# Poisson Hypothesis Testing(CIE A Level Maths: Probability & 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
• 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
• Using the formula for this can be time consuming so only use this method if you need to
• otherwise compare the probability of the random variable being at least as extreme as the observed value with the significance level
• For a two-tailed test you will need to find both critical values, one at each end of the distribution
• Take extra care when finding the critical region in the upper tail, you will have to find the probabilities for less than and subtract from one

#### 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 an increase in the number of likes he gets. Over a 6-hour period chosen at random Mr Viajo’s travel blog receives 5 likes.

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

(ii)
Find the rejection region for the test.

(iii)
Find the probability of a Type I error.

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

#### Exam Tip

• Take extra careful when working in the upper tail in Poisson distribution questions, this is where its easy to make mistakes.

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