Poisson Hypothesis Testing (DP IB Applications & Interpretation (AI)): Revision Note

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 

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 < m₀ or 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 H­₀ then 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 < m₀ the p-value is 

  • For H₁ : m > m₀ the p-value is 

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 < m₀ the critical region is  where c is the critical value

  • c is the largest integer such that 

    • Check that 

• For H₁ : m > m₀ the critical region is  where c is the critical value

  • c is the smallest integer such that 

    • Check that