# How to interpret coefficient standard errors in linear regression?

I'm wondering how to interpret the coefficient standard errors of a regression when using the display function in R.

For example in the following output:

lm(formula = y ~ x1 + x2, data = sub.pyth)
coef.est coef.se
(Intercept) 1.32     0.39
x1          0.51     0.05
x2          0.81     0.02

n = 40, k = 3
residual sd = 0.90, R-Squared = 0.97


Does a higher standard error imply greater significance?

Also for the residual standard deviation, a higher value means greater spread, but the R squared shows a very close fit, isn't this a contradiction?

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