statistics questions?

A lurking variable is also known as a confounding variable. It is a variable that is related to a predictor variable of interest and to an outcome of interest but is not in the causal pathway of interest. It is because of lurking or confounding variables that correlation does not imply causation. It may turn out that two variables are correlated but their correlation is mediated by a third variable and there is no causal relationship. For example:

The number of murders in a town is correlated with the number of churches. But more murders don't cause more churches nor do more churches cause more murders. There's more murders and more churches simply because of a third lurking variable: the size of the population! The higher the population the more churches. The higher the population the more murders.

An example of two variables with linear correlation are the amount of time I spend studying and how high my school grades are. The line slants up from left to right - as I study more my grades are higher. The line hits the y-axis not much above zero - if I don't study at all I get a very low grade. y-hat is the expected grade I'd get if I studied for the corresponding x hours. b0 is the grade I expect to get if I don't study at all. b1 is the expected increase in my grade for each hour of additional study time.

my favorite example for a lurking variable, or more commonly called confounding factor, is that shoe size and vocabulary in grade school students are strongly correlated. The larger the shoe size, the larger the vocabulary. But it is clear that having a big foot doesn't make anyone have a larger vocab (just look at most pro basketball players). Nor does having a large vocab imply a large shoe. The confounding factor is age. as a grade school student ages both their shoe size and vocabulary will increase.

a great example for your second question is Hook's Law for finding a spring constant. A spring will deform relative to the load pulling on it. y = k * x + b where y = the deformation, k = spring constant, x = load, b = initial deformation or initial spring length.

in the regression model, y-hat is the measured deformation

β0 is the estimate of k, the spring constant and β1 is the estimate of the initial deformation. β1 is the y-axis intercept, β0 is the slope.

Amazed that I found this question already answered! it's like you've read my thoughts!