Question

1.      The U.S. Department of Agriculture publishes data annually on various selected farm products. Shown here are the unit production figures (in millions of bushels) for three farm products for 10 years during a 20-year period. (leave 3 decimal places)

Corn	Soybeans	Wheat
4152	1127	1352
6639	1798	2381
4175	1636	2420
7672	1861	2595
8876	2099	2424
8226	1940	2091
7131	1938	2108
4929	1549	1812
7525	1924	2037
7933	1922	2739

(1). If we want to predict the corn production by the production of soybeans and wheat. What is the dependent variable and what are the independent variables? (Written)

(2). Write down the multiple linear regression model. How to interpret the coefficients in the regression? (Written)

Answer 1

1. Here we want to predict the corn production by the production of soybeans and wheat.

So independent variable is soybeans and wheat and dependent variable is corn

2.

Sum of X₁ = 17794
Sum of X₂ = 21959
Sum of Y = 67258
Mean X₁ = 1779.4
Mean X₂ = 2195.9
Mean Y = 6725.8
Sum of squares (SS_X1) = 700612.4
Sum of squares (SS_X2) = 1494276.9
Sum of products (SP_X1Y) = 3814621.8
Sum of products (SP_X2Y) = 3477858.8
Sum of products (SP_X1X2) = 739466.4

Regression Equation = ŷ = b₁X₁ + b₂X₂ + a

b₁ = ((SP_X1Y)*(SS_X2)-(SP_X1X2)*(SP_X2Y)) / ((SS_X1)*(SS_X2)-(SP_X1X2)*(SP_X1X2)) = 3128341511432.1/500098368444.6 = 6.25545

b₂ = ((SP_X2Y)*(SS_X1)-(SP_X1X2)*(SP_X1Y)) / ((SS_X1)*(SS_X2)-(SP_X1X2)*(SP_X1X2)) = -384153649078.4/500098368444.6 = -0.76816

a = M_Y - b₁M_X1 - b₂M_X2 = 6725.8 - (6.26*1779.4) - (-0.77*2195.9) = -2718.35776

ŷ = 6.25545X₁ - 0.76816X₂ - 2718.35776

So for every increase in x1(=soyabeans), y (=corn) will increase to 6.25545

For every increase in x2(=wheat), y(=corn) will decrease to 0.76816