In a Poisson test, what are the hypotheses written in terms of?

The population parameter \lambda (or \mu), the mean rate.

When is a test one-tailed?

When H1 says the parameter has increased or decreased: \lambda\lambda0 or \lambda<\lambda0.

When is a test two-tailed?

When H1 says only that the parameter has changed: \lambda\neq\lambda0.

What is the decision rule for a one-tailed test using a probability?

Reject H0 if the probability of the result or one more extreme is less than the significance level.

What is a critical region?

The set of values of the test statistic that leads to rejecting H0.

What is the actual significance level?

The probability of the test statistic falling in the critical region when H0 is true.

X\simPo(6), H1:\lambda<6 at 5\%. Critical region?

X\leq1, since P(X\leq1)=0.0174 and P(X\leq2)=0.0620.

How should you phrase a conclusion when H0 is not rejected?

There is insufficient evidence, at the stated level, that the mean rate has changed (in context).

Why might a Poisson model fail in context?

Events may not be independent or the rate may not be constant, for example bursts of arrivals.

Hypothesis test for the mean of a Poisson distributionEdexcel A-Level Further Maths: Flashcards

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State the conditions for events to follow a Poisson distribution.

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State the conditions for events to follow a Poisson distribution.: Events occur randomly, independently and at a constant average rate.
A rate is $1.2$ errors per page. What is the distribution of the number of errors on $5$ pages?: $\mathrm{Po}(6)$
In a Poisson test, what are the hypotheses written in terms of?: The population parameter $\lambda$ (or $\mu$ ), the mean rate.
When is a test one-tailed?: When $\mathrm{H}_1$ says the parameter has increased or decreased: $\lambda>\lambda_0$ or $\lambda<\lambda_0$ .
When is a test two-tailed?: When $\mathrm{H}_1$ says only that the parameter has changed: $\lambda\neq\lambda_0$ .
How do you find $\mathrm{P}(X\geq x)$ from cumulative tables?: $1-\mathrm{P}(X\leq x-1)$
What is the decision rule for a one-tailed test using a probability?: Reject $\mathrm{H}_0$ if the probability of the result or one more extreme is less than the significance level.
For a two-tailed test at $10\%$ , how much goes in each tail?: $5\%$ in each tail.
What is a critical region?: The set of values of the test statistic that leads to rejecting $\mathrm{H}_0$ .
What is the actual significance level?: The probability of the test statistic falling in the critical region when $\mathrm{H}_0$ is true.
$X\sim\mathrm{Po}(6)$ , $\mathrm{H}_1:\lambda<6$ at $5\%$ . Critical region?: $X\leq1$ , since $\mathrm{P}(X\leq1)=0.0174$ and $\mathrm{P}(X\leq2)=0.0620$ .
How should you phrase a conclusion when $\mathrm{H}_0$ is not rejected?: There is insufficient evidence, at the stated level, that the mean rate has changed (in context).
Why might a Poisson model fail in context?: Events may not be independent or the rate may not be constant, for example bursts of arrivals.

Exam questions on Hypothesis test for the mean of a Poisson distribution

A help desk historically receives calls at a mean rate of $3$ per hour, modelled by a Poisson distribution. After a publicity campaign, the manager records $7$ calls in a randomly chosen hour and wants to test, at the $5\%$ significance level, whether the mean rate of calls has increased.
Write down the conclusion of the test, in context.2 marks
In a long-running textbook, typing errors occur at a mean rate of $1.2$ per page, modelled by a Poisson distribution. A new editor checks $5$ randomly chosen pages and finds $2$ errors in total. A test is carried out at the $5\%$ significance level to see whether the mean rate of errors has decreased.
Find the critical region for this test, and state the actual significance level of the test.2 marks
A factory's machines have historically broken down at a mean rate of $4$ per week. During a randomly chosen two-week period there are $14$ breakdowns. The manager believes that the weekly rate has changed. Breakdowns are assumed to occur at random, independently and at a constant average rate, and the test is carried out at the $10\%$ significance level.
State suitable hypotheses for the test, where $\lambda$ is the mean number of breakdowns per week, and the distribution of the number of breakdowns in the two-week period under $\mathrm{H}_0$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).