 2003 The blue numbers are the ln returns which are in general the ones you want to use for analysis such as com 13 Beta's and it is also the data needed to implement the Option Pricing models. Adjusted Cash 1+ Return = 14 Date Close Dividend (Pt+Dt)/Pt-1 ln(1+Return) 15 #NAME? 16 Feb-98 51.81 17 Mar-98 51.53 0.99460 -0.542% #NAME? 18 Apr-98 57.48 1.11547 10.927% 19 May-98 58.29 1.01409 1.399% 20 Jun-98 56.96 0.97718 -2.308% 21 Jul-98 65.73 1.15397 14.321% 22 Aug-98 55.87 0.84999 -16.253% 23 Sep-98 63.75 1.14104 13.194% 24 Oct-98 73.67 1.15561 14.463% 25 Nov-98 81.92 1.11199 10.615% 26 Dec-98 91.47 1.11658 11.027% 27 Jan-99 90.91 0.99388 -0.614% 28 Feb-99 84.21 0.92630 -7.656% 29 Mar-99 87.93 1.04418 4.323% 30 Apr-99 103.78 1.18026 16.573% 31 May-99 115.09 1.10898 10.344% 32 Jun-99 128.24 1.11426 10.819% 33 Jul-99 124.71 0.97247 -2.791% 34 Aug-99 123.59 0.99102 -0.902% 35 Sep-99 120.06 0.97144 -2.898% 36 Oct-99 97.48 0.81193 -20.834% 37 Nov-99 102.26 1.04904 4.787% 38 Dec-99 107.03 1.04665 4.559% 39 Jan-00 111.37 1.04055 3.975% 40 Feb-00 101.95 0.91542 -8.838% 41 Mar-00 117.45 1.15204 14.153% 42 Apr-00 110.63 0.94193 -5.982% 43 May-00 106.47 0.96240 -3.833% 44 Jun-00 108.71 1.02104 2.082% 45 Jul-00 111.37 1.02447 2.417% 46 Aug-00 130.99 1.17617 16.226% 47 Sep-00 111.74 0.85304 -15.895% 48 Oct-00 97.73 0.87462 -13.397% 49 Nov-00 92.77 0.94925 -5.209% 50 Dec-00 84.34 0.90913 -9.527% 51 Jan-01 111.13 1.31764 27.584% 52 Feb-01 99.12 0.89193 -11.437% 53 Mar-01 95.43 0.96277 -3.794% 54 Apr-01 114.24 1.19711 17.991% 55 May-01 110.93 0.97103 -2.940% 56 Jun-01 112.61 1.01514 1.503% 57 Jul-01 104.39 0.92700 -7.580% 58 Aug-01 99.17 0.95000 -5.130% 59 Sep-01 91 0.91762 -8.598% 60 Oct-01 107.23 1.17835 16.412% 61 Nov-01 114.69 1.06957 6.726% 62 Dec-01 120.02 1.04647 4.543% 63 Jan-02 107.05 0.89193 -11.436% 64 Feb-02 97.36 0.90948 -9.488% 65 Mar-02 103.19 1.05988 5.816% 66 Apr-02 83.11 0.80541 -21.641% 67 May-02 79.82 0.96041 -4.039% 68 Jun-02 71.44 0.89501 -11.092% 69 Jul-02 69.85 0.97774 -2.251% 70 Aug-02 74.96 0.15 1.07530 7.260% 71 Sep-02 57.98 0.77348 -25.686% 72 Oct-02 78.5 1.35392 30.300% 73 Nov-02 86.59 0.15 1.10497 9.982% 74 Dec-02 77.21 0.89167 -11.466% 75 Jan-03 77.91 1.00907 0.903% 76 Feb-03 77.81 0.15 1.00064 0.064% 77 Mar-03 78.29 1.00617 0.615% 78 Apr-03 84.74 1.08239 7.917% 79 May-03 86.18 1.01699 1.685% 80 81 Monthly Average Return = 0.817% #NAME? 82 Monthly Standard Deviation = 11.317% #NAME? 83 84 Annual Average Return = 9.800% #NAME? 85 Annual Standard Deviation = 39.203% #NAME? 86 87 H I y convenient for statistical analysis. d the reason in a variety of ways but the basic reason is that stock