My client needs an estimate of the percentage of taxes he is going to pay, based on what happened in the recent past.

He needs something like "the percentage P will be between 1.8% and 2.0% with a 95% probability".

So I guess this is not a confidence interval but a probability interval.

Let's see what I have:

The client sells a product that comes in 3 varieties (A,B,C)

For each of the last 10 months I know what percentage of the sales was represented by each variety.

For instance, in February he sold 50% A, 30% B and 20%C while in March I sold 45% A, 35% B and 20%C. Of course these 3 numbers always add up to 100%

He has to pay taxes on what he sold: 1% over A, 2% over B and 3% over C.

Let's say all the products cost the same, 1$, and every month he always sells 100 products.

So, in February, he has to pay 1% of 50, 2% of 30 and 3% of 20. This gives 0.5$+0.6$+0.6$ = 1.7$ of taxes.

That is, He has to pay P = 1.7% of taxes in February and P = 1.75% of taxes in March.. .and so on.

Therefore, P, the percentage he pays every month, varies. My client needs an estimate of this P, with a reasonable "error".

There is NO seasonal effect on sales.

One thing I can do is calculating P in the past for each month. I calculate the mean M and the Standard Deviation SD.

I would say to the client: your P is approximately M and with a probability of 95% your P will lie between P-2*M and P+2*M.

**Is this correct?**One possible problem is that I have less than 30 months to work with and I have no idea of the normality of the distribution of the values of P.

I have another way of reasoning.

I can examine how to different sell percentages (A%, B%, C%) varies over time. I could calculate mean and StDev of these 3 quantities.

But in this case I don't know if (and how) I can propagate the error to the calculation of P. Is the error on P a weighted sum of the errors on A,B and C?

It seems to me that the estimate on P could be more precise if I examine more data (the various A,B,C) but since I will have an error on each of these, I don't know how to combine them to get a reasonable error on P.

I had to simplify somehow my situation but I hope I gave enough info. I'll be glad to add more if needed.

Thank you for Your time.

Wentu