Determining Critical Regions in Poisson Distribution Tests | 泊松分布检验中临界区域的确定方法

📚 Determining Critical Regions in Poisson Distribution Tests | 泊松分布检验中临界区域的确定方法

In a hypothesis test, the critical region (also called the rejection region) is the set of all values of the test statistic for which the null hypothesis H₀ is rejected. For a Poisson distribution X ~ Po(λ), where X is a discrete count variable, the critical region is always a set of integers at the lower end, the upper end, or both ends of the distribution. Mastering how to locate this region is an essential skill for Edexcel A-Level Further Mathematics.

在假设检验中，临界区域（又称拒绝域）是使得检验统计量落于其中时我们拒绝原假设 H₀ 的所有取值集合。对于泊松分布 X ~ Po(λ)，由于 X 是离散型计数变量，临界区域总是位于分布低端、高端或两端的整数集合。熟练确定临界区域是 Edexcel 进阶数学考试中的核心技能。

1. Setting Up Hypotheses for a Poisson Test | 泊松检验假设的建立

Before finding a critical region, we must formally state the hypotheses. The Poisson mean is denoted by λ. There are three standard arrangements. A one-tailed lower test examines whether the mean has decreased, giving H₀: λ = λ₀ against H₁: λ < λ₀. A one-tailed upper test examines whether the mean has increased, giving H₀: λ = λ₀ against H₁: λ > λ₀. A two-tailed test examines whether the mean has changed in either direction, giving H₀: λ = λ₀ against H₁: λ ≠ λ₀.

在确定临界区域之前，我们必须正式写出假设。泊松均值记为 λ，检验有三种标准形式。单尾下限检验考察均值是否减小，即 H₀: λ = λ₀ 对 H₁: λ < λ₀。单尾上限检验考察均值是否增大，即 H₀: λ = λ₀ 对 H₁: λ > λ₀。双尾检验考察均值是否朝任一方向变化，即 H₀: λ = λ₀ 对 H₁: λ ≠ λ₀。

The significance level α (usually 5% or 1%) is the maximum probability of wrongly rejecting H₀ when it is true. Because the Poisson distribution is discrete, the actual probability of falling in the critical region is often less than α; we state this actual significance level when drawing conclusions.

显著性水平 α（通常取 5% 或 1%）是在 H₀ 为真时错误拒绝 H₀ 的最大概率。由于泊松分布是离散的，实际落入临界区域的概率往往小于 α；在作结论时需要说明实际显著性水平。

2. Finding a Lower-Tail Critical Region | 下尾临界区域的确定

For a lower-tail test at significance level α, we seek the largest integer c such that P(X ≤ c) ≤ α. The critical region is then X ≤ c. The procedure is to scan the cumulative probability P(X ≤ x) from x = 0 upward: keep increasing x while the cumulative probability remains no greater than α; the last such x is the critical value c.

对于显著性水平为 α 的下尾检验，我们要找最大的整数 c，使得 P(X ≤ c) ≤ α。此时临界区域为 X ≤ c。操作方法是从 x = 0 开始向上扫描累积概率 P(X ≤ x)：只要累积概率仍不超过 α 就继续增大 x；最后一个满足条件的 x 即为临界值 c。

Example. Let X ~ Po(3), and test H₀: λ = 3 against H₁: λ < 3 at the 5% level. We compare P(X ≤ 0) = e⁻³ ≈ 0.0498 ≤ 0.05 and P(X ≤ 1) = 0.1991 > 0.05. Hence c = 0, so the critical region is X = 0. If we observe X = 0, we reject H₀ and conclude that the mean has decreased.

例。设 X ~ Po(3)，在 5% 显著性水平下检验 H₀: λ = 3 对 H₁: λ < 3。比较 P(X ≤ 0) = e⁻³ ≈ 0.0498 ≤ 0.05，而 P(X ≤ 1) = 0.1991 > 0.05。因此 c = 0，临界区域为 X = 0。若观测到 X = 0，则拒绝 H₀，认为均值确实减小了。

3. Finding an Upper-Tail Critical Region | 上尾临界区域的确定

For an upper-tail test, we seek the smallest integer c such that P(X ≥ c) ≤ α. Since the Poisson cumulative tables give P(X ≤ x), we use the identity P(X ≥ c) = 1 − P(X ≤ c − 1). Therefore we need the smallest c such that P(X ≤ c − 1) ≥ 1 − α. In practice, we scan P(X ≤ x) downward from large x until the cumulative probability first falls below 1 − α; the value just above is c − 1, and hence c is found.

