Maths › Further Statistics 1 › Hypothesis tests for Poisson and geometric models
Hypothesis tests for Poisson and geometric models
The testing machinery does not change when the distribution does: state hypotheses about the parameter, compute the tail probability of what you saw, and compare it with the significance level.
Builds on Geometric and negative binomial distributions and Hypothesis testing with the binomial.
IN THIS TOPIC
- State hypotheses about λ for a Poisson model or p for a geometric one.
- Find the tail probability of the observed value and compare with the level.
- Identify critical regions and report conclusions in context.
WHAT YOU PROBABLY THINK
Because the Poisson distribution has no fixed number of trials, hypothesis testing cannot be applied to it.
Testing a rate
A Poisson test asks whether the underlying rate has changed. The hypotheses name the parameter, not the data: H₀: λ = λ₀ against a one- or two-tailed alternative. Nothing about the method needs a fixed number of trials, so the opening claim is looking for a requirement that was never there. Compute the probability of a result at least as extreme as the one observed, assuming H₀, and compare with the significance level.
WORKED EXAMPLE
A rate under suspicion
Complaints arrive at a mean of 5 per week. After a change of policy, 10 arrive in a week. Test at the 5% level whether the rate has increased.
H₀: λ = 5, H₁: λ > 5, one-tailed at 5%.
Assuming H₀, P(X ≥ 10) = 1 − P(X ≤ 9) = 0.0318.
0.0318 < 0.05, so reject H₀: there is evidence at the 5% level that the complaint rate has risen. Note P(X ≥ 9) would be 0.068, so 10 is the smallest value in the critical region.
Testing a waiting time
For a geometric model the parameter is p, and a long wait is evidence that p is smaller than claimed. The upper tail is unusually easy here: P(X ≥ x) = (1 − p)x-1, since needing at least x trials means the first x − 1 all failed. No summation is required at all.
WORKED EXAMPLE
A machine that keeps missing
A process is claimed to succeed with probability 0.25 per attempt. The first success comes on the 12th attempt. Test at the 5% level whether p is smaller than claimed.
H₀: p = 0.25, H₁: p < 0.25. A small p means a long wait, so the evidence is in the upper tail of X.
P(X ≥ 12) = 0.75¹¹ = 0.0422.
0.0422 < 0.05, so reject H₀: there is evidence that the success probability is below 0.25.
YOUR TURN
A two-tailed Poisson test
Faults occur at Po(8) per shift. A new shift records 3 faults. Test at the 5% level whether the rate has changed.
Show the working
H₀: λ = 8, H₁: λ ≠ 8, two-tailed, so each tail carries 2.5%.
The observed value is low, so compute the lower tail: P(X ≤ 3) = 0.0424.
0.0424 > 0.025, so do not reject H₀: there is insufficient evidence at the 5% level that the fault rate has changed. Halving the level for a two-tailed test is what changes the verdict here.
THE EXAM BIT
- State hypotheses in terms of λ or p, never in terms of the sample value.
- Halve the significance level for each tail of a two-tailed test, and say that you have.
- For geometric upper tails use (1 − p) to the power x − 1 rather than summing terms.
- Finish in context: name the rate or process rather than stopping at 'reject H₀'.
CHECK YOURSELF
For a Poisson model with H₀: λ = 4 and H₁: λ > 4, the observed value is 9 and P(X ≥ 9) = 0.0214. State the conclusion at the 5% level.
Show a hint
Compare the tail probability with 0.05.
Show the answer
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Poisson tests fix hypotheses on λ; compute the tail probability of the observed count under H₀.
Geometric upper tails are (1 − p) to the power x − 1: a long wait is evidence for a smaller p.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- State hypotheses about λ for a Poisson model or p for a geometric one.
- Find the tail probability of the observed value and compare with the level.
- Identify critical regions and report conclusions in context.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.