How can I do a hypothesis test with an estimated null hypothesis? The situation I have is with data that (I'm assuming for now) follows a poisson distribution. I want to know if this year had an unusual number of a certain type of fatality compared to previous years. I could use a hypothesis test to evaluate if this result is unusual. If the average rate in previous years was 8 fatalities per year, and last year there were 13 fatalities:
Null hypothesis: λ = 8
Alternative hypothesis: λ > 8
Using a cumulative poisson distribution table, I can see that P[x>=13] = 0.0342 when λ = 8 is, so the result is statistically significant at the 0.05 level. 
However, I only have a few years of data with which to estimate my null hypothesis λ = 8. I know that the standard error for my estimate is sqrt(λ/n) - how do I include this extra uncertainty in my hypothesis test?
Thanks!
Edit: Thanks everyone for your responses so far. Some have asked for specifics of the data I am looking at. I actually want to stay away from solutions for specific data since this is a business question that comes up a lot in with different types of accidents and with different numbers of years available, and I'm looking for a good general solution. If it helps, lets say I always have between 7 and 15 years of data to give me my null hypothesis lambda. 
 A: This is a difficult situation with not much data, but I think your
analysis is reasonable. However, it seems you have a computational error.
Suppose $40$ accidents of a particular kind over the last $5$ years. (The annual numbers may
have been 7. 11, 5, 9, 8.) It seems reasonable to model these accidents
according to a Poisson distribution and the estimated Poisson mean is
$\hat \lambda = 40/5 = 8.$
Now in the current year just finished, there have been $13$ accidents
of this kind. It is certainly possible that one of the previous five
years might have had $13$ accidents. The annual numbers might have been
7, 8, 13, 9, 3. 
Generally speaking, it is certainly possible for a Poisson distribution to produce an occasional large count. 
How likely is that? The result from R is $P(X \ge 13) = 1 - P(X \le 12) = 0.064.$ So the P-value of your test is about 6.4% > 5% and you cannot reject $H_0: \lambda = 8$ in favor of $H_a: \lambda > 8$ at the 5% level of significance.
1 - ppois(12, 8)
[1] 0.0637972


If the count for the most recent year had been $14$ then the P-value would be about 3.4%, leading to rejection at the 5% level.
1 - ppois(13, 8)
[1] 0.0341807

Notes: With so little data and with 'borderline' P-values, subjective factors are bound to affect the believability of this test:
(a) In order for this test procedure to work, it is necessary for the
correct model to be Poisson with the same mean $\lambda$ holding true
throughout the 5-year reference period. With only five years of data
for reference, there is no way to check whether the Poisson model is
correct or whether $\lambda$ has been constant. 
If the numbers for
the past five years were something like 5, 7, 9, 8, 11, then I would worry
that there might be an increasing accident rate across all six years. In that case it wouldn't be surprising for the sixth year to be larger.
(b) If something occurred just before the beginning of the last year
(with its 13 accidents) that might be expected to increase the accident rate, and that occurrence triggered doing the test,  then I would be more inclined to trust the test. (E.g., heavier traffic on a road due to a new factory nearby, or
major construction with narrower traffic lanes at unpredictable places throughout the year.)
By contrast, if you just notice a year with 13 accidents and decide to do a test based on that alone, I would be less inclined to trust the test.
A: My first-nanosecond, Malcolm-Gladwell-Blink-level reaction asks why you're using the cumulative Poisson as the distribution of your test statistic. If you've got a small number of samples from any distribution (including the Poisson), the Central Limit Theorem says you can use the t- or F- distribution (via a t-test or an ANOVA) to test your hypothesis. The small sample size would be worked into your test in computing the degrees of freedom.
So, 
H0: λ = 8, HA: λ ≠ 0, and since you have a small sample ("a few data points"), compute a t-statistic with df = a few − 1.
If there's a reason to use a Poisson CDF, I'm not seeing it at the moment. Tell us more, or give us more data or more context.
