Poisson Hypothesis Testing (DP IB Maths: AI HL)

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