# Mean interval estimation when data are not binomial

When a set of data are binomial I have a lot of methods available to me (Wilson score interval, Wilson score interval with continuity correction, Jeffreys interval, Clopper-Pearson interval, Agresti-Coull Interval) other than the standard Wald interval estimation method (see Interval Estimation for a Binomial Proportion Author(s): Lawrence D. Brown, T. Tony Cai and Anirban DasGupta).

However, I can't use these methods for data that is itself proportional in the sense of it sitting in [0,1].

For example consider a sample of n respondents, giving a score for a question X as:

r_1 = 0.4
r_2 = 0.3
...
r_n = 0.2


How would I calculate the confidence interval for a mean based on this data? Is there any way other than bootstrapping?

• Do you mean that the response to the question must lie between $0$ and $1$? Also, how large is the sample? If it's big enough (practically speaking $>30$ with a reasonable range of answers), asymptotics should allow you calculate it on something approaching normal... – dardisco Jan 15 '14 at 8:59
• Yes it lies between 0 and 100 or 0 and 1 rather. The sample size is variable, we decided to only consider >30. For less than 30 we have considered a bootstrap. – dominic Jan 15 '14 at 9:12
• I am wary of the CLT confidence interval interval for growing sample size - the paper above showed that the coverage probability varies greatly away from the nominal confidence level as n grows. So I'm really interested in a mean confidence interval method for non binomial data (apart from CLT CI). – dominic Jan 15 '14 at 10:41
• You don't have bimomial data. You have discrete data on a scale of 0-100. Any scenario you think of for the actual population will have the sample means behaving like a normal distribution for n>30. You should run some simulations on your own to convince yourself. – SaraWong123 Jan 15 '14 at 16:38
• Your answer echoes the very title: the O.P. says these are indeed not binomial. How, then, does this reply address his question about calculating a CI for the mean? – whuber Jan 15 '14 at 18:49