# Poisson Hypothesis Testing(Edexcel International A Level Maths: Statistics 2)

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Amber

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Maths

## Poisson Hypothesis Testing

#### How is a hypothesis test carried out for the mean of a Poisson distribution?

• The population parameter being tested will be the mean, λ , in a Poisson distribution Po(λ)
• As it is the population mean, sometimes μ will be used instead
• A hypothesis test is used when the mean is questioned
• The null hypothesis, H0  and alternative hypothesis, H1 will be given in terms of λ (or μ)
• Make sure you clearly define λ before writing the hypotheses
• The null hypothesis will always be H0 : λ = ...
• The alternative hypothesis will depend on if it is a one-tailed or two-tailed test
• A one-tailed test would test to see if the value of λ has either increased or decreased
• The alternative hypothesis, will be H1 will be  H1 : λ > ...or H1 : λ < ...
• A two-tailed test would test to see if the value of λ has changed
• The alternative hypothesis, H1 will be  H1 : λ  ≠ ...
• To carry out a hypothesis test with the Poisson distribution, the random variable will be the mean number of occurrences of the event within the given time/space interval
• Remember you may need to change the mean to fit the interval of time or space for your observed value
• When defining the distribution, remember that the value of λ  is being tested, so this should be written as λ in the original definition, followed by the null hypothesis stating the assumed value of λ
• The Poisson distribution will be used to calculate the probability of the random variable taking the observed value or a more extreme value
• The hypothesis test can be carried out by
• either calculating the probability of the random variable taking the observed or a more extreme value and comparing this with the significance level
• or by finding the critical region and seeing whether the observed value of the test statistic lies within it
• Finding the critical region can be more useful for considering more than one observed value or for further testing

#### How is the critical value found in a hypothesis test with the Poisson distribution?

• The critical value will be the first value to fall within the critical region
• The Poisson distribution is a discrete distribution so the probability of the observed value being within the critical region, given a true null hypothesis may be less than the significance level
• This is the actual significance level and is the probability of incorrectly rejecting the null hypothesis (a Type I error)
• For a one-tailed test use the formula to find the first value for which the probability of that or a more extreme value is less than the given significance level
• Check that the next value would cause this probability to be greater than the significance level
• For H1 : λ < ...   if  and  then c is the critical value
• For H1 : λ > ... if  and  then c is the critical value
• For a two-tailed test you will need to find both critical values, one at each end of the distribution

#### What steps should I follow when carrying out a hypothesis test with the Poisson distribution?

Step 1.  Define the mean, λ

Step 2.  Write the null and alternative hypotheses clearly using the form

H0 : λ = ...

H1 : λ = ...

Step 3.  Define the distribution, usually   where λ is the mean to be tested

Step 4.  Calculate the probability of the random variable being at least as extreme as the observed value

• Or if told to find the critical region

Step 5.  Compare this probability with the significance level

• Or compare the observed value with the critical region

Step 6.  Decide whether there is enough evidence to reject H0 or whether it has to be accepted

Step 7.  Write a conclusion in context

#### Worked example

Mr Viajo believes that his travel blog receives an average of 8 likes per day (24 hour period).  He tries a new advertising campaign and carries out a hypothesis test at the 5% level of significance to see if there is a change in the number of likes he gets. Over a 12-hour period chosen at random Mr Viajo’s travel blog receives 7 likes.

(i)
State null and alternative hypotheses for Mr Viajo’s test.

(ii)
Find the critical regions for the test.

(iii)
Find the actual level of significance.

(iv)
Carry out the hypothesis test, writing your conclusion clearly.

#### Exam Tip

• You might have to change the value of λ based on the length of the interval
• Always make it clear to the examiner which value of λ you are using when you calculate probabilities

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