Question

7

7. The following 12 data pairs relate variable x_i, the amount of fertilizer, to variable Y_i, the amount of wheat harvested:             

                    x: 30 30 30 50 50 50 70 70 70 90 90 90

                    Y: 9 11 14 12 14 23 19 22 31 29 33 35                                                        

such that :

∑x = 720   ∑y = 252 ,   ∑ xy =17240,    ∑x² =49200,      ∑y² = 6228

a) Find equation of linear regression line: Y = A + BX. b) 95% 2 sided confidence interval for B. c) Is there regression on input variable? d) Find 2-sided 99% prediction interval for response if x₀ = 40. e) Calculate R², explain its meaning. [5+5+5+5+5 = 25]

Answer 1

Given that

x: Amount of fertilizer

y: Amunt of wheat harvested

∑x = 720   ∑y = 252 ,   ∑ xy =17240,    ∑x2 =49200,      ∑y2 = 6228

b=Sxy/Sxx=2120/6000=0.3533

a=ybar-b*xbar=21-0.3533*60= -0.2

The regression equation is Y^=a+bx=> Y^=(-0.2)+0.3533x

To find 95% confidence interval for b ,

let us find all summary

SStotal=Syy=936, SSreg =b*Sxy=0.3533*2120=749.0667

Therefore SSE=SStotal-SSreg= 936-749.0667 = 186.9333

MSE=SSE/n-2= 186.933/10= 18.6933

Also From t table Tcrit=Tn-2,0.05/2= 2.228

Therefor 95% confidence interval for b is

(b Tcrir*Sb1)

(0.3533 2.228*0.1244)

(0.3533 0.1244)

(0.2289,0.4777)

d) To find Prediction interval at x=40

y=(-0.2)+0.3533x

yh=-0.2+0.3533*40= 13.9333

99% Prediction Interval is:

(3.9219,23.9448)

e) R^2=SSreg/SStotal= 749.0667/936= 0.8033

R^2*100=0.8003*100= 80.03% variation is explained wheat harvested amount is explained by the fertilizers amount