Poisson & Geometric Hypothesis Testing

• Hypothesis testing is a formal way of checking whether an observed value is consistent with a claimed parameter (λ for Poisson, p for Geometric), or whether it's suspiciously extreme.
• Both tests follow the exact same 5-step skeleton: hypotheses → rescale if needed → find the tail probability → compare to α → conclude in context.
• A critical region is a pre-calculated set of "surprising" values of X. If your observed X falls inside it, you automatically reject H₀ — no probability calculation needed at that point.
• Because Poisson and Geometric are discrete distributions, you can almost never hit α exactly. The probability you actually land on is called the actual significance level.
• For Geometric tests, the inequality direction flips between H₁ and the critical region — this is the single most common source of exam errors.

Imagine you run a small shop, and you know from years of data that you get an average of 3 customer complaints per week. That's your Poisson mean, λ = 3. One week, after training your staff differently, you get 7 complaints. Is that just a bad-luck week, or has something actually changed?

This is exactly the question Poisson hypothesis testing answers. If X ~ Po(λ), we test the mean λ using:

You'll always be given an observed value, x, and a significance level, α (commonly 5% = 0.05, sometimes 1% or 10%). The whole test then boils down to one question:

"Assuming H₀ is true, what's the probability of getting a result this extreme, or more extreme?"

If that probability is small (less than α, or α/2 for a two-tailed test), your observed result was too unlikely to be explained by chance alone — so you reject H₀. If the probability isn't that small, the result is still plausible under H₀, so you don't reject H₀.