Poisson & Geometric Hypothesis Testing
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Poisson & Geometric
Hypothesis Testing
Summary — What This Chapter Covers
- 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.
Topic 1 — Poisson Hypothesis Testing
Setting Up the Test
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.
Watch the Time/Space Interval — Rescaling λ
Mr Viajo believes his travel blog receives an average of 8 likes per day (24 hours). After a new advertising campaign, he tests at the 5% significance level whether this has changed. Over a randomly chosen 12-hour period, the blog receives 7 likes.
H₀: λ = 8 H₁: λ ≠ 8 (two-tailed, since we're testing for "a change" in either direction)
Lower tail — want P(X ≤ c) as close to 2.5% as possible without exceeding it:
P(X = 0) = 0.0183 < 2.5% ✓ P(X ≤ 1) = 0.0916 > 2.5% ✗
→ Critical region (lower): X = 0
Upper tail — want P(X ≥ c) as close to 2.5% as possible without exceeding it:
P(X ≥ 9) = 1 − P(X ≤ 8) = 1 − 0.9786 = 0.0214 < 2.5% ✓
P(X ≥ 8) = 1 − P(X ≤ 7) = 1 − 0.9489 = 0.0511 > 2.5% ✗
→ Critical region (upper): X ≥ 9
Full critical region: X = 0 or X ≥ 9
Finding the Critical Region
A critical region is just the set of "extreme enough to reject H₀" values, worked out in advance — before you even see the observed data. Once you have it, testing is instant: is x inside the region or not?
Actual Significance Level
Because Poisson only takes whole-number values (0, 1, 2, 3, ...), you can never land exactly on α — there's always a "gap" between your critical value and the target. Whatever probability you actually end up with in the tail(s) is called the actual significance level.
Example: if P(X ≤ 6) = 0.0298 and P(X ≤ 7) = 0.0510, with α = 0.05 — normally X ≤ 6 is chosen (doesn't exceed α). But if the question asks for the region "as close as possible to α", you'd pick X ≤ 7, giving an actual significance level of 0.0510.
A call centre receives an average of 5 complaints per hour. Following a new training scheme, the manager tests at the 5% significance level whether the mean has decreased. In a randomly chosen hour, only 1 complaint is received. (a) State the hypotheses. (b) Find the critical region. (c) State the conclusion.
A website usually gets 6 sign-ups per day. After a redesign, 11 sign-ups occur in one day. Test at the 5% level whether the mean has increased, using the critical region method.
Topic 2 — Geometric Hypothesis Testing
Setting Up the Test
Picture flipping what's supposed to be a fair coin (p = ½) and counting how many flips it takes to get your first tail. If it takes 5 flips, is the coin still fair, or is it biased? Geometric hypothesis testing answers exactly this kind of question — but instead of counting how many events happen in an interval (Poisson), we're counting how many trials it takes to get the first success.
The Flip Rule — Most Important Idea in This Topic
| H₁ says… | …extreme (surprising) values of X are… |
|---|---|
| p < claimed value | X ≥ x (took unusually many trials) |
| p > claimed value | X ≤ x (took unusually few trials) |
Finding the Critical Region
Just like Poisson, you can pre-calculate the critical region — but the useful formulas here are slightly different, and based on the geometric tail probabilities:
Palamedes builds a spinner numbered 1–40 and claims P(landing on '1') = 1/40 exactly. Odysseus is suspicious and runs a two-tailed test at the 10% level (5% in each tail), counting how many spins it takes to land on '1' for the first time.
Lower tail: solve 1 − (39/40)^c₁ = 0.05 → (39/40)^c₁ = 0.95 → c₁ = log(0.95)/log(39/40) = 2.026
Testing c₁ = 2: P(X ≤ 2) = 0.04938 (under 0.05, good) c₁ = 3: P(X ≤ 3) = 0.0731 (over)
→ c₁ = 2 is the critical value.
Upper tail: solve (39/40)^(c₂ − 1) = 0.05 → c₂ − 1 = log(0.05)/log(39/40) = 118.325 → c₂ = 119.325
Testing c₂ = 119: P(X ≥ 119) = 0.05041 (over, but closer) c₂ = 120: P(X ≥ 120) = 0.04915 (under)
→ c₂ = 119 is the critical value (chosen since the question asks for "as close as possible", even though it slightly exceeds 5%).
Critical region: X ≤ 2 or X ≥ 119
A gardener claims that 1 in 5 seeds of a certain flower germinate (p = 0.2). A friend is suspicious that the true germination probability is lower. They plant seeds one at a time and record that it takes 9 seeds to get the first one to germinate. Test at the 5% significance level.
A basketball player claims her free-throw success probability is p = 0.6. A coach suspects it's actually higher. In a session, she scores on her very first attempt. Using the critical region approach at the 5% level, is there sufficient evidence to support the coach's suspicion? (Hint: find the critical region X ≤ c first.)
What to Memorise
| Concept | Poisson X ~ Po(λ) | Geometric X ~ Geo(p) |
|---|---|---|
| Parameter tested | Mean rate, λ | Success probability, p |
| Observed value x means | Count of events in an interval | Trial number of first success |
| Sanity check | Rescale λ to match x's interval | Compare x with E(X) = 1/p |
| H₁: parameter < claim → extreme X | X ≤ x (fewer events) | X ≥ x (more trials needed) — flipped! |
| H₁: parameter > claim → extreme X | X ≥ x (more events) | X ≤ x (fewer trials needed) — flipped! |
| Critical region formula | Use cumulative Poisson tables/calculator | P(X≥c)=(1−p)^(c−1), P(X≤c)=1−(1−p)^c |
| Two-tailed split | α/2 in each tail | α/2 in each tail |
2. Rescale if needed (Poisson only) or compute E(X) = 1/p (Geometric only).
3. Find P(X as extreme as observed, or more extreme) under H₀.
4. Compare to α (or α/2 for two-tailed). Smaller → reject H₀.
5. Write the conclusion in context, in words, referring back to the original scenario.
Concepts Checklist
Exam Tips & Common Mistakes
- Forgetting to rescale λ. If the question gives a rate over one interval (e.g. per day) but the observed data is over a different interval (e.g. per 12 hours), you must rescale λ before doing anything else. Marking schemes award a method mark just for this step.
- Mixing up P(X ≥ c) and P(X ≤ c − 1). Cumulative tables/calculators usually only give P(X ≤ k) directly. Remember P(X ≥ c) = 1 − P(X ≤ c − 1). Students very often accidentally compute 1 − P(X ≤ c) instead, which is actually P(X ≥ c + 1) — off by one.
- The Geometric "flip rule" gets ignored. Students instinctively write X ≤ x for H₁: p < ... (copying the Poisson pattern), when actually a lower p needs more attempts, so the extreme region is X ≥ x. Always physically reason it through: "lower success chance → more tries needed → higher X is surprising."
- Exceeding α without realising it. When finding critical values, you must choose the value that does not exceed α (or α/2), unless the question specifically says "as close as possible even if it means exceeding α" — read the question wording carefully every time.
- Not stating a conclusion in context. "Reject H₀" alone loses marks. You must translate it back into the scenario: e.g. "there is sufficient evidence that the mean number of accidents per hour has increased."
- Forgetting to split α for two-tailed tests. Two-tailed tests always use α/2 in each tail, not α in each tail — a very common slip that inflates the critical region incorrectly.
- Sign errors when using logs. Since 0 < 1 − p < 1, log(1 − p) is negative. Dividing an inequality by a negative number flips its direction — easy to forget mid-calculation.
- Exam Tips & Common Mistakes
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