Multiple linear regression. In multiple linear regression, we use two or more explanatory variables to predict the value of the response variable. If we wish to predict the value of a random variable $\displaystyle {z}$ using the values of random variables $\displaystyle {x}$ and $\displaystyle {y}$, then we use a regression equation of the form $\displaystyle {z}={a}+{b}{x}+{c}{y}$. The easiest method for determing the coefficents is to use the principle of least squares. This involves minimizing the sum

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

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1. Determine the system of linear equations which must be satisfied by the coefficients for multiple linear regression.
   
   

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

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

   =

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

   
   
2. The data set provided below contains the heights (feet), diameters (inches) and volumes (cubic feet) of 31 black cherry trees from the Allegheny National Forest in Pennsylvania. To access the data, click on the button below. Click on the button a second time to hide the data.
   
   Data
   
   

   Diameter	Height	Volume
   7.3	72	10.3
   8.6	65	11.3
   7.8	64	11.2
   10.5	70	16.4
   9.7	80	19.8
   11.8	83	20.7
   12	68	16.6
   11	77	17.2
   11.1	78	21.6
   11.2	74	18.9
   11.3	78	23.2
   12.4	76	21
   12.4	78	22.4
   11.7	69	21.3
   13	77	19.1
   12.9	73	22.2
   11.9	83	33.8
   13.3	87	27.4
   12.7	69	26.7
   14.8	65	23.9
   13	76	33.5
   14.2	81	31.7
   13.5	72	36.3
   17	74	39.3
   15.3	75	43.6
   17.3	83	54.4
   16.5	81	55.7
   16.9	80	58.3
   18	79	51.5
   18	79	50
   20.6	88	76

   
   
   
1. Enter the numerical coefficients of the linear system which must be satisfied by the coefficients of the multiple linear regression equation.
   
   

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

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

   =

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

   
   
2. Provide the multiple linear regression equation. Enter the coefficients rounded to 4 decimal places.
   
   $\displaystyle {y}=$ Preview Question 6 Part 25 of 28  
   
   
3. Use the multiple linear regression equation to predict the usable volume of timber in a black cherry tree with a diameter of 12 inches and a height of 72 feet. Round your answer to 1 decimal place.
   
   
   
   
4. What is correct interpretation of the coefficient $\displaystyle {b}={4.3226}$ in the multiple linear regression equation?
   
   
   • If the diameter of the tree increases by 1 inch, then the volume of the tree increases by 4.3226 cubic feet.
   • If the diameter of the tree increases by 4.3226 inches, then the volume of the tree increases by 1 cubic foot.
   
   
5. What is correct interpretation of the coefficient $\displaystyle {c}={0.4211}$ in the multiple linear regression equation?
   
   
   • If the height of the tree increases by 1 foot, then the volume of the tree increases by 0.4211 cubic feet.
   • If the height of the tree increases by 0.4211 feet, then the volume of the tree increases by 1 cubic foot.