# Standard error on an estimate derived from 2 estimated proportions - Survey data

I have survey data where I want to calculate a derived variable, based on 2 estimated proportions. (e.g., "Did you visit the web-page?" If Yes, "Did you place an order?", and I want to estimate total orders placed.)

Let's say I have a known population of 10,000, randomly survey 1,000 people, and get 80% Yes to the first question;

Some R code of the example;

N=10000
n_1= 1000
p_1= 0.8
prop_SE<- function(p,n) {sqrt(p*(1-p) / n)} # function for SE of a proportion
p1_SE= prop_SE(p_1,n_1)


0.80 +/- 0.0126

20% of those Yes respondents to Q1, respond Yes to the follow-up question. I treat those 800 as the sample size for calculating SE for that proportion.

n_2= p_1*n_1
p_2= 0.2
p2_SE= prop_SE(p_2,n_2)


0.2 +/- 0.0141

Then I want to estimate 'total orders placed', which would be N * p_1 * p_2 = 1600.

This seems like a simple problem, but I haven't found an explanation of how to calculate the standard error for this estimate. Is it simply the sum of the 2 SEs? p1_SE * n_1 + p2_SE * n_2= 23.96? Why or why not?