Hypothesis testing with the Poisson | CAIE 9709 Probability & Statistics 2 Notes

Chapters

Probability & Statistics 2

Mathematics9709

Put the claimed rate in H-zero

Write $H_0:\lambda=\lambda_0$ and choose $H_1:\lambda>\lambda_0$ , $H_1:\lambda<\lambda_0$ or $H_1:\lambda\ne\lambda_0$ from the claim.
Scale the null rate to the total time before calculating.
The alternative hypothesis chooses the tail.

Under H₀, how surprising is 4 or more?

red tail = P(X ≥ 4) = 0.0383

H₁ chooses the upper tail; observed 4 chooses where it begins.

The alternative hypothesis chooses the tail; the observed count chooses where that tail begins.

Key idea

The p-value includes the observed count and all results more extreme in the direction of

H_1

Calculate the exact tail under H-zero

Worked example

Current-paper late-arrival test

Before a timetable change, girls and boys arrive late at mean rates 0.10 and 0.15 per day. In 5 days after the change, 4 students are late. Test at 5% whether the total mean rate has increased.

\lambda_0=5(0.25)=1.25

H_0:\lambda=1.25

H_1:\lambda>1.25

p=P(X\ge4)

p=1-P(X\le3)

p=0.0383

Since $0.0383<0.05$ , reject $H_0$ .

Conclusion: There is sufficient evidence, at the 5% level, that the mean number of late students has increased. (9709/62/F/M/24 Q5(d))

Common mistake

Do not calculate only

P(X=4)

. An upper-tail p-value for an observed count of 4 is

P(X\ge4)

For a large null mean, use a corrected normal tail

If the null mean exceeds 15, use $N(\lambda_0,\lambda_0)$ with a continuity correction. The variance still comes from $H_0$ , not from the observed count.

Your turnIndependent Poisson-rate test [5 marks]

Machine faults occur at a mean rate of 1.5 per week. After an overhaul, 28 faults occur in 26 weeks. Use a suitable approximation to test at 5% whether the mean rate has fallen.

Show worked answer

H_0:\lambda=39,\qquad H_1:\lambda<39

X\approx N(39,39)

z=\frac{28.5-39}{\sqrt{39}}=-1.681

p=0.0463

Since $0.0463<0.05$ , reject $H_0$ .

Conclusion: There is sufficient evidence that the mean weekly fault rate has fallen.

Examiner note

A conclusion must be non-definite and in context: write “there is sufficient evidence that…”, not “the claim is proved”.