What is a hypothesis test using a Poisson distribution?
- You can use a Poisson distribution to test whether the mean number of occurrences for a given time period within a population has increased or decreased
- These tests will always be one-tailed
- You will not be expected to perform a two-tailed hypothesis test with the Poisson distribution
- A sample will be taken and the test statistic x will be the number of occurrences which will be tested using the distribution
What are the steps for a hypothesis test of a Poisson proportion?
- STEP 1: Write the hypotheses
- H0 : m = m0
- Clearly state that m represents the mean number of occurrences for the given time period
- m0 is the assumed mean number of occurrences
- You might have to use proportion to find m0
- H1 : m < m0 or H1 : m > m0
- H0 : m = m0
- STEP 2: Calculate the p-value or find the critical region
- See below
- STEP 3: Decide whether there is evidence to reject the null hypothesis
- If the p-value < significance level then reject H0
- If the test statistic is in the critical region then reject H0
- STEP 4: Write your conclusion
- If you reject H0 then there is evidence to suggest that...
- The mean number of occurrences has decreased (for H1 : m < m0)
- The mean number of occurrences has increased (for H1 : m > m0)
- If you accept H0 then there is insufficient evidence to reject the null hypothesis which suggests that...
- The mean number of occurrences has not decreased (for H1 : m < m0)
- The mean number of occurrences has not increased (for H1 :m > m0)
How do I calculate the p-value?
- The p-value is determined by the test statistic x
- The p-value is the probability that ‘a value being at least as extreme as the test statistic’ would occur if null hypothesis were true
- For H1 : m < m0 the p-value is
- For H1 : m > m0 the p-value is
How do I find the critical value and critical region?
- The critical value and critical region are determined by the significance level α%
- Your calculator might have an inverse Poisson function that works just like the inverse normal function
- You need to use this value to find the critical value
- The value given by the inverse Poisson function is normally one away from the actual critical value
- For H1 : m < m0 the critical region is where c is the critical value
- c is the largest integer such that
- Check that
- For H1 : m > m0 the critical region is where c is the critical value
- c is the smallest integer such that
- Check that
The owner of a website claims that his website receives an average of 120 hits per hour. An interested purchaser believes the website receives on average fewer hits than they claim. The owner chooses a 10-minute period and observes that the website receives 11 hits. It is assumed that the number of hits the website receives in any given time period follows a Poisson Distribution.