### Difference between confidence intervals and prediction intervals

• For a prediction interval in linear regression you still use $\hat{E}[Y|x] = \hat{\beta_0}+\hat{\beta}_{1}x$ to generate the interval. You also use this to generate a confidence interval of $E[Y|x_0]$. What's the difference between the two?

$\hat{E}[Y|x] = \hat{\beta_0}+\hat{\beta}_{1}x$ doesn't "generate the interval".

I do not see a reason for the divergence between the two methods in any of the answers above. Regression results are typically estimated based upon parametric Student's t distribution parameters and typically regression, especially from poorly matched to the data regression models, lead to residuals that are not studentized, e.g., skewed but especially with heavy tails typically (if not always) making the parametric measures of data dispersion larger than their corresponding anticipated measured quantiles. A rule of thumb I have found useful: If I see residuals with outliers, long tails, and u

9 years ago

Your question isn't quite correct. A confidence interval gives a range for $\text{E}[y \mid x]$, as you say. A prediction interval gives a range for $y$ itself. Naturally, our best guess for $y$ is $\text{E}[y \mid x]$, so the intervals will both be centered around the same value, $x\hat{\beta}$.

As @Greg says, the standard errors are going to be different---we guess the expected value of $\text{E}[y \mid x]$ more precisely than we estimate $y$ itself. Estimating $y$ requires including the variance that comes from the true error term.

To illustrate the difference, imagine that we could get perfect estimates of our $\beta$ coefficients. Then, our estimate of $\text{E}[y \mid x]$ would be perfect. But we still wouldn't be sure what $y$ itself was because there is a true error term that we need to consider. Our confidence "interval" would just be a point because we estimate $\text{E}[y \mid x]$ exactly right, but our prediction interval would be wider because we take the true error term into account.

Hence, a prediction interval will be wider than a confidence interval.