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

Poisson Hypothesis Testing

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

• If , test for the mean, , using the following hypotheses

  • 

  • or or

  • with significance level 

    • For example,  for 5%

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

  • This is what is being tested against 

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

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

• Assuming 

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

  • If the total probability of these values is (or  for two-tailed tests)

    • Write that "there is sufficient evidence to reject "

  • If not, write that "there is insufficient evidence to reject " 

• 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  

  • Assume that

  • Then test different integer values, , to get  as close to 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 is

  • The critical region is

• If

  • It's the same process, but with  as close to as possible, without exceeding it

    • Beware of integers with discrete inequalities

      • means 

      • You may have found what  is, not what  is!

  • The critical region is 

• If

  • The critical region is  or

    • is as close to  as possible, without exceeding it

    • is as close to 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  exactly

  • That's because  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  has the critical region 

  • Then will be just less than 

    • It's value is the actual significance level

    • It represents the probability of rejecting  incorrectly (when was actually true)

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

  • For example, if and  where 

    • Then  is the critical region that's as close to  as possible

    • The actual significance level is 0.0510