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There is no contradiction, nor could there be. It is just the standard deviation of your sample conditional on your model. The residual standard deviation has nothing to do with the sampling distributions of your slopes. Thus, larger SEs mean lower significance. To calculate significance, you divide the estimate by the SE and look up the quotient on a t table. 05 in this case, is the standard deviation of that sampling distribution. 51, but without knowing how much variability there is in it's corresponding sampling distribution, it's difficult to know what to make of that number. In your example, you want to know the slope of the linear relationship between x1 and y in the population, but you only have access to your sample. That's what the standard error does for you. I’ve read about and have completed the categorical coding for regression and the linear regression analysis using Real Statistics Using Excel. I have several categorical variables and some continuous ones. We need a way to quantify the amount of uncertainty in that distribution. I’m trying to determine the effects of several factors on the results of a finite element analysis.
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The central limit theorem suggests that this distribution is likely to be normal. In fact, if we did this over and over, continuing to sample and estimate forever, we would find that the relative frequency of the different estimate values followed a probability distribution. Moreover, neither estimate is likely to quite match the true parameter value that we want to know. For example, if we took another sample, and calculated the statistic to estimate the parameter again, we would almost certainly find that it differs.
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However, there are certain uncomfortable facts that come with this approach. The population parameters are what we really care about, but because we don't have access to the whole population (usually assumed to be infinite), we must use this approach instead. Parameter estimates, like a sample mean or an OLS regression coefficient, are sample statistics that we use to draw inferences about the corresponding population parameters.