For the simple linear regression model:
$Y_i = \beta*X_i + \epsilon_i$
I want to find CI for $\beta x$ which is $E(Y_i)$ when $x_i=x$
I find that $\hat\beta$ ~ $N(\beta, \frac{\sigma^2}{\sum(X_i^2)})$, so the distribution of $\hat\beta x$ is $N(\beta x$, $\frac{X^2\sigma^2}{\sum(X_i^2)})$. If the variance is known I can use $P(-z_{\alpha/2}$ $\leq$ $\frac{\hat\beta x - \beta x}{x\sigma/\sum(x_i^2)}$ $\leq$ $z_{\alpha/2})$.
If the variance is unknown, what unbiased estimator should I use for $\sigma^2$? Is it the sample variance? What is it in this case?