21.1.1 Poisson Hypothesis Testing

Introduction

Hypothesis testing for the mean of a Poisson distribution involves determining whether the observed data supports a specified value of the population mean $\lambda$ (or equivalently $\mu$ , where $\mu = \lambda$ ). This note focuses on:

Setting up null and alternative hypotheses.
Performing one-tailed and two-tailed tests.
Using critical regions to make decisions about hypotheses.

The Poisson Distribution

For a random variable $X \sim \text{Po}(\lambda)$

P(X = x) = \frac{\lambda^x e^{-\lambda}}{x!}, \quad x = 0, 1, 2, \dots

where $\lambda > 0$ is the mean and variance of the distribution.

Hypotheses for Poisson Tests

In hypothesis testing, the hypotheses are expressed in terms of $\lambda$ (the mean of the Poisson distribution):

Null Hypothesis ( $H_0$ ):

Assumes a specific value for $\lambda$ , e.g. $H_0: \lambda = \lambda_0$

Alternative Hypothesis ( $H_1$ ):

Specifies a different value or range for $\lambda$ , depending on the test:

One-tailed (e.g., $H_1: \lambda > \lambda_0$ or $H_1: \lambda < \lambda_0$ )
Two-tailed (e.g., $H_1: \lambda \neq \lambda_0$ ).

Test Statistic

The test statistic is the observed value of $X$ , which follows $\text{Po}(\lambda)$ .

The test compares $X$ to the critical region determined by $H_0$ .

Significance Level and Critical Region

The significance level ( $\alpha$ ) is the probability of rejecting $H_0$ when $H_0$ is true.

The critical region is the range of $X$ values that lead to rejecting $H_0$ . It is determined by ensuring that:

P(\text{Reject } H_0 | H_0 \text{ true}) \leq \alpha

Worked Examples

infoNote

Example 1: One-Tailed Test

Problem

A shop receives an average of 55 complaints per day. A new policy is introduced, and the manager believes that complaints have decreased. A random sample shows 22 complaints on a given day.

Test, at the 5% significance level, whether the mean number of complaints has decreased.

Step 1: Define Hypotheses

$H_0: \lambda = 5$ (mean complaints per day remain the same),
$H_1: \lambda < 5$ (mean complaints per day have decreased).

Step 2: Identify Test Statistic and Distribution

Under $H_0, X \sim \text{Po}(5)$

Step 3: Determine the Critical Region

The test is one-tailed, so the critical region is for low values of $X$ .

Find $r$ such that:

P(X \leq r | \lambda = 5) \leq 0.05

Using Poisson probabilities:

P(X \leq 0) = e^{-5} \times \frac{5^0}{0!} = e^{-5} \approx 0.0067

P(X \leq 1) = P(X = 0) + P(X = 1)

= e^{-5} + e^{-5} \times \frac{5^1}{1!} \approx 0.0067 + 0.0337 = 0.0404

P(X \leq 2) = P(X \leq 1) + P(X = 2)

= 0.0404 + e^{-5} \times \frac{5^2}{2!} \approx 0.0404 + 0.0842 = 0.1246

At $r=1, P(X \leq 1) \approx 0.0404 < 0.05$ , so the critical region is X ≤ 1

Step 4: Compare Test Statistic to Critical Region

The observed value is $X = 2$ .

Since $2 \not\in [0, 1], X$ is not in the critical region.

Step 5: Conclusion

There is insufficient evidence to reject $H_0$ at the $5\%$ significance level.

The mean number of complaints has not significantly decreased.

infoNote

Example 2: Two-Tailed Test

Problem

A factory produces items with an average of 88 defects per day. After equipment changes, 55 defects are observed on a given day.

Test, at the 10% significance level, whether the mean number of defects has changed.

Step 1: Define Hypotheses

$H_0: \lambda = 8$ (mean number of defects remains the same),
$H_1: \lambda \neq 8$ (mean number of defects has changed).

Step 2: Identify Test Statistic and Distribution

Under $H_0, X \sim \text{Po}(8)$

Step 3: Determine the Critical Region

The test is two-tailed.

Split the significance level into two tails: $\alpha = 0.1$ , so each tail has α/2 = 0.05

Find $r_1$ and $r_2$ such that:

P(X \leq r_1 | \lambda = 8) \leq 0.05, \quad P(X \geq r_2 | \lambda = 8) \leq 0.05

Left Tail:

Using cumulative probabilities for $X \sim \text{Po}(8)$

P(X \leq 5) = 0.1912, \quad P(X \leq 4) = 0.0902

Critical value: $r_1 = 4$

Right Tail:

Using complementary probabilities:

P(X \geq 12) = 1 - P(X \leq 11) = 1 - 0.9329 = 0.0671.

Critical value: $r_2 = 12$

Critical region: X ≤ 4 or X ≥ 12

Step 4: Compare Test Statistic to Critical Region

The observed value is $X = 5$ .

Since $5 \not\in [0, 4] \cup [12, \infty), X$ is not in the critical region

Step 5: Conclusion

There is insufficient evidence to reject $H_0$ at the $10\%$ significance level.

The mean number of defects has not significantly changed.

Note Summary

infoNote

Common Mistakes

Misinterpreting hypotheses: Always frame $H_0$ and $H_1$ in terms of $\lambda$ , the mean of the Poisson distribution.
Incorrect critical region: Ensure cumulative probabilities correspond to the correct tail(s) for one-tailed or two-tailed tests.
Using the wrong distribution: Verify that the test statistic follows a Poisson distribution under $H_0$

infoNote

Key Formulas

Poisson Probability:

P(X = x) = \frac{\lambda^x e^{-\lambda}}{x!}

Critical Region:

One-tailed: $P(X \leq r | H_0) \leq \alpha$ or $P(X \geq r | H_0) \leq \alpha$
Two-tailed: Split $\alpha$ between the two tails.

Mean and Variance of Poisson:

\mathbb{E}[X] = \lambda, \quad \text{Var}(X) = \lambda

Poisson Hypothesis Testing Simplified Revision Notes for A-Level Edexcel Further Maths Further Statistics 1

21.1.1 Poisson Hypothesis Testing

Introduction

The Poisson Distribution

Hypotheses for Poisson Tests

Null Hypothesis ( $H_0$ ):

Alternative Hypothesis ( $H_1$ ):

Test Statistic

Significance Level and Critical Region

Worked Examples

Example 1: One-Tailed Test

Example 2: Two-Tailed Test

Note Summary

Common Mistakes

Key Formulas

500K+ Students Use These Powerful Tools to Master Poisson Hypothesis Testing For their A-Level Exams.

Other Revision Notes related to Poisson Hypothesis Testing you should explore

Poisson Hypothesis Testing Simplified Revision Notes for A-Level Edexcel Further Maths Further Statistics 1

21.1.1 Poisson Hypothesis Testing

Introduction

The Poisson Distribution

Hypotheses for Poisson Tests

Null Hypothesis (H0H_0H0​):

Alternative Hypothesis (H1H_1H1​):

Test Statistic

Significance Level and Critical Region

Worked Examples

Example 1: One-Tailed Test

Example 2: Two-Tailed Test

Note Summary

Common Mistakes

Key Formulas

500K+ Students Use These Powerful Tools to Master Poisson Hypothesis Testing For their A-Level Exams.

Other Revision Notes related to Poisson Hypothesis Testing you should explore

Null Hypothesis ( $H_0$ ):

Alternative Hypothesis ( $H_1$ ):