Question

Consider the data.

xi	3	12	6	20	14
yi	55	35	45	10	15

The estimated regression equation for these data is ŷ = 62.25 − 2.75x.

(a) Compute SSE, SST, and SSR using equations SSE = Σ(y_i − ŷ_i)², SST = Σ(y_i − y)², and SSR = Σ(ŷ_i − y)².

SSE=

SST=

SSR=

(b) Compute the coefficient of determination r².(Round your answer to three decimal places.)

r² =

Comment on the goodness of fit. (For purposes of this exercise, consider a proportion large if it is at least 0.55.) Chose one of the following.

1.The least squares line provided a good fit as a large proportion of the variability in y has been explained by the least squares line.

2.The least squares line did not provide a good fit as a large proportion of the variability in y has been explained by the least squares line.    

3.The least squares line provided a good fit as a small proportion of the variability in y has been explained by the least squares line.

4.The least squares line did not provide a good fit as a small proportion of the variability in y has been explained by the least squares line.

(c) Compute the sample correlation coefficient. (Round your answer to three decimal places.)

Answer 1

	X	Y	X * Y	X2	Y2	Ŷ	SSE =Σ(Y - Ŷ)2	SST = Σ(Yi - Y̅ )2	SSR = Σ( Ŷ - Y̅ )2
	3	55	165	9	3025	54	1	529	484
	12	35	420	144	1225	29.25	33.0625	9	7.5625
	6	45	270	36	2025	45.75	0.5625	169	189.0625
	20	10	200	400	100	7.25	7.5625	484	612.5625
	14	15	210	196	225	23.75	76.5625	289	68.0625
Total	55	160	1265	785	6600	160	118.75	1480	1361.25

X̅ = Σ (Xi / n ) = 55/5 = 11
Y̅ = Σ (Yi / n ) = 160/5 = 32

Equation of regression line is Ŷ = a + bX
b = ( n Σ(XY) - (ΣX* ΣY) ) / ( n Σ X² - (ΣX)² )
b = ( 5 * 1265 - 55 * 160 ) / ( 5 * 785 - ( 55 )²)
b = -2.75

a =( ΣY - ( b * ΣX ) ) / n
a =( 160 - ( -2.75 * 55 ) ) / 5
a = 62.25
Equation of regression line becomes Ŷ = 62.25 - 2.75 X

Part a)

SSE =Σ(Y - Ŷ)² = 118.75

SST = Σ(Yi - Y̅ )² = 1480

SSR = Σ( Ŷ - Y̅ )²  = 1361.25

Part b)

Coefficient of Determination
R² = r² = 0.92
Explained variation = 0.92* 100 = 92%
Unexplained variation = 1 - 0.92* 100 = 8%

1.The least squares line provided a good fit as a large proportion of the variability in y has been explained by the least squares line.

Part c)

r = -0.959