What does a confidence interval with a negative endpoint mean?

Question

I have 10 iid r.v. with Bernoulli distribution with $X_{i} = 1$ for a positive result. I'm given $\sum_{i=1}^{10} X_i= 1$ and need to find a two-sided 99% confidence interval for $\theta$.

So $\alpha = 0.01$ and $z_{\alpha/2} = 2.575$ based on the standard normal distribution. The variance is given $\sigma^{2} = \hat{\theta}(1-\hat{\theta})$, with $\theta = 1/10$ from the given information. The confidence interval is: $$ \left(\hat{\theta} - z_{\alpha/2}\frac{\sigma}{\sqrt{n}}, \hat{\theta} + z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\right) $$ so my confidence interval is $(-0.143, 0.343)$. Have I messed up a calculation somewhere? I'm not sure what this confidence interval tells me ... perhaps that the sample size is too small to really get any useful information on $\theta$?

Answer 1

When the procedure you have used to calculate a confidence interval gives an interval including impossible values, that is an indication of problems with the method. In your case, you have used a normal (central limit theorem-based) CI with so few observations that the approximation is invalid. You can test that easily in R, say:

We have plotted the loglikelihodd function for your case. If this is (close to) quadratic, the normal approximation will be good. That is clearly not the case here!

As @Glen_b says in a comment, you need to read up on binomial confidence intervals, see for instance Wikipedia or Binomial confidence interval estimation - why is it not symmetric?.

R code used for the plot:

make_loglik <- function(n, x) {
     function(p) dbinom(x, n, p, log=TRUE)  
    }

loglik <- make_loglik(10, 1)

plot(loglik, from=0, to=1, xlab="p", col="blue", main="log likelihood function\nBernoulli, n=10, x=1")