Converting linear regression estimates to percentages?

#1
Hi all.
Is there a method to convert two estimates that have been calculated using linear regression, to a percentage?
eg, I have two X scores, two corresponding estimated Y scores, and a standard error of estimate.
Is it valid to say a given X will score higher than another given X, a specified percentage of the time?

I guess to rephrase the question...
Is there a way to calculate the percentage likelihood that the true value of the estimate for a given X score will be larger than that for another given X score?
i.e %chance that Y¹ > Y² for given X¹ and X²

example data:
X¹ is predicted as Y¹= 50
X² is predicted as Y² = 60
Standard error of estimate = 10
so, 95% of the time the true value of Y¹ will be between 30 and 70.
and 95% of the time the true value of Y² will be between 40-80.
What would be the percentage chance that Y¹ will be higher than Y², or vice versa?
(Assuming we ignore the other 5%, or so, of the time).

I've tried working it out manually, and its doing my head in..
Eg, I can work out using only 1 std deviation, ie, 65% of the time, that, for the give figures above, Y¹ will be higher than Y² 25% of the time, and therefore Y² will be higher 75% of the time... but taking it to 2 std devs gets more complicated, and I'm sure there's some sort of confidence interval equation I could be using.

I do realise that the 65% of the time is for one Y value only, however, for brevity, I'm ignoring that that figure applies to all X's..
Hope this makes sense, and any help or suggestions appreciated.
 
#2
Ai Josh,

You started in the right direction.

The regression will predict the mean of Y for the relevant X.
So you may convert the question to a new question:
you should check the distribution and Sigma.
Under an assumption of Normal distribution and that the standard deviation is not dependent on X
(those assumptions are almost ..the linear regression assumption. Homoscedasticity but the normality is only for the residuals)

Y1~N(m1,Sigma) Y2~N(m2,Sigma)

D=Y2-Y1

You want to find the probability that Y2>Y2 => the probability that D>0

From here I assume you can continue.

The example was for a normal distribution, but you should check the distribution.
If for example, you don't know the standard deviation and the population distribute normally, you should use T distribution.
 
#3
Thanks obh.
I appreciate your help.
I'll take some time to work out what you have said and try to make some sense of it. (I didn't study enough maths when I was at school).
Yes, the data is normally distributed.
Thank you.