Is there a relation between confidence interval of Linear Regression parameters and confidence interval of a prediction? I know how to calculate the confidence interval of the estimates $\widehat{\beta}_0$ and $\widehat{\beta}_1$ in $Y = \widehat{\beta}_0 + \widehat{\beta}_1\cdot x$, but I'm having some trouble understanding the confidence interval of a single prediction. To me, it would be simply be: $$ \left[ {\widehat{\beta}_0}_\text{lower} + {\widehat{\beta}_1}_\text{lower}\cdot x_i, {\widehat{\beta}_0}_\text{upper} + {\widehat{\beta}_1}_\text{upper}*x_i \right],$$
with lower and upper referring to the lower and upper bounds of the confidence interval of the regression parameters.
Why is this wrong?

Even if you knew the parameter values with complete certainty, so that the lengths of their confidence intervals were zero, there would still be as much uncertainty in the next observation as there is in every individual observation; i.e. the variance of that observation would be as big as the variance of every individual observation.
