Test for a Poisson mean

Test for a Poisson mean

Overview of the Poisson Mean Test

  • The Poisson distribution is commonly applied in probability theory and statistics, particularly in scenarios where events occur randomly and independently.

  • The test for a Poisson mean is used in order to determine how likely the mean of a given sample could have come from a population with a Poisson distribution.

  • In the context of the Poisson, the mean (λ) is used as the average rate of occurrence.

Interpreting Poisson Scenarios

  • The Poisson distribution is typically used to model rare events, such as radioactive decay and phone calls to a call centre.

  • Always read scenarios carefully to identify whether they are underpinned by a Poisson distribution. Look out for indicators like “random”, “independent”, “average rate” or “over a fixed interval”.

Conducting the Test for a Poisson Mean

  • The first step is to state the null hypothesis, Ho, and alternative hypothesis, H1.

  • The null hypothesis, Ho, assumes the data follows a Poisson distribution. The alternative hypothesis, H1, assumes that it does not.

  • Next, determine the sample mean (x̄) and sample size (n). The sample mean can be used as both the sample and population parameter in a Poisson test, due to the Poisson distribution having a single parameter.

  • Conduct a goodness of fit test between your observed (O) and expected (E) results. The goodness of fit test allows you to see how well your expected distribution fits with your observed data.

Evaluating Results

  • Calculate the test statistic by using the Chi-square Test statistic formula: χ² = Σ [(O - E)² / E].

  • Compare the test statistic to the critical value from the Chi-square distribution table. If the test statistic is greater than the critical value, then reject the null hypothesis.

  • Be able to justify the critical region based on the significance level used, which is typically 0.05 or 5%.

Endnote

  • Mastering the steps in conducting a test for a Poisson mean is necessary to solve relevant problem sets.

  • Regular practice with different Poisson scenarios will deepen your understanding of this statistical concept.