If the standard error of estimate is zero there is a

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based on the regression equation. If the standard error of estimate is zero, there is a perfect match between the observed and estimated values and, hence, it is a case of perfect correlation. A short-cut formula for the determination of the standard error of estimate is given by 2 2 Y a Y b XY n All factors, except 2 Y , have been computed earlier for estimating the values of a and b. Example - 5 In a study of how wheat yield depends on the fertilizer input, ten experimental plots were selected for study. The yields (Y) at different levels of fertilizer input (X) were observed as given table. Plot no. Fertilizer input (X), kg/acre Yield (Y) MT/acre Plot no. Fertilizer input (X), kg/acre Yield (Y) MT/acre 1 2 3 4 5 15 25 30 40 50 15.0 20.0 22.5 25.0 28.0 6 7 8 9 10 60 75 80 90 100 30.0 32.0 35.0 37.5 40.0 Calculate the coefficient of determination. Also determine the standard error of estimate. Solution
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174 The estimated regression line for the sample data for fitting the least squared regression line, we need to obtain the values of a and b by solving the normal equations given in equation gives the values of 2 , , i i i X Y X and i i X Y . Plot No. Fertilizer Input (X), kg/acre Yield (Y) MT/acre 2 X XY Y 1 2 3 4 5 6 7 8 9 10 15 25 30 40 50 60 75 80 90 100 15.0 20.0 22.5 25.0 28.0 30.0 32.0 35.0 37.5 40.0 225 625 900 1600 2500 3600 5625 6400 8100 10000 225.0 500.0 675.0 1000.0 1400.0 1800.0 2400.0 2800.0 3375.0 4000.0 17.295 19.995 21.345 24.045 26.745 29.445 33.495 34.845 37.545 40.245 565 285.0 39575 18175.0 285.000 Also, we have 565 56.50 10 285.0 28.50 10 i i X X n Y Y n 2 2 i i i i i i n X Y X Y b n X X 2 10 1875 (565)(285) 20725 0.27 76725 10 39575 (565) b a Y bX 285 (565) 0.27 13.245 10 10 a Therefore, the estimated regression line is 13.245 0.27 i i Y X We have also obtained the values of Y for given values of X on the basic of the aforementioned regression line. These values and other required calculations are given in table. Plot No. Fertilizer input (X), kg/acre Estimated Y, Y Y Y 2 Y Y
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175 1 2 3 4 5 6 7 8 9 10 15.0 20.0 22.5 25.0 28.0 30.0 32.0 35.0 37.5 40.0 17.295 19.995 21.345 24.045 26.745 29.445 33.495 34.845 37.545 40.245 -2.295 0.005 1.155 0.955 1.255 0.555 -1.495 0.155 -0.045 -0.245 5.267 0 1.334 0.912 1.575 0.308 2.235 0.024 0.002 0.060 The coefficient of determination is given by 2 2 2 11.717 1 1 0.98 573.0 Y Y R Y Y Thus, the coefficient of determination is 0.98, very close to one. The standard error of estimate is given by 2 11.717 1.21 2 8 Y Y n The standard error of estimate is 1.21. 18.3.3. MULTIPLE REGRESSION ANALYSIS In the simple regression analysis, we have used the coefficient of determination as a measure to determine the strength of the relationship of Y on X. It is quite natural for the researcher to generate more precise estimates, say, of wheat production, than obtained by the simple regression method. A logical step to do this is by including additional independent variables in the analysis. The regression analysis that uses more than one independent variable is termed as multiple regression analysis. There are very few new concepts in multiple regression analysis. Most of the analysis carried out in the simple regression analysis can be extended directly to the multiple regression analysis. For example, the concept of
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