Question

The following table gives the bushels of corn per acre, y resulting from the use of the various amounts of fertilizer in pounds per acre, x, produced on a farm in each of 10 years from 1971 to 1980.  

Year	n	y	x
1971	1	40	6
1972	2	44	10
1973	3	46	12
1974	4	48	14
1975	5	52	16
1976	6	58	18
1977	7	60	22
1978	8	68	24
1979	9	74	26
1980	10	80	32

A) Calculate the SST, SSE, and SSR of the model.

B) Calculate the estimated variances and the standard errors of and .

C) Calculate the R² term of the estimated model and interpret the R² term.

Answer 1

X	Y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
6	40	144	289	204
10	44	64	169	104
12	46	36	121	66
14	48	16	81	36
16	52	4	25	10
18	58	0	1	0
22	60	16	9	12
24	68	36	121	66
26	74	64	289	136
32	80	196	529	322

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	180	570	576	1634	956
mean	18	57	SSxx	SSyy	SSxy

sample size ,   n =   10      
here, x̅ =   18       ȳ =   57
              
SSxx =    Σ(x-x̅)² =    576      
SSxy=   Σ(x-x̅)(y-ȳ) =   956      
a)

SSE=   (Sx*Sy - S²xy)/Sx =    47.31
SSR=   S²xy/Sxx =   1586.69

SST=SSE+SSR=47.31+1586.69=1634.00

b)estimated variance=SSE/(n-2)=5.91319

estimate of std error=Se =    √(SSE/(n-2)) =    2.4317

c)

R² =    (Sxy)²/(Sx.Sy) =    0.9710
97.10% variation in observation of variable y, is explained by variable x.