Simplicio collects some experimental data consisting of a great many pairs (X,Y) and with high confidence finds a correlation of 0.6 between X and Y. So given the value y of Y, his best prediction for the value of X is 0.6y.
Eh?
That’s just not what correlation means. (If we have, say, X=0.6Y or X=100Y or X=0.0001Y, exactly in each case, then the correlation is 1. The correlation tells you nothing about the coefficient in the relationship.)
Yes, that was an error. I was thinking of the case where X and Y are both normalised to have s.d. 1, in which case the regression line is indeed Y = cX, but that isn’t the case here. In general, the line is Y = bcX/a where the standard deviations of X and Y are a and b.
He said “best prediction”—there’s no implication of an exact relationship. But he did get the expression wrong (or he assumed that the variables were standardized to a mean of zero and a variance of one).
Eh?
That’s just not what correlation means. (If we have, say, X=0.6Y or X=100Y or X=0.0001Y, exactly in each case, then the correlation is 1. The correlation tells you nothing about the coefficient in the relationship.)
Presumably X and Y have been converted to canonical form with mean 0, sd 1.
Yes, that was an error. I was thinking of the case where X and Y are both normalised to have s.d. 1, in which case the regression line is indeed Y = cX, but that isn’t the case here. In general, the line is Y = bcX/a where the standard deviations of X and Y are a and b.
He said “best prediction”—there’s no implication of an exact relationship. But he did get the expression wrong (or he assumed that the variables were standardized to a mean of zero and a variance of one).