# Confidence Interval derivation

I am predicting a "yearly cumulative variable" from monthly results.

I use $Y_j = \sum_{i=1}^j (X_i) / f_i$ where $j$ is the current month.

I know the $f$ from history; e.g. January = .073, February = .070, March = .087, April = .076, May = .085, ...., December = .124)

So, $\sum (f_j) = 1.0$

Suppose X1 = 1000, then Y = 1000 / .073 = 13699

Suppose X2 = 950, then Y = (1000 + 950) / (.073 + .070) = 13636 etc

My question is how do I create a confidence / prediction interval on the prediction for Y? The prediction in January (11 months remaining) should be more variable than that in November (1 month remaining). Once December results are available, we are finished as Y is known and then we start over for the next year.

• Are your weights, $f$ and $X$ correlated or derived from the same data in any way? Dec 7, 2016 at 17:49
• the f are the historical average fraction of the ∑ (Xi) for each month compared to the eventual known total. So, the f and X are not correlated but the f and ∑ (Xi) are.
– Jeff
Dec 8, 2016 at 16:04