Question

a. Develop the estimated regression equation by using the formula for computing the values for bo and b1. “ SHOW YOUR WORK” means WRITE the formula and create the columns as needed below follow the steps through. If you need more space below, create them by pressing ENTER
b. Compute SSE, SST, and SSR using only Computing formulas.
c. Compute the coefficient of determination r (The same as R ). Comment on the goodness of fit (That is, what does the value of r tell you about the goodness of the fit). Explain why.
d. Compute the sample correlation coefficient (r). (Hint: You do not need to use the formula for r !)
e. Determine if the slope is significant at Alpha = 0.05ant

y   -3 -1   0   0   7   5   13
x   1   2   3   4   5   6   7

Answer 1

X	Y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
1	-3	9	36	18
2	-1	4	16	8
3	0	1	9	3
4	0	0	9	0
5	7	1	16	4
6	5	4	4	4
7	13	9	100	30

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	28	21	28	190	67
mean	4	3	SSxx	SSyy	SSxy

sample size ,   n =   7      
here, x̅ =   4   ,   ȳ =   3
              
SSxx =    Σ(x-x̅)² =    28      
SSxy=   Σ(x-x̅)(y-ȳ) =   67      

a)

slope ,    ß1 = SSxy/SSxx = 67/28 = 2.393          
                  
intercept,   ß0 = y̅-ß1* x̄ = 3-2.393*4 = -6.571          
                  
so, regression line is   Ŷ =   -6.5714   +   2.3929   *x

b)

SSE=   (Sx*Sy - S²xy)/Sx = (28*190-67²)/28 = 29.68

SST=SSyy = 190

SSR=SST-SSE=190-29.68=160.32

c)

coefficient of determination,R² = SSR/SST=0.8438

84.38% of obeservation of Y is explained by X, so, it is a good estimate.

d)

sample correlation coefficient (r). =√R² =√0.8438 = 0.9186

e)

std error ,Se =    √(SSE/(n-2)) =    2.4363

slope hypothesis test              

Ho:   ß1=   0          
H1:   ß1╪   0          
n=   7              
alpha=   0.05              
estimated std error of slope =Se(ß1) =                s/√Sxx = 2.4363/√28 = 0.4604
                  
t stat =    ß1 /Se(ß1) = 2.3929/0.4604 = 5.197   
                  
df=n-2 = 5   
p-value =    0.0035              
decision :    p-value<α , reject Ho

so, there is enough evidence that slope is significant at α=0.05