IB DP Maths: AI HL复习笔记4.12.4 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

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

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