# Confidence interval of the mean for a beta distribution when alpha and beta are estimated [duplicate]

I have elementary knowledge in statistics. I'm trying to estimate the confidence interval for mean of a beta distribution as specified in this article using log likelihood estimation given alpha, beta and sample size. This is closely related to this question. The main difference is I have the alpha, beta and the sample size estimated using log likelihood from the data.

Lets suppose I have estimated $(\alpha,\beta)$ for two different data with different sample sizes:

1. ($\alpha,\beta$) = (0.688,3.806) using a sample size $n=500$ and
2. ($\alpha,\beta$) = (0.704,1.182) using a sample size $n=1500$

The mean of a beta distribution is $$E(X) = \frac{\alpha}{(\alpha+\beta)}$$ and the variance is given by $$Variance(x) = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$$

the mean and variance for 2 data are as follows:

1. mean = 0.15 and variance = 0.0236
2. mean = 0.37 and variance = 0.0810

Edit:

Using Bootstrap we could estimate the confidence interval of the beta distribution, This is based on the answer provided earlier to a similar question:

# Sample size
n = 50

# Parameters of the beta distribution
alpha = 0.688
beta = 3.806

# Simulate some data
set.seed(1)
x = rbeta(n, alpha, beta)

curve(dbeta(x,alpha,beta))

library(simpleboot)

x.boot = one.boot(x, mean, R=10^4)
hist(x.boot)
boot.ci(x.boot, type="bca")


Below is the confidence interval for beta distribution when $\alpha = 0.688$ and $\beta = 3.806$ with a mean of 0.15.

Intervals :
Level       BCa
95%   ( 0.1159,  0.2082 )
Calculations and Intervals on Original Scale


My question is around using the alpha, beta, and estimated mean and variance to estimate the CI of the mean as opposed to using bootstrap estimates. For normal distribution you could estimate CI given a mean, variance and sample size. How do I do the same for Beta distribution ?

How do I estimate the confidence interval for the mean, given the alpha, beta, and estimated mean and variance?