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Let's say that I want to estimate the number of cars that pass my house every minute. I count 30 cars in 10 minutes, and I want to use this to generate a confidence interval. How should I do this?
The point estimate would be 3 cars/minute, but I don't know how inaccurate this estimate could be. How would I go about computing a confidence interval?
I have the feeling that 30 cars in 10 minutes may not be enough data. Would I have to record the amount of time between each car?
This could plausibly be thought of as a Poisson process, i.e. the probability that a car comes along in the next millisecond does not depend on how long it's been since the last car. That would mean the time from each car to the next has an exponential distribution. With the exponential distribution, the variance is the square of the mean; hence the standard deviation is the same as the mean. If you estimate the mean time at $1/3$ minute, and estimate the standard deviation at $1/3$ minute, then you could say the sample mean is approximately normal because of the central limit theorem. Let $c$ be the number for which $\Pr(-c<Z<c)=1-\alpha$, and then $\dfrac13\pm c\dfrac13$ are the endpoints of a $100(1-\alpha)\%$ confidence interval for the average time.
