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, H_1?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?H_1?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,?H_1?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
    • 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 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