Question

An agricultural scientist used four test plots to determine the relationship between wheat yield y (in bushels per acre) and the amount of fertilizer x (in hundreds of pounds per acre). The table shows the results.

Fertilizer, x	Yield, y
1.0	31
1.5	39
2.0	47
2.5	55

(a) Find the least squares regression line

y = ax + b

for the data by solving the system for a and b

4b	+	7.0a	=	172
7b	+	13.5a	=	321

y =

(b) Use the linear model from part (a) to estimate the yield for a fertilizer application of 156 pounds per acre.
bushels per acre

Answer 1

a) Given, 4b+7.0a = 172..........(i)

7b+13.5a = 321..........(ii)

Multiplying (i) by 7 and (ii) by 4 and then subtracting we get,

(28b+49a)-(28b+54a) = 1204-1284

i.e., -5a = -80

i.e., 5a = 80

i.e., a = 80/5

i.e., a = 16

Putting this value in (i) we get, 4b+(7*16) = 172

i.e., 4b+112 = 172

i.e., 4b = 172-112

i.e., b = 60/4

i.e., b = 15

Therefore, a = 16 and b = 15.

Hence, the least squares regression line is, y = 16x+15.

b) Using the linear model from part (a), we are going to estimate the yield for a fertilizer application of 156 pounds per acre.

Since x is in hundreds of pounds per acre, therefore given value of x = 156/100 ,i.e., x = 1.56

Putting x = 1.56 in y = 16x+15 we get,

y = (16*1.56)+15

i.e., y = 39.96

Therefore, required wheat yield is 39.96 bushels per acre.