Question

Suppose a researcher converts all of their recorded throwing distances to inches by multiplying every Y value by 12. The transformed data are shown in the table below. Compute the equation of the regression line (the regression of Y on X) as well as the standard error of estimate. Does the regression equation and the standard error of estimate change or remain unchanged when one of the variables is transformed? (Compared to if the Y values were left in their original feet measurement.)

Distance jumped in inches (X)	Distance of discus throw in inches (Y)
3	12
1	36
4	60
1	12
3	48
5	12
1	24

Answer 1

The regression equation is given by

Y = a+bX

where a and b are given by

Calculating the values

	Distance jumped in inches (X)	Distance of discus throw in inches (Y)	XY	X2	Y2
	3	12	36	9	144
	1	36	36	1	1296
	4	60	240	16	3600
	1	12	12	1	144
	3	48	144	9	2304
	5	12	60	25	144
	1	24	24	1	576
Sum	18	204	552	62	8208

N= 7, we have

Substituting the values in the equation of b, we get

So, the Regression equation is

Standard Error of the estimate is given by

where Y hat is the predicted value

Distance jumped in inches (X)	Distance of discus throw in inches (Y)	Y hat	Y -Y hat	(Y-Y hat)2
3	12	29.89	-17.89	320.08
1	36	26.40	9.60	92.16
4	60	31.64	28.36	804.50
1	12	26.40	-14.40	207.36
3	48	29.89	18.11	327.94
5	12	33.38	-21.38	457.18
1	24	26.40	-2.40	5.76
			Sum	2214.98

Y hat s calculated from the equation for each values of x

So, the standard error of the estimate is 21.05

What happens if a variable is transformed (If we transform Y)

If we transform Y, ie if we multiply Y values by a constant k, then observing the formula for b, in the numerator, we have the effect of k in both the terms and we can take that k out. So b would be also multiplied by k

Similarly for a, both the numerator terms will have the effect of k, so a would also be multiplied by k

And for Standard error also, the error term would be multiplied by k, giving the new SE as multiplied by k

So, if Y becomes K*Y

=> a = K*a

=> b =K*b

=>SE = K*SE

So, yes the value of regression and standard error would change if Y is transformed and for the earlier case, we would have the values of a, b and Standard error as 1/12th of the calculated values above for each case