Given 1000 observations that come from a distribution that is bounded between 0 and 1. How do you calculate correct 95% Confidence intervals for the MEAN when dealing with a bounded distribution?

Ciolan3u 2022-09-14 Answered
Confidence interval for a bounded distribution
Given 1000 observations that come from a distribution that is bounded between 0 and 1. How do you calculate correct 95% Confidence intervals for the MEAN when dealing with a bounded distribution?
R code
set.seed(10)
data =runif(1000, min = 0, max = 1)
mean(data)
mean(data)+1.96*sd(data)/sqrt(length(data)) #usual CIs
mean(data)-1.96*sd(data)/sqrt(length(data)) #usual CIs
Are there any references on how to calculate 95% CIs for bounded distributions?
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Answers (1)

detegerex
Answered 2022-09-15 Author has 16 answers
Step 1
You are using a Central Limit Theorem approximation for the distribution of the mean, justified by a finite variance of the original distribution (which is 1 12 in this case) and a large sample size (which is 1000 here)
So unless your original random variable has particularly extreme features (it does not in this case), you do not need to worry much about whether the original random variable has bounded or unbounded support
Step 2
Your example code gives a sample mean of about 0.505 and a confidence interval of about [0.487,0.523], which contains the true population mean of 0.5. Using different seeds and your 1.96 multiple, you would expect about 95% of examples of confidence intervals to contain the true population mean. I think that for example seeds of 28, 44, 100, 140, 141, 147, 153, 154, or 160 would not

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