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I was using the softeware PASS to calculate the sample size and power of a one Poisson mean problem.

Solve For ................................................ Power
Ha (Alternative Hypothesis).................... Ha: λ0 < λ1
Alpha....................................................... 0.025
n (Sample Size) ...................................... 12 24
λ0 (Null or Baseline) ............................... 1.0
λ1 (Alternative)........................................ 1.1 1.4 1.8 2.2 2.5

In the PASS manuscript they explained that how the power and sample size was calculated. It is based on "Guenther, William C. 1977. Sampling Inspection in Statistical Quality Control. Griffin's Statistical Monographs. Macmillan, NY. Pages 25-29." See the scree shot below. You can also refer to the link. Can some one explain what is the rationale of this sample size method? I cannot figure it out.

enter image description here https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/PASS/Tests_for_One_Poisson_Mean.pdf

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    $\begingroup$ As for why the $\chi^2$ appears in the formula, see WIkipedia's Poisson distribution # Related distributions (specifically the last bullet-point in the section) $\endgroup$ – Glen_b Jan 27 at 5:16
  • $\begingroup$ thanks, that answers my question regarding the chi-square. But I am still a little confused about the hypothesis testing here. The example shown in the PASS manuscript was based on the H_a: \lambda >\lamba_1. What if I would like to test the H_a:\lamda<\lambda_1? how can I construct the sample size formula? $\endgroup$ – Justice Jan 28 at 2:58

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