Confidence Interval for the Mean Response

The Confidence Interval for the Mean Response in the context of a linear regression corresponds to the calculated confidence interval for the mean predicted response \(\mu_{Y|X_0}\) for a given value \(X = X_0\).

So, this confidence interval gives us a credible set in which we expect the find the average response \(Y\), for a fixed predictor value \(X = X_0\)

How do you calculate this confidence interval

First, we need to know the mean squared error (\(\hat{\sigma}^2\)), for which you use the following formula:

\[\hat{\sigma}^2 = \displaystyle \frac{SSE}{n-2}\]

The mean squared error is a type of standard error which gives you the variability of the response variable for different times you evaluate at \(X = X_0\), and it is used as the basis for the confidence interval.

In other words, this standard error plays the same role as the standard deviation plays on the calculation of the confidence interval for the mean \(\mu\).

Formula for the Confidence Interval Formula for the Mean Response

Ok, we have everything we need now, so we go into the confidence interval formula: Based on this information, the \(1-\alpha)\times 100 \)% confidence interval for the mean response \(\mu_{Y|X_0}\) is given by

\[CI = \displaystyle \left( \hat\mu_{Y|X_0} - t_{\alpha/2; n-2} \sqrt{ \hat{\sigma}^2 \left(\frac{1}{n} + \frac{\left(X_0 - \bar X\right)^2}{SS_{XX}}\right) }, \hat\mu_{Y|X_0} + t_{\alpha/2; n-2} \sqrt{ \hat{\sigma}^2 \left(\frac{1}{n} + \frac{\left(X_0 - \bar X\right)^2}{SS_{XX}}\right) } \right)\]

As it is the case with most confidence intervals (not all, though), the interval is symmetric around a center point, which in this case, is the actual predicted Y value for \(X = X_0\).

This center value of the confidence interval is found by simply plugging the value of \(X = X_0\) into the estimated regression model.

More regression calculators

It is important to notice that here we have shown how to calculate confidence interval of regression prediction mean response. If you are interested rather in a confidence interval for the prediction itself, please use instead this prediction interval calculator for regression predictions .

Naturally, if we are talking about regression, you can check this linear regression calculator for the case you have one predictor, or this multiple linear regression calculator when you have many predictors.

One interesting application is the case of the polynomial regression , in which there is one dependent variable Y and one predictor X, but actually we use also the powers of X as predictors, so it is technically a multiple regression.

Regression analysis is truly important in Statistics and we cannot really overstate its importance. Now, it is crucial to ensure that the regression results found are valid, for which reason is highly advisable to analyze the regression residuals , as they will be crucial at the time of assessing whether the regression assumptions are met.