Hypothesis Tests – Poisson Distribution - Upper tail test

Introduction to Hypothesis Tests – Poisson Distribution - Upper Tail Test

• The Poisson distribution is a probability distribution used in statistical modelling where events occur independently and at a constant average rate.
• A hypothesis test is a statistical test used to determine if there is enough evidence in a sample of data to infer that a particular condition is true for the entire population.
• An upper tail test in this context refers to testing the probability of events exceeding a certain level. It is also known as a one-tailed test.

Formulating the Hypothesis

• Every hypothesis test contains two opposing hypotheses about a population parameter: the null hypothesis denoted as H0 and the alternative hypothesis denoted as H1.
• For a Poisson Distribution, the null hypothesis H0 usually states that the mean λ (lambda) equals a specified value.
• The alternative hypothesis H1 indicates what researchers think is true about the population. In an upper tail test, H1 usually states that the mean λ is greater than the specified value.

Setting the Significance Level

• The significance level (designated as α) represents the probability of rejecting the null hypothesis when it is true. Commonly, α is set to 0.05.
• If the generated p-value is less than the significance level α, we reject the null hypothesis.

Conducting the Test and Calculating the p-value

• The p-value is a probability that measures the evidence against H0. For an upper tail test, it’s the probability that a Poisson random variable is greater than or equal to the observed value given that H0 is true.
• To compute the p-value, you must determine the observed value, then find the sum of the probabilities of that value and all values more extreme (higher) under the assumed Poisson distribution.

Interpretation of the Results

• The strength of the evidence against H0 in favour of H1 is measured by the p-value. The smaller the p-value, the stronger the evidence against H0.
• If the p-value is less than or equal to the preferred significant level, you reject H0 and conclude that the evidence strongly supports H1.

Understanding Assumptions

• The assumptions for this particular type of hypothesis test include: events are occurring independently, with a constant mean rate, and within a fixed period of time or region.
• Violating any of these assumptions can significantly affect the result of the test.

Considerations and Applications

• This technique is commonly used in fields such as manufacturing, quality control, telecommunications, finance, and environmental studies.
• Care should be taken while interpreting the results, as the reality of the assumption of independence and constant rate of occurrences might not be entirely accurate in the real world.

Concluding Thoughts

• Mastering the upper tail test of Poisson Distribution provides a powerful tool for making inferences and decisions in diverse areas.
• Working through a range of diverse problems is an excellent way to become proficient in applying these tests.