I am measuring airborne bacteria in a new high-risk room. The bacteria are measured with Petri plates that have area 19,5 cm$^2$. I have 30 old measurements results, that give average bacteria counts 2,8 CFU/plate from other High-Risk rooms. The probability of the bacteria count is somewhat Poisson distributed. I want to use the old measurements to determine how many samples(plates) I need to use to show with 95 – 97% confidence that the new room conforms to the old ones. Max bacteria allowed is 50 CFU/plate for a High-Risk room.
When I started looking into your answer I noticed that you used 3.8 instead of 2.8 from the original question. So, I recalculated your answer with 2.8 instead and tried to figure the whole thing out. I wanted to ask you if my recalculations are in the vicinity of correct and if not, where I might have faltered?
Test. Suppose you decide to use n=8 plates. Assuming a Poisson distribution, you currently have λ=2.8 CFU per plate, which would be λT=8(2.8)=22.4 CFU per 8 plates. You could test H0: λT=22.4 against Ha: λT>22.4 at the 5% level of significance. To do so, you would reject H0 in favor of Ha, if the number T of CFU per dozen plates has T≥30. That is, P(T≥30|H0)=1−P(T≤30|H0)=0.043<0.05, assuming the null distribution T|H0∼Pois(λT=8(2.8)=22.4). [Computations in R.]
ppois(30, 2.8*8)  0.951017 1 - ppois(30, 2.8*8)  0.04898297
Then the power against the particular alternative λa=8(3.75)=30 is P(X≥31|λa=)=0.99. So, if the true rate per plate is λ=6 CFU, our test has a 99% chance of detecting that much increase.
1 - ppois(30, 6*8)  0.9963514
In the plot below, the null Poisson distribution (blue bars) is sufficiently different from the alternative distribution, for a satisfactory test using critical value 34 (dotted vertical line).
R code for plot:
t = 20:100; PDF = dpois(t, 22.4); PDF.a = dpois(t, 48) plot(t-.1, PDF, type="h", lwd=2,col="blue", main="Null and Alternative Distn's") lines(t+.1, PDF.a, type="h", lwd=2, col="brown") abline(h=0, col="green2") abline(v = 34, lwd=2, lty="dotted")
Confidence interval. A reasonably accurate 95% confidence interval for a Poisson rate based on T observed counts is of the form T+2 ±1.96T+1−−−−−√. If you observe T=48, then a 95% CI for λT is (22.4, 48) and a 95% CI for the rate λ per plate is (2.8, 6).
38 + qnorm(c(.025,.975))*sqrt(37)  26.0 - 49.9 (38 + qnorm(c(.025,.975))*sqrt(37))/12  2.17 - 4.16