prices cannot be negative. mately normally distributed if you take the log (natural) of the return. t to use for analysis such as computing ls. Other applications of logarithms: Trends ROW/COL B C D E F G H I 5 Other applications of logarithms 6 7 Suppose you have the following series of numbers for which you are trying to forecast the next one 8 10% 9 Year Sales Year ln(sales) 10 1 $100.00 1 4.6052 #NAME? 11 2 $110.00 2 4.7005 #NAME? 12 3 $121.00 3 4.7958 #NAME? 13 4 $133.10 4 4.8911 #NAME? 14 5 $146.41 5 4.9864 #NAME? 15 6 $161.05 6 5.0817 #NAME? 16 7 $177.16 7 5.1770 #NAME? 17 8 $194.87 8 5.2723 #NAME? 18 9 $214.36 9 5.3677 #NAME? 19 10 $235.79 10 5.4630 #NAME? 20 11 $259.37 11 5.5583 #NAME? 21 12 $285.31 12 5.6536 #NAME? 22 13 ? 13 ? 23 The linear trend forecasts: 24 Forecast Series ln(series) Exp(ln forecast) 25 13 $286.25 13 5.7489 $313.84 #NAME? 26 #NAME? #NAME? 27 28 It is clear that the process is not linear. Thus a linear estimate would not be expected to produce good estim 29 30 31 32 Sales 33 34 $350.00 6.0000 35 36 Forecast 5.8000 $300.00 37 5.6000 38 $250.00 5.4000 39 Sales ($) 40 5.2000 ln(Sales) $200.00 41 5.0000 42 $150.00 Trend Line 43 4.8000 44 $100.00 4.6000 45 Raw Data 4.4000 46 $50.00 47 4.2000 48 $0.00 4.0000 49 0 5 10 15 0 5 10 15 50 Time 51 52 53 54 ast the next one d to produce good estimate Natural Log of Sales 6.0000 Forecast 5.8000 5.6000 5.4000 5.2000 Trend Line 5.0000 4.8000 4.6000 4.4000 4.2000 4.0000 0 5 10 15 0 5 10 15 Time Multiplicative Demand Functions ROW/COL B C D E F G 5 For demand equations we can also use the following: 6 7 Q = aPcAdYe 8 This is called a multiplicative demand function. 9 The exponents are the elasticities of the variables. 10 11 12 Ln(Q) = a + c*ln(P) + d*ln(A) + e*ln(Y) 13 14 15 Some data: 16 17 Q P A Y 18 500 105.00 100.00 500.00 19 1800 90.00 195.00 576.54 20 900 75.00 95.00 600.00 21 1400 130.00 145.00 706.23 22 1800 59.00 120.00 750.00 23 1200 114.00 169.00 810.00 24 1300 105.00 95.00 795.00 25 975 165.00 152.00 815.00 26 2000 134.00 202.00 900.00 27 28 A linear fit using the above data 29 30 Q= f + g*P + h*A + i*Y 31 32 33 SUMMARY OUTPUT 34 35 Regression Statistics 36 Multiple R 0.888864171 37 R Square 0.790079515 38 Adjusted R Square 0.664127224 39 Standard Error 283.3424136 40 Observations 9 41 42 ANOVA 43 df SS MS F Significance F 44 Regression 3 1510807.61 503602.5 6.2728475 0.0379091 45 Residual 5 401414.617 80282.92 46 Total 8 1912222.22 47 48 CoefficientsStandard Error t Stat P-value Lower 95% 49 Intercept -142.008306 572.243073 -0.24816 0.8138798 -1613.00355 50 X Variable 1 -9.98905184 3.65624537 -2.73205 0.0411796 -19.3877144 51 X