Quadratic regression. In quadratic regression, we find the parabola which best fits the data. The easiest method for finding such a parabola is to use the principle of least squares. In mathematical terms, the parabola $\displaystyle {y}={a}{x}^{{2}}+{b}{x}+{c}$ is fitted to the data $\displaystyle {\left({x}_{{1}},{y}_{{1}}\right)},{\left({x}_{{2}},{y}_{{2}}\right)},\ldots,{\left({x}_{{n}},{y}_{{n}}\right)}$ by minimizing the sum

$\displaystyle {S}={\sum_{{{i}={1}}}^{{n}}}{\left[{y}_{{i}}-{\left({a}{{x}_{{i}}^{{2}}}+{b}{x}_{{i}}+{c}\right)}\right]}^{{2}}$

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1. Determine the system of linear equations which must be satisfied by the least squares parabola.
   
   

   $\displaystyle {\left[\begin{array}{c} \\\\\\\\\\\end{array}\right.}$ $\displaystyle {\sum_{{{i}={1}}}^{{n}}}$ $\displaystyle {\sum_{{{i}={1}}}^{{n}}}$ $\displaystyle {\sum_{{{i}={1}}}^{{n}}}$ $\displaystyle {\left.\begin{array}{c} \\\\\\\\\\\end{array}\right]}$ $\displaystyle {\sum_{{{i}={1}}}^{{n}}}$ $\displaystyle {\sum_{{{i}={1}}}^{{n}}}$ $\displaystyle {\sum_{{{i}={1}}}^{{n}}}$ $\displaystyle {\sum_{{{i}={1}}}^{{n}}}$ $\displaystyle {\sum_{{{i}={1}}}^{{n}}}$ $\displaystyle \ \ \ \ \ \ \ \$

   $\displaystyle {\left[\begin{array}{c} \\\\\\\\\\\end{array}\right.}$ $\displaystyle {a}$ $\displaystyle {\left.\begin{array}{c} \\\\\\\\\\\end{array}\right]}$ $\displaystyle {b}$ $\displaystyle {c}$

   =

   $\displaystyle {\left[\begin{array}{c} \\\\\\\\\\\end{array}\right.}$ $\displaystyle {\sum_{{{i}={1}}}^{{n}}}$ $\displaystyle {\left.\begin{array}{c} \\\\\\\\\\\end{array}\right]}$ $\displaystyle {\sum_{{{i}={1}}}^{{n}}}$ $\displaystyle {\sum_{{{i}={1}}}^{{n}}}$

   
   
2. Consider the nitrogen/corn yield data set shown below which contains the pound of nitrogen per acre and the number of bushels of corn per acre for eight plots. Source: P.R. Johnson (1953). "Alternative functions for Analyzing a Fertilizer-Yield Relationship", Journal of Farm Economics, Vol. 35, #4, pp 519-529.
   
   

   Nitrogen	Yield
   0	24.9
   20	43
   40	50.5
   60	63.2
   80	73.6
   120	83.1
   160	95.6
   180	90.1

   
   
1. Enter the numerical coefficients of the linear system which must be satisfied by the least squares parabola for the corn yield data set.
   
   

   $\displaystyle {\left[\begin{array}{c} \\\\\\\\\\\end{array}\right.}$ $\displaystyle {\left.\begin{array}{c} \\\\\\\\\\\end{array}\right]}$

   $\displaystyle {\left[\begin{array}{c} \\\\\\\\\\\end{array}\right.}$ $\displaystyle {a}$ $\displaystyle {\left.\begin{array}{c} \\\\\\\\\\\end{array}\right]}$ $\displaystyle {b}$ $\displaystyle {c}$

   =

   $\displaystyle {\left[\begin{array}{c} \\\\\\\\\\\end{array}\right.}$ $\displaystyle {\left.\begin{array}{c} \\\\\\\\\\\end{array}\right]}$

   
   
2. Provide the equation of the least squares parabola. Enter the coefficients rounded to 4 decimal places.
   
   $\displaystyle {y}=$ Preview Question 6 Part 25 of 26  
   
   
3. Use the least square parabola to predict the corn yield for a plot fertilized at a rate of 100 pounds per acre. Round your answer to 1 decimal place.