barcode font reporting services Curve Fitting, Regression, and Correlation in Software

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CHAPTER 8 Curve Fitting, Regression, and Correlation
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to 1, 2, 3, 4, 5 and 1, 2, 3, 4, 5, respectively. With this choice gx and gx3 are zero and equations (2) are simplified. The work involved in computation can be arranged as in Table 8-22. The normal equations (2) become 11a (3) 110a From the second equation in (3), b Then the required equation is (4) where the origin, x y 110c 110b 1958c 886.8 1429.8 9209.0 76.64, c 0.3974.
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13.00; from the first and third equations, a 76.64 13.00x 0.3974x2
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0, is July 1, 1900, and the unit of x is 10 years.
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(b) The trend values, obtained by substituting x 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5 in (4), are shown in Table 8-23 together with the actual values. It is seen that the agreement is good.
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Table 8-22
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Year 1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 gx x 5 4 3 2 1 0 1 2 3 4 5 0 y 23.2 31.4 39.8 50.2 62.9 76.0 92.0 105.7 122.8 131.7 151.1 gy 886.8 x2 25 16 9 4 1 0 1 4 9 16 25 gx2 110 x3 125 64 27 8 1 0 1 8 27 64 125 gx3 0 x4 625 256 81 16 1 0 1 16 81 256 625 gx4 1958 xy 116.0 125.6 119.4 100.4 62.9 0 92.0 211.4 368.4 526.8 755.5 gxy 1429.8 x2y 580.0 502.4 358.2 200.8 62.9 0 92.0 422.8 1105.2 2107.2 3777.5 gx2y 9209.0
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Table 8-23
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Year Trend Value Actual Value x 1850 21.6 23.2 5 x 1860 31.0 31.4 4 x 1870 41.2 39.8 3 x 1880 52.2 50.2 2 x 1890 64.0 62.9 1 x 0 x 1 x 2 x 3 x 4 x 5 1900 1910 1920 1930 1940 1950 76.6 76.0 90.0 92.0 104.2 105.7 119.2 122.8 135.0 131.7 151.6 151.1
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(c) 1945 corresponds to x
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4.5, for which y
76.64 76.64
13.00(4.5) 13.00(6)
0.3974(4.5)2 0.3974(6)2
(d) 1960 corresponds to x 6, for which y too well with the actual value, 179.3.
168.9. This does not agree
(e) 1840 corresponds to x 6, for which y 76.64 13.00( 6) 0.3974( 6)2 12.9. This does not agree with the actual value, 17.1. This example illustrates the fact that a relationship which is found to be satisfactory for a range of values need not be satisfactory for an extended range of values.
CHAPTER 8 Curve Fitting, Regression, and Correlation
8.57. The average prices of stocks and bonds listed on the New York Stock Exchange during the years 1950 through 1959 are given in Table 8-24. (a) Find the correlation coefficient, (b) interpret the results. Table 8-24 Year Average Price of Stocks (dollars) 1950 35.22 1951 39.87 1952 1953 1954 1955 53.29 1956 1957 1958 1959
41.85 43.23 40.06
54.14 49.12 40.71 55.15
Average Price of 102.43 Bond (dollars)
100.93 97.43 97.81 98.32 100.07 97.08 91.59 94.85 94.65
Source: New York Stock Exchange.
(a) Denoting by x and y the average prices of stocks and bonds, the calculation of the correlation coefficient can be organized as in Table 8-25. Note that the year is used only to specify the corresponding values of x and y.
Table 8-25
x 35.22 39.87 41.85 43.23 40.06 53.29 54.14 49.12 40.71 55.15 gx 452.64 x # 45.26 y 102.43 100.93 97.43 97.81 98.32 100.07 97.08 91.59 94.85 94.65 gx 975.16 y # 97.52 xr x x # yr y xr2 y # 100.80 29.05 11.63 4.12 27.04 64.48 78.85 14.90 20.70 97.81 gxr 449.38
xryr 49.30 18.38 0.31 0.59 4.16 20.48 3.91 22.89 12.15 28.38 gxryr 94.67
yr2 24.11 11.63 0.01 0.08 0.64 6.50 0.19 35.16 7.13 8.24 gyr2 93.69
10.04 5.39 3.41 2.03 5.20 8.03 8.88 3.86 4.55 9.89
4.91 3.41 0.09 0.29 0.80 2.55 0.44 5.93 2.67 2.87
Then by the product-moment formula, r B a xryr Q a xr2 R Q a yr2 R 94.67 !(449.38)(93.69) 0.4614
(b) We conclude that there is some negative correlation between stock and bond prices (i.e., a tendency for stock prices to go down when bond prices go up, and vice versa), although this relationship is not marked. Another method Table 8-26 shows the ranks of the average prices of stocks and bonds for the years 1950 through 1959 in order of increasing prices. Also shown in the table are the differences in rank d and gd 2.
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