Variable 2 8.198651835 2.7095159 3.025873 0.0292191 1.23363086 52 X Variable 3 1.933355427 0.88602668 2.182051 0.0809034 -0.34424493 53 54 Estimating the results 55 #NAME? Estimated 56 Q P A Y Q Actual 57 500 105.00 100.00 500.00 596 500 58 1800 90.00 195.00 576.54 1672 1800 59 900 75.00 95.00 600.00 1048 900 60 1400 130.00 145.00 706.23 1114 1400 61 1800 59.00 120.00 750.00 1702 1800 62 1200 114.00 169.00 810.00 1671 1200 63 1300 105.00 95.00 795.00 1125 1300 64 975 165.00 152.00 815.00 1032 975 65 2000 134.00 202.00 900.00 1916 2000 66 67 68 Now try the non-linear fit 69 70 Take the logarithms of the data 71 72 ln(Q) ln(P) ln(A) ln(Y) 73 6.2146 4.6540 4.6052 6.2146 74 7.4955 4.4998 5.2730 6.3570 75 6.8024 4.3175 4.5539 6.3969 76 7.2442 4.8675 4.9767 6.5599 77 7.4955 4.0775 4.7875 6.6201 78 7.0901 4.7362 5.1299 6.6970 79 7.1701 4.6540 4.5539 6.6783 80 6.8824 5.1059 5.0239 6.7032 81 7.6009 4.8978 5.3083 6.8024 82 83 The multiplicative fit 84 85 SUMMARY OUTPUT 86 87 Regression Statistics 88 Multiple R 0.889270086 89 R Square 0.790801287 90 Adjusted R Square 0.665282059 91 Standard Error 0.251221433 92 Observations 9 93 94 ANOVA 95 df SS MS F Significance F 96 Regression 3 1.19286622 0.397622 6.3002402 0.03759558 97 Residual 5 0.31556104 0.063112 98 Total 8 1.50842726 99 100 CoefficientsStandard Error t Stat P-value Lower 95% 101 Intercept -1.81142129 3.01281415 -0.60124 0.573897 -9.55609396 102 X Variable 1 -0.84307889 0.32376373 -2.60399 0.048017 -1.67533869 103 X Variable 2 0.922450092 0.34014869 2.711902 0.0421803 0.04807146 104 X Variable 3 1.266559782 0.51433052 2.462541 0.0570503 -0.05556676 105 106 107 108 Remember that the above results are in logs, so the answer will be the log of sales and you would 109 need to take the anti-log (exp()) of the number ot get the resulting quantity sold 110 111 #NAME? Estimated Estimated 112 ln(Q) ln(P) ln(A) ln(Y) ln(Q) exp(Q) 113 6.2146 4.6540 4.6052 6.2146 6.3841 592 114 7.4955 4.4998 5.2730 6.3570 7.3105 1496 115 6.8024 4.3175 4.5539 6.3969 6.8514 945 116 7.2442 4.8675 4.9767 6.5599 6.9842 1079 117 7.4955 4.0775 4.7875 6.6201 7.5518 1904 118 7.0901 4.7362 5.1299 6.6970 7.4099 1652 119 7.1701 4.6540 4.5539 6.6783 6.9242 1017 120 6.8824 5.1059 5.0239 6.7032 7.0081 1106 121 7.6009 4.8978 5.3083 6.8024 7.5716 1942 122 123 #NAME? 124 125 H I gnificance F Upper 95% 1328.986939 -0.590389268 15.16367281 4.210955787 #NAME? Error 96 #NAME? -128 148 -286 -98 471 -175 57 -84 gnificance F Upper 95% 5.933251388 -0.010819089 1.796828719 2.588686326 of sales and you would sold Actual error 500 92 #NAME? 1800 -304 900 45 1400 -321 1800 104 1200 452 1300 -283 975 131 2000 -58