Question

Given below are four observations collected in a regression study on two variables x (independent variable) and y (dependent variable). *NOOO a. Develop the least squares estimated regression equation. b. At 95% confidence, perform at test and determine whether or not the slope is significantly different from zero. C. Perform an F test to determine whether or not the model is significant. Let a = 0.05. d. Compute the coefficient of determination.

Answer 1

x	y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
2	4	20.25	9.00	13.50
6	7	0.25	0.00	0.00
9	8	6.25	1.00	2.50
9	9	6.25	4.00	5.00

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	26	28	33	14.000	21.000
mean	6.500	7.000	SSxx	SSyy	SSxy

a)

sample size ,   n =   4          
here, x̅ = Σx / n=   6.50   ,     ȳ = Σy/n =   7.00  
                  
SSxx =    Σ(x-x̅)² =    33.0000          
SSxy=   Σ(x-x̅)(y-ȳ) =   21.0          
                  
estimated slope , ß1 = SSxy/SSxx =   21.0   /   33.000   =   0.63636
                  
intercept,   ß0 = y̅-ß1* x̄ =   2.86364          
                  
so, regression line is   Ŷ =   2.863636   +   0.636364   *x

b)

Ho:   ß1=   0          
H1:   ß1╪   0          
n=   4              
alpha =   0.05              
estimated std error of slope =Se(ß1) = Se/√Sxx =    0.564   /√   33.00   =   0.0982
                  
t stat = estimated slope/std error =ß1 /Se(ß1) =    0.6364   /   0.0982   =   6.4807
                  
Degree of freedom ,df = n-2=   2              
p-value =    0.0230              
decison :    p-value<α , reject Ho              
Conclusion:   Reject Ho and conclude that slope is significantly different from zero              

c)

Anova table
variation	SS	df	MS	F-stat	p-value
regression	13.3636	1	13.3636	42.00	0.0230
error,	0.6364	2	0.3182
total	14.0000	3

F=42

p value=0.0230

p-value<α , reject Ho

there is enough evidence that model is significant

d)

R² =    (Sxy)²/(Sx.Sy) =    0.9545