对于上尾检验，我们要找最小的整数 c，使得 P(X ≥ c) ≤ α。由于泊松累积表给出的是 P(X ≤ x)，我们利用恒等式 P(X ≥ c) = 1 − P(X ≤ c − 1)。因此需要找满足 P(X ≤ c − 1) ≥ 1 − α 的最小 c。实际操作中，从较大的 x 向下扫描 P(X ≤ x)，直到累积概率首次低于 1 − α；前一个值就是 c − 1，从而确定 c。

Example. Let X ~ Po(3), test H₀: λ = 3 against H₁: λ > 3 at the 5% level. We need P(X ≤ c − 1) ≥ 0.95. From tables, P(X ≤ 6) = 0.9665 ≥ 0.95, but P(X ≤ 5) = 0.9161 < 0.95. Hence c − 1 = 6, so c = 7. The critical region is X ≥ 7, and the actual significance level is P(X ≥ 7) = 1 − 0.9665 = 0.0335 = 3.35%.

例。设 X ~ Po(3)，在 5% 显著性水平下检验 H₀: λ = 3 对 H₁: λ > 3。需要 P(X ≤ c − 1) ≥ 0.95。查表得 P(X ≤ 6) = 0.9665 ≥ 0.95，但 P(X ≤ 5) = 0.9161 < 0.95。因此 c − 1 = 6，即 c = 7。临界区域为 X ≥ 7，实际显著性水平为 P(X ≥ 7) = 1 − 0.9665 = 0.0335 = 3.35%。

4. Two-Tailed Critical Regions | 双尾临界区域的确定

In a two-tailed test, the significance level is split equally between the two tails: each tail receives α/2. We find a lower critical value c₁ such that P(X ≤ c₁) ≤ α/2, and an upper critical value c₂ such that P(X ≥ c₂) ≤ α/2, i.e. P(X ≤ c₂ − 1) ≥ 1 − α/2. The overall critical region is X ≤ c₁ or X ≥ c₂.

在双尾检验中，显著性水平被均分为两部分：每条尾巴各得 α/2。我们需要找到下临界值 c₁ 使得 P(X ≤ c₁) ≤ α/2，以及上临界值 c₂ 使得 P(X ≥ c₂) ≤ α/2，即 P(X ≤ c₂ − 1) ≥ 1 − α/2。总临界区域为 X ≤ c₁ 或 X ≥ c₂。

An important subtlety is that one tail may have no critical value at all. If even P(X = 0) exceeds α/2, then no lower critical region exists; similarly for the upper tail. In that case the two-tailed test effectively becomes one-sided in the opposite direction. You must still report this honestly in your conclusion.

一个重要的细节是：某一条尾可能根本不存在临界值。如果连 P(X = 0) 都大于 α/2，那么就没有下尾临界区域；上尾同理。此时双尾检验实际上只在另一个方向有效，在结论中必须如实说明这一点。

5. Worked Example — Lower-Tail Test | 完整例题——下尾检验

A shop claims that the mean number of defective items per batch is λ = 4. A quality-control manager suspects the number has decreased. Test at the 5% significance level.

某商店声称每批次的次品均数为 λ = 4。质量控制经理怀疑次品数已经减少。在 5% 显著性水平下进行检验。

Let X be the number of defects per batch, with X ~ Po(λ). We set H₀: λ = 4 and H₁: λ < 4. Since this is a one-tailed lower test, we seek the largest c with P(X ≤ c) ≤ 0.05.

设 X 为每批次次品数，X ~ Po(λ)。建立假设 H₀: λ = 4，H₁: λ < 4。由于这是单尾下限检验，我们找满足 P(X ≤ c) ≤ 0.05 的最大 c。

P(X ≤ 0) = 0.0183 ≤ 0.05, but P(X ≤ 1) = 0.0916 > 0.05

The cumulative probabilities for λ = 4 are shown below.

λ = 4 的累积概率如下表所示。