Poisson Distribution Confidence Interval Suppose I have 6 pieces of data from a poisson experiment:
4, 6, 8, 10, 12, and 14.
$\hatλ$ = 9.  I want to establish a 95% confidence interval for this poisson experiment, and z* = 1.960.  So I do the following:
Upper boundary:
$λ+1.960\sqrt{λ}$
$9+1.960\sqrt{9}$
$14.88$
Lower boundary:
$λ-1.960\sqrt{λ}$
$9-1.960\sqrt{9}$
$3.12$
Is this work mathematically sound?
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As for interpreting the confidence interval, I'm lost.  What does the confidence interval mean?
 A: What you have is not a confidence interval. A confidence interval should
be based on data. If you know that $X_i \sim \mathsf{Pois}(\lambda = 9),$
what is the point of getting an interval estimate of $\lambda?$ 
From the corrected data  ($4,6,8,10,12,14$), you have
$\bar X = \hat \lambda = 9.0.$ 
One simple style of approximate confidence interval for $\lambda$ is the so-called Wald 95% CI. It is of the form
$$\bar X \pm 1.96\sqrt{\bar X/n}.$$
This would be an approximate confidence interval based on data, and would give
a somewhat similar result: $(6.60, 11.40).$
A somewhat better style of 95% CI for the Poisson mean $\lambda$ uses
$T = n\bar X$ to get a CI for $n\lambda$ as
$$T + 2  \pm 1.96\sqrt{T + 1},$$
and then divide the endpoints by $n$. The resulting 95% CI is $(7.1, 11.9).$
There are many styles of Poisson confidence intervals, you should use the
one given in your text. (But I don't understand what the t distribution has
to do with a Poisson CI.)
Note: Per Comments, I'm showing a graph of the PDF of $\mathsf{Pois}(9)$ compared with the density curve of $\mathsf{Norm}(\mu=9,\,\sigma=3).$

