Hello Alba, why the cartoon picture ? Read this link it may give you some info. http://www.statsoft.com/textbook/multiple-regression/?button=2

Deborah Hilton Statistics Online

http://sites.google.com/site/deborahhilton/

Hi Alba, I an not entirely sure I understand your question. Could you be more clear what you mean by a "determined value"? I wonder what you mean by "close to" also. The traditional test associated with a regression coefficient is done by a t-ratio. In that case the null hypotheses is that the regressoin coefficient is equal to zero. The question is how many standard errors is it away from zero, so we divide the regression coefficient by zero to get the t-ratio. We compare that to the critical values of T for the degrees of freedom (which is the size of the sample minus the degrees of freedom for parameters estimated by the regression), which can be looked up on a table or found by Googling critical values of T on the Internet. You pick the critical value based on the alpha level you want, for example, .05. When the sample is large enough the critical values of T are the same as Z, so that critical value is 1.96, as many people know. Of course, our statistical programs do all that work for us. So perhaps you are asking if your regression coefficient is significantly different from some number other than zero. Here what you would do is subtract that other number from your regression coefficient and divide the result by the standard error, to get the t-ratio and then compare that to the critical value of T. But you have to remember, particularly if you have a small sample, that not rejecting the null hypothesis may not be very strong evidence. You have rejected the hypothesis that the result you have came from a sample where the two coefficients were different, but you might not have had the statistical power to find anything other than a very large difference, so you might not be saying much. Bob

The value of the parameter (in your example for "b") of a statistical model (in your case a regression) is always an estimate. This estimate is determined in a way that your expectation about the deviations of other such data from the estimate is minimized. This estimated value is a "point estimate", it is your "best guess".

It is a (common but) illegitimate question for a "prove" that a guess is "close to" any given value. There is no prove in statistics.

What you can ask is: "How precise is my guess?" (how much different values would you stll consider a "good" guess?, how much different values would you consider a "bad" guess) - the answers always refer to the data you actually have used for the guess. This precision is often represented by the standard error of the estimate or, better, by a confidence interval (and best by the entire likelihood profile).

You may calculate a confidence interval (a so-called "intervall estimate") for the parameter and interpret what it means for you whether or not the given, determined value falls inside this interval.

This whole procedure is very related to everything Roberd said above, just seen from a little different standpoint. Roberd explained a solution for a specific case: the assumed probability distribution of the errors is normal, so one can use the t-distribution (which is a kind of a normalized the likelihood profile) to calculate the confidence interval, and instead of actually calculating the interval Robert explained how to calculate the probability of an interval including the detemined value. He further suggested to use the criterion whether or not the value is withing the 95% confidence interval to make a yes/no decision for the question "am I able to statistically distiguish my estimate from the determined value with 95% confidence?".

Thanks, Jochen, although I didn't mean to imply that it had to be a 95% confidence interval as opposed to any other probability of choice, and I certainly didn't mean to imply that it would be proven, quite the contrary, only not rejected. You are also very right to point out that anything based on the standard errors is laden with assumptions, too. Of course we are still not sure what that "determined value" means. Bob

Robert, no worries :) I think your answer was clear and correct. I just wanted to mke some additional remarks and provide a slightly different was of interpretation. I wanted to supplement your answer, not "correct" it.