Confidence intervals for the probabilities of each outcome in a multinomial I have a survey that includes questions where you can choose one option from a list.
For example: "How often do you go to the supermarket?"
A. Every day
B. 2-6 times per week
C. Once per week or less
D. Never

Say 1000 people respond to the survey and 100 people choose option A, 400 option B, 400 option C and 100 option D. How do you calculate the confidence intervals of these proportions? Are they simply binomial confidence intervals (i.e. just normal proportions)? Or is it something more complicated, because the proportions are related to each other?
 A: I tried bootstrapping confidence intervals (using R). The results are similar to standard binomial confidence intervals - which I think suggests that the standard confidence intervals are OK? (Though not sure of the theory behind that ...)
# make sample data

set.seed(4)
n <- 1000
d <- sample(1:4, size = n, replace = T, prob = c(0.1, 0.4, 0.4, 0.1))

# binomial confidence intervals

rbind(tabulate(d) / n, 
      sapply(table(d), function(x) prop.test(x, n)$conf.int[1:2]))

#              1         2         3         4
#[1,] 0.09300000 0.4260000 0.3780000 0.1030000
#[2,] 0.07606873 0.3951986 0.3479774 0.0851969
#[3,] 0.11313228 0.4573769 0.4089719 0.1239218

# bootstrapped confidence intervals

b <- 10000
bs <- sample(d, size = n * b, replace = T)
bs <- matrix(bs, ncol = b)
bs <- sapply(1:4, function(x) colSums(bs == x)) / n
bs <- apply(bs, 2, quantile, prob = c(0.025, 0.975))
rbind(tabulate(d) / n, bs)

#       [,1]  [,2]  [,3]  [,4]
#       0.093 0.426 0.378 0.103
# 2.5%  0.075 0.396 0.348 0.084
# 97.5% 0.111 0.456 0.408 0.122

