Poisson Hypothesis Testing

What is a hypothesis test using a Poisson distribution?

You can use a Poisson distribution to test whether the mean number of occurrences for a given time period within a population has increased or decreased

These tests will always be one-tailed
You will not be expected to perform a two-tailed hypothesis test with the Poisson distribution

A sample will be taken and the test statistic x will be the number of occurrences which will be tested using the distribution $X ~ Po (m)$

What are the steps for a hypothesis test of a Poisson proportion?

STEP 1: Write the hypotheses
- H₀: m = m₀
  - Clearly state that m represents the mean number of occurrences for the given time period
  - m₀ is the assumed mean number of occurrences
  - You might have to use proportion to find m₀
- H₁: m < m₀or H₁: m > m₀

STEP 2: Calculate the p-value or find the critical region

See below

STEP 3: Decide whether there is evidence to reject the null hypothesis

If the p-value < significance level then reject H₀
If the test statistic is in the critical region then reject H₀

STEP 4: Write your conclusion

If you reject H₀ then there is evidence to suggest that...

The mean number of occurrences has decreased (for H₁: m < m₀)
The mean number of occurrences has increased (for H₁: m > m₀)

If you accept H₀then there is insufficient evidence to reject the null hypothesis which suggests that...

The mean number of occurrences has not decreased (for H₁: m < m₀)
The mean number of occurrences has not increased (for H₁:m > m₀)

How do I calculate the p-value?

The p-value is determined by the test statistic x
The p-value is the probability that ‘a value being at least as extreme as the test statistic’ would occur if null hypothesis were true

For H₁: m < m₀the p-value is $P (X \leq x | m = m_{0})$
For H₁: m > m₀the p-value is $P (X \geq x | m = m_{0})$

How do I find the critical value and critical region?

The critical value and critical region are determined by the significance level α%
Your calculator might have an inverse Poisson function that works just like the inverse normal function

You need to use this value to find the critical value
The value given by the inverse Poisson function is normally one away from the actual critical value

For H₁: m < m₀the critical region is $X \leq c$ where c is the critical value

c is the largest integer such that $P (X \leq c | m = m_{0}) \leq α %$

Check that $P (X \leq c + 1 | m = m_{0}) > α %$

For H₁: m > m₀the critical region is $X \geq c$ where c is the critical value

c is the smallest integer such that $P (X \geq c | m = m_{0}) \leq α %$

Check that $P (X \geq c - 1 | m = m_{0}) > α %$

Examiner Tip

In an exam it is very important to state the time period for your variable
Make sure the mean used in the null hypothesis is for the stated time period

Worked example

The owner of a website claims that his website receives an average of 120 hits per hour. An interested purchaser believes the website receives on average fewer hits than they claim. The owner chooses a 10-minute period and observes that the website receives 11 hits. It is assumed that the number of hits the website receives in any given time period follows a Poisson Distribution.

State null and alternative hypotheses to test the purchaser’s claim.

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Find the critical region for a hypothesis test at the 5% significance level.

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Perform the test using a 5% significance level. Clearly state the conclusion in context.

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Poisson Hypothesis Testing (DP IB Maths: AI HL)

Revision Note