Calculating the confidence interval for simple linear regression coefficient estimates

I have a data set of paired measurements $(x_1,y_1),(x_2,y_2),...,(x_n,y_n)$. I need to fit a linear regression line $y=ax+b$ to this data. Therefore, I have to estimate the parameters $a$ and $b$.

How can I then calculate the confidence interval for these estimated parameters?

I referred to the wiki article which says http://en.wikipedia.org/wiki/Simple_linear_regression:

Normal assumption

Under the first assumption above, that of the normality of the error terms, the estimator of the slope coefficient will itself be normally distributed with mean $\beta$ and variance $\sigma^2/\sum(x_i-\bar > x)^2$.

I didn't get how this formula was derived.

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If this is homework, please tag it as such. –  JohnRos Jun 30 '12 at 19:05
I referred to the wiki article. But I didn't get some of the stuff. I have reposted in the question –  user34790 Jun 30 '12 at 19:59
–  cardinal Jun 30 '12 at 21:06

Under normality assumptions for the error term in the model the formulas for the least squares estimates are:
$\beta_0=\bar{y}-\beta_1\bar{x}$ (where $\bar{x}$ is the mean of the $x_i$s and $\bar{y}$ is the mean of the $y_i$s) and $\beta_1=(\sum x_iy_i-n\bar{x}\bar{y})/(\sum x_i^2-n\bar{x}^2)$.

Both $\beta_0$ and $\beta_1$ are then normally distributed and when divided by their estimated standard deviations have t distributions under the null hypothesis that the true value is 0. Given this you can construct confidence intervals based on the t distribution.

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+1. Michael, given your many expert contributions on this site, it would help users if you would learn and use mathjax so that your answers are easier to read and follow. You can infer a lot about how it works by looking at the edits people have made on your answers. You can also see how any expression on CV was produced by right-clicking on it and then selecting 'Show Math As' -> 'TeX Commands'. I have also found this to be a very comprehensive reference (albeit slow to load). Thanks again. –  gung Jul 1 '12 at 15:04