I have two questions regarding simple linear regression that I was hoping
someone could help me with.
1) According to what I have learned so far, the levels of X are "fixed," so
that only Y is the random variable ( error is random as well). My question
is, what if X is a random variable as well? It seems like this could be the
case with some of my textbook examples. Does simple model of y=a+bx+e still
hold? Are assumptions the same, such as conditional distributions of Y are
normal with same variance, E(Y) is a straight line function of X, and
independence/normality of error terms? Also, in repeated sampling the sample
slope is normal because Y is normal. However, if X also varies from sample
to sample, is the sample slope still normally distributed (sampling
distribution)?
2) My second question regards the prediction interval. I can perform this on
a computer, but it is difficult for me to conceptualize. If you are using
Y-hat (the mean of estimated regression function) to estimate a future
response, does this mean that the difference,
(Y(future response)-Y hat), is a statistic that has a sampling distribution,
from which you can derive the standard error? It seems like this might be
the case, but there is no parameter. I don't even know if what I just said
makes any sense.
I understand that my questions are long, and perhaps not in any logical
order, but I would greatly appreciate any help with these conceptual
matters.
Thank you
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