How do I find a 95% percent ML confidence interval of the probability of category 5 strength Let ${\bf X} = (X_1, \dots, X_n)$ denote the log-intensities of all $n = 291$ tropical cyclones from the north Atlantic basin between 1980 and 2006. Log-intensity is defined as the logarithm of its maximum sustained windspeed (measured in knots). A tropical cyclone is a category 5 hurricane if its maximum sustained windspeed is 137 knots or more, i.e. if its log-intensity is 4.92 or more. We are interested in the probability $p \in [0,1]$ of a tropical cyclone attaining category 5 strength. Consider the following model: $X_i \overset{{\rm iid}}{\sim} \mathsf{Normal}(\mu, \sigma^2)$, where $\sigma = 0.33, \mu \in (-\infty, \infty)$. Under this model $p = \Phi\left({\mu - 4.92 \over \sigma}\right)$ where $\Phi(x)$ denotes the standard normal CDF. Find the 95% ML confidence interval for $p$ based on the following summaries of the data $$\bar x = 4.21 \quad \sqrt{n} = 17.06 \quad \#\{x_i \ge 4.92\} = 9$$

My Work
The formula for an ML interval for $X_i \overset{{\rm iid}}{\sim} \mathsf{Normal}(\mu, \sigma^2)$ with known $\sigma$ is $B_c(x) = [\bar X \mp c{\sigma \over \sqrt{n}}]$. Calculating along these lines, for $c = 1.96$ as the $z$ value that leaves $0.05$ probability in the two tails combined of the normal pdf, I get $B_{1.96}(x) = [4.172, 4.247]$, but when I try to evaluate the endpoints of this interval for $p$ given that $\Phi(x)$ is a monotone transformation, I get nonsense answers. What am I doing wrong?
Also, this will never be handed in for any class, I simply came across this problem and wondered how it could be solved. 
 A: Your logic is correct.  Based on the estimated mean and standard deviation that you have for the wind speeds I checked that you constructed a 95% confidence interval for the mean correctly on the log scale.  Using the exponential transformation on the scale of the original speed measurement (knots)  the interval I get is approximately [64.85, 69.96].
What do you think is wrong with that?  Note that on the log scale 4.92 is far above the upper limit of the interval.  Also only 9 of the 291 observations are above 4.92. which means on the scale of knots only 9 cases are abobe 137 knots.  Hurricane speeds are much higher than normal wind speeds so most of the other observed speeds are going to be much lower than 137 (apparently more likely to be in the range 60 to 70 knots).
If you are confused because 137 is far above 69.96 don't be.  4.92 is far above 4.25 on the log scale also.  All this is telling you is that category 5 hurricanes have winds speeds far above the average speeds observed in the north Atlantic basin over the time period from 1980 to 2006.  There should be nothing surprising about that conclusion.
But if you want to answer the question in the title of your problem the confidence interval for the mean doesn't answer your question.  You should take the fitted normal distribution and calculate the probability for the interval from [4.92, ∞) to estimate that probability.
