Geometric Hypothesis Tests

A geometric distribution is a model used when you have two outcomes (usually “success” and “failure”), the probability of success is constant for each trial, and you’re interested in the number of trials needed to get one success.
Geometric hypothesis tests involve using the geometric distribution to test a claim about the success probability in a geometric setting, denoted as p.
The five steps to perform the Geometric Hypothesis Test are as follows: State the hypothesis, identify the test statistic and its distribution, compute the p-value, state the conclusion and context.
The Null Hypothesis (H0) states that the population parameter (p) is equal to a specific value. The Alternative Hypothesis (H1) affirms that population parameter is not equal, less than, or greater than the specific value.
The test statistic is computed from the sample data. For the geometric distribution, the relevant statistic is X, the number of trials until the first success.
The significance level, denoted by alpha, is the probability of rejecting the null hypothesis when it is true. A common choice is 0.05, meaning there is a 5% risk of concluding that a difference exists when there is no actual difference.
The p-value is the probability that the test statistic will take a value as extreme or more extreme than the one observed, assuming the null hypothesis is true. This value is compared with the significance level to determine if the null hypothesis can be rejected.
If the p-value is less than the significance level, we reject the null hypothesis (say there is sufficient evidence to support the claim). If the p-value is greater than the significance level, we do not reject the null hypothesis (say there is not sufficient evidence to support the claim).
Contextual understanding is important for the completion of hypothesis testing. The conclusion reached from a statistical test must be communicated in the context of the problem, not just in terms of p-values and hypothesis terms.
Don’t forget to check assumptions! Geometric distribution assumes independence of trials, constant probability of success, and a binary outcome for each trial. If these assumptions are not met, conclusions drawn from hypothesis testing may not be reliable.