Poisson Hypothesis Testing (Edexcel A Level Further Maths: Further Statistics 1): Revision Note

Poisson hypothesis testing

How do I test for the mean of a Poisson distribution?

• If $X~\mathrm{Po}\left(\lambda \right)$, test for the mean, $\lambda$, using the following hypotheses

  • ${\mathrm{H}}_0:\lambda =...$

  • ${\mathrm{H}}_1:\lambda \ne ...$ or $\lambda <...$or $\lambda >...$

  • with significance level $\alpha$

    • For example, $\alpha =0.05$ for 5%

• You will be given an observed value, $x$, in the question

  • This is what is being tested against $\lambda$

    • For example, "There's usually 3 accidents per hour ($\lambda =3$), but last week there was 5 accidents per hour ($x=5$)"

  • You may need to rescale $\lambda$ to fit in the same interval of time or space as $x$

• Assuming ${\mathrm{H}}_0:\lambda =...$

  • Find the probability that $X$ is the observed value $x$, or more extreme than that

  • If the total probability of these values is $<\alpha$ (or $<\frac{\alpha }{2}$ for two-tailed tests)

    • Write that "there is sufficient evidence to reject ${\mathrm{H}}_0$"

  • If not, write that "there is insufficient evidence to reject ${\mathrm{H}}_0$" 

• Write a conclusion in context

  • For example

    • "the mean number of accidents has increased from 3 per hour"

    • or "the mean number of accidents has not changed from 3 per hour"

How do I find the critical region for a Poisson hypothesis test?

• If ${\mathrm{H}}_1:\lambda <...$ 

  • Assume that ${\mathrm{H}}_0:\lambda =...$

  • Then test different integer values, $c$, to get $\mathrm{P}\left(X\le c\right)$ as close to $\alpha$as possible, without exceeding it

    • Use cumulative Poisson tables or a calculator to help

    • The integer that's the nearest is called the critical value

    • Checking one integer higher should show that $\mathrm{P}\left(X\le c+1\right)$ is $>\alpha$

  • The critical region is $X\le c$

• If ${\mathrm{H}}_1:\lambda >...$

  • It's the same process, but with $\mathrm{P}\left(X\ge c\right)$ as close to $\alpha$ as possible, without exceeding it

    • Beware of integers with discrete inequalities

      • $\mathrm{P}\left(X\ge c\right)$ means $1-\mathrm{P}\left(X\le c-1\right)$

      • You may have found what $c-1$ is, not what $c$ is!

  • The critical region is $X\ge c$

• If ${\mathrm{H}}_1:\lambda \ne ...$

  • The critical region is $X\le c_1$ or $X\ge c_2$

    • $\mathrm{P}\left(X\le c_1\right)$ is as close to $\frac{\alpha }{2}$ as possible, without exceeding it

    • $\mathrm{P}\left(X\ge c_2\right)$ is as close to $\frac{\alpha }{2}$ as possible, without exceeding it

What is the actual significance level?

• As the Poisson model is discrete, it's not possible to get a critical region whose probability sums to $\alpha$ exactly

  • That's because $X$ can only take integer values

• Whatever it does sum to is called the actual significance level

  • The actual amount of probability in the tail (or tails)

• For example, if ${\mathrm{H}}_1:\lambda <...$ has the critical region $X\le c$

  • Then $\mathrm{P}\left(X\le c\right)$ will be just less than $\alpha$

    • It's value is the actual significance level

    • It represents the probability of rejecting ${\mathrm{H}}_0$ incorrectly (when ${\mathrm{H}}_0$ was actually true)

• Some questions want a critical region that's as close to $\alpha$ as possible, even if that means probabilities that exceed $\alpha$

  • For example, if $\mathrm{P}\left(X\le 6\right)=0.0298$ and $\mathrm{P}\left(X\le 7\right)=0.0510$ where $\alpha =0.05$

    • Then $X\le 7$ is the critical region that's as close to $\alpha$ as possible

    • The actual significance level is 0.0510