Test for the parameter p of a Geometric Distribution

Introduction to Testing the Parameter p of a Geometric Distribution

In statistics, a Geometric Distribution is a type of probability distribution that focuses on the number of trials needed to get the first success in repeated Bernoulli trials. In the Geometric Distribution, the success/failure probability remains constant from trial to trial, and each trial is independent of the others.
The parameter p of a Geometric Distribution corresponds to the probability of success on each trial.
A Hypothesis Test is a statistical method used to make inferences or draw conclusions about the population from which a sample has been drawn. For the parameter p of a Geometric Distribution, it’s often used to establish whether a given value of p is plausible.

Performing a Test for the Parameter p

The first step to perform a test for the parameter p of a Geometric Distribution is to state the null hypothesis H0 and the alternative hypothesis H1. The null hypothesis usually asserts that p equals to a specified value, while the alternative hypothesis asserts that p is less than, greater than, or not equal to the specified value, depending on the question.
Next, choose a suitable test statistic, T. For Geometric Distributions, T is usually the number of trials up to and including the first success.
Determine the critical region for the test, which is the set of values of T for which we would reject H0. The critical region depends on the significance level of the test, usually denoted by α. If the observed value of T falls in the critical region, we reject H0 in favour of H1.
The decision of whether to reject or not reject the null hypothesis is made based on the observed value of the test statistic and the critical region.

Interpreting the Outcome of the Test

If the observed value of T falls in the critical region, that indicates that the result is ‘statistically significant’ at the level α, and we reject H0 in favour of H1. This would suggest that our data provides enough evidence the true probability of success, p, differs from the value stated in H0.
If the observed value of T does not fall in the critical region, we do not have sufficient evidence to reject H0. This does not prove H0 is true, rather, it suggests that our data is consistent with H0 being true.

Considerations and Limitations

The outcome of a hypothesis test is not a definitive proof but a statement about what is suggested by the data in the context of the test.
The conclusions from the test can change with a different significance level α or a different sample.
Hypothesis testing for the parameter p of a Geometric Distribution assumes the assumption of independent and identically distributed Bernoulli trials. If this assumptions do not hold, the test might not be appropriate.