Is it possible to approximate the standard error of the prediction based on the standard errors of intercept and coefficients?
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Since you do not give any specific context, so I just assume that you are talking about linear regression. Suppose that we are working with the following model: $$y=\beta_0 + \beta_1x + \varepsilon,$$ where $\varepsilon\sim N(0, \sigma^2)$.
Now suppose that for $x=5$, we would like to predict $E(y|x=5) = \beta_0 + 5\beta_1$, denoted by $pre$.
We note that $pre$ is just a value if we know $\beta_0$ and $\beta_1$. However, they are unknow and each of them has its own sampling distribution (where the sd of this distribution is called the se). Therefore, the standard error associated with $pre$ is computed as: $$Var(pre) = Var(\beta_0 + 5\beta_1) = Var(\beta_0) + 10 Cov(\beta_0, \beta_1) + 25Var(\beta_1).$$ Now you can see that you cannot compute this variance if you do not know the covariance $Cov(\beta_0, \beta_1)$.