[pgreenwood]

Ok I re-ran setting T properly for both cases. The results were similar:

> poisson.test(c(1, 11800), c(3, 1000000), alternative = c("two.sided"),conf.level = .93)

Comparison of Poisson rates

  data:  c(1, 11800) time base: c(3, 1e+06)
  count1 = 1, expected count1 = 0.035403, p-value = 0.03478
  alternative hypothesis: true rate ratio is not equal to 1
  93 percent confidence interval:
     1.006334 146.142032
  sample estimates:
  rate ratio 
    28.24859

The lower bound of the CI approaches a rate ratio = 1 for a 93% confidence interval.

Interestingly, if you multiply the CI I claimed before by the rate ratio instead of the expected rate, you get almost exactly the same CI as here.

  > ci <- c(0.03562718, 5.17251332)
  > 28.24859 * ci
  [1]   1.006418 146.116208
 

* Note 11800 is about two years of pedestrian deaths and time units are in millions of miles. https://crashstats.nhtsa.dot.gov/Api/Public/ViewPublication/...