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I understand that the confidence interval for the mean is completely different from the confidence interval band for the regression model. I also understand that the confidence interval bad represents the confidence interval for a mean for each value of predictor variable. My question here, can I interpret the graph of the confidence band as a goodness fit measure of the best fitted regression line? That is, if the confidence interval very close to the fitted regression lime then the regression line is well fit the data.

Any help with an example, please?

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No, if the confidence band is very close to the fitted line then you can say that your model is very confident of the (linear) relationship. However you cannot say much about "goodness of fit", if your model has overfit you might also get a very narrow band. Goodness of fit has to be evaluated differently.

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