Question

country	stork pairs(102)	birth rate (104/year)
Albania	1	8
Austria	3	9
Greece	25	11
Portugal	15	12
Romania	50	35
Spain	80	45

g) Calculate the average error for the predicted birth rates

h) Calculate the proportion of variance accounted for and explain it verbally.

Answer 1

let X : Stork pairs(102), Y : birth rate(104/year), we are going to find the linear regression model such that Y= a+bx, with a and b being the intercept and the slope(regression coefficient) parameters.

sl. no.
1	1	8	784	144	336
2	3	9	676	121	286
3	25	11	16	81	36
4	15	12	196	64	112
5	50	35	441	225	315
6	80	45	2601	625	1275
Total	174	120	4714	1260	2360
mean	29	20

hence, Y= 5.4815 + 0.5006 x

Sl. no.	X	Y	Y(predicted)	( Ypredicted- Y)2
1	1	8	5.9821	4.07192041
2	3	9	6.9833	4.06707889
3	25	11	17.9965	48.95101225
4	15	12	12.9905	0.98109025
5	50	35	30.5115	20.14663225
6	80	45	45.5295	0.28037025
			squared sum of error	78.4981043
			average	13.08301738

Hence g) average error for the predicted birth rates = 13.083 (approximately)

MSE = SSE/ (n-2)= 78.4981/(6-2)= 19.6245

h) note that, for a simple linear regression problem, r = correlation coefficient and R²= coefficient of determination

and  ,

= 0.9377 (approx.)

Converting the r² into a percentage we get 93.77%, hence, about 93.77% of the total variation in the response variable is explained by the simple linear regression model. It is a very high value and indicates a better goodness of fit for the observations. The R²  is used as a measure for how well , change in the response variable can be explained by changes in the predictor variable.