I understand the basics of confidence intervals, the central limit theorem, etc, to be able to know things like given N samples of random variable, we're 68/95/99.7 percent sure the variable is within these two values.
I'm wondering about how this generalizes to cases where we're trying to derive a relationship between two variables. Ie, if we're trying to measure how some variable Y depends on a variable X, one thing I guess we could do is set X to 1, then get a 95% confidence interval for X = 1. Then set X += 0.1 and do that again, and again again etc, to derive a 95% "confidence relationship", or something.
This seems kind of sloppy though. Is there some other statistical mechanism designed for doing things like this (ie, finding a relationship between X and Y that we are 95% sure is the right relationship).
edit: Some more info on what I'm looking for:
Say I have a bunch of (x,y) pairs. I'm trying to express the y's as a function of x. x values might repeat. I also don't necessarily care about expressing the relationship algebraically. So, the naive thing to do would be to average over the ys for every possible x value we have data for, and plot that. (so if the dataset was {(1,2), (1,3), (2, 2)}, I'd plot the points {(1,2.5), (2,2)}).
This totally disregards data about how much y data we got for each x value, though. It also seems to be ignoring the fact that we suspect the data to be correlated in some way. Like if our data points were {(1 + 1e-10, 4), (1, 17)}, my naive approach would plot {(1,17), (1 + 1e-10, 4)}. But considering how close 1 and 1e-10 are, we'd think that the values 4 and 17 should be close to each other. So maybe something like {(1, 10.5), (1 + 1e-10, 10.5)} would be more accurate.
Or, in another sense, if our dataset was {(1,16), (1+1e-10,16.2}), the fact that we got two values so close to 16 both times should seem to "inspire our confidence". But since the x values are technically different we won't notice this.
