Can confidence intervals be added together? I have 95% confidence intervals for a figure that represents the higher-than expected number of failures for a particular machine where the expected number comes from using data for months 0 - 13 to predict the number of failures for months 14, 15 and 16. The observed number of failures is for months 14, 15 and 16.
At the moment, I have the difference between expected and observed for months 14, 15 and 16 individually.
However, I would like to aggregate months 14, 15 and 16 together to assess whether the number of failures was higher-than-expected for the entire 14-16 month period rather than each month individually.
Can I just add the confidence intervals for months 14, 15 and 16 to eachother for this?
NOTE: I have read a previous and related question but I don't think it quite covers mine.
 A: No you can't add confidence limits . You can add variances to get the variance of the sum but the variables being added must be independent of each other otherwise complications (cross-covariance) must be factored in. Your data is time series and the forecast variances are thusly correlated (unless the ARIMA model is trivial) You need to compute the variance of a sum of predictions which are cross-correlated. This feature is available in some but not all forecasting packages.  
RESPONSE TO OP'S QUESTION REQUESTING SPECIFICS OF HOW TO DO THIS.
 Steps to take to compute the variance of an aggregated ARIMA forecast



*

*Build a model that has a white noise error process . No pulses/step shifts/seasonal pulses/local time trends in the error process  thus model parameters are robust. Parameters for the model are “proven” to be  constant over time. Model  error variance  is “proven” to be constant over time.

*Develop baseline forecasts and forecast variances based upon a) possibilities of pulses arising in the future and b) the concept that the model parameters that you have estimated may not be  the population parameters.  Note that standard ARIMA forecast confidence limits naively assume equivalence of sample and population values as only the psi weights are used.  See http://www.autobox.com/cms/index.php/blog/entry/you-should-have-50-confidence-in-your-confidence-limits for a detailed discussion of this.

*Compute the variance -covariance matrix of the observed series up to length NF where NF is the length of the forecast. Compute correlation matrix from this matrix to approximate  cross-correlations between forecasted values .

*Take forecast variances obtained in step 2 and use the correlation matrix from step 3 to estimate the variance of a sum of forecasts.
