3.2.2 Poisson Hypothesis Testing

How is a hypothesis test carried out for the mean of a Poisson distribution?

• The population parameter being tested will be the mean, λ , in a Poisson distribution
  • As it is the population mean, sometimes μ will be used instead
• A hypothesis test is used when the mean is questioned
• The null hypothesis, H₀  and alternative hypothesis, H₁ will be given in terms of λ (or μ)
  • Make sure you clearly define λ before writing the hypotheses
  • The null hypothesis will always be H₀ : λ = ...
  • The alternative hypothesis will depend on if it is a one-tailed or two-tailed test
  • A one-tailed test would test to see if the value of λ has either increased or decreased
    • The alternative hypothesis, will be H₁ will be  H₁ : λ > ...or H₁ : λ < ...
  • A two-tailed test would test to see if the value of λ has changed
    • The alternative hypothesis, H₁ will be  H₁ : λ  ≠ ...
• To carry out a hypothesis test with the Poisson distribution, the random variable will be the mean number of occurrences of the event within the given time/space interval
  • Remember you may need to change the mean to fit the interval of time or space for your observed value
• When defining the distribution, remember that the value of λ  is being tested, so this should be written as λ in the original definition, followed by the null hypothesis stating the assumed value of λ
• The Poisson distribution will be used to calculate the probability of the random variable taking the observed value or a more extreme value
• The hypothesis test can be carried out by
  • either calculating the probability of the random variable taking the observed or a more extreme value and comparing this with the significance level
  • or by finding the critical region and seeing whether the observed value of the test statistic lies within it
    • Finding the critical region can be more useful for considering more than one observed value or for further testing

How is the critical value found in a hypothesis test with the Poisson distribution?

• The critical value will be the first value to fall within the critical region
  • The Poisson distribution is a discrete distribution so the probability of the observed value being within the critical region, given a true null hypothesis may be less than the significance level
  • This is the actual significance level and is the probability of incorrectly rejecting the null hypothesis (a Type I error)
• For a one-tailed test use the formula to find the first value for which the probability of that or a more extreme value is less than the given significance level
  • Check that the next value would cause this probability to be greater than the significance level
    • For H₁ : λ < ...   if  and  then c is the critical value
    • For H₁ : λ > ... if  and  then c is the critical value
  • Using the formula for this can be time consuming so only use this method if you need to
    • otherwise compare the probability of the random variable being at least as extreme as the observed value with the significance level
• For a two-tailed test you will need to find both critical values, one at each end of the distribution
• Take extra care when finding the critical region in the upper tail, you will have to find the probabilities for less than and subtract from one

What steps should I follow when carrying out a hypothesis test with the Poisson distribution?

Step 1.  Define the mean, λ

Step 2.  Write the null and alternative hypotheses clearly using the form

H₀ : λ = ...

H₁ : λ = ...

Step 3.  Define the distribution, usually   where λ is the mean to be tested

Step 4.  Calculate the probability of the random variable being at least as extreme as the observed value

• • Or if told to find the critical region

Step 5.  Compare this probability with the significance level

• • Or compare the observed value with the critical region

Step 6.  Decide whether there is enough evidence to reject H₀ or whether it has to be accepted

Step 7.  Write a conclusion in context