Two-tailed tests | 双尾检验

📚 Two-tailed tests | 双尾检验

A two-tailed test is a hypothesis test where the alternative hypothesis does not specify a direction of the effect. Instead of testing whether a parameter is strictly greater or strictly less than a claimed value, a two-tailed test checks for any significant difference — either higher or lower. In A‑Level Edexcel Mathematics, you will encounter two‑tailed tests with binomial, Poisson, normal and t‑distributions. Understanding how to set up the hypotheses, identify rejection regions on both sides of the distribution, calculate critical values and interpret p‑values is essential for success in Statistics.

双尾检验是一种不指定效应方向的假设检验。备择假设不判断参数是严格大于还是严格小于原假设值，而是检验是否存在任何显著差异——无论偏高还是偏低。在Edexcel A‑Level数学中，你将遇到基于二项分布、泊松分布、正态分布和t分布的双尾检验。掌握如何建立假设、在分布两端确定拒绝域、计算临界值并解读p值，是统计部分取得好成绩的关键。

1. Introduction to Two-tailed Tests | 双尾检验简介

A two‑tailed test is used when we want to detect a change in either direction from a hypothesised population parameter. For example, a machine that fills bottles with 500 ml of water might be tested to see if the mean volume has changed, without specifically looking for over‑filling or under‑filling. In such cases, the alternative hypothesis is of the form “not equal to”, written as H₁: θ ≠ θ₀, where θ₀ is the value stated in the null hypothesis H₀: θ = θ₀.

当我们希望检测总体参数是否存在任何方向的变化时，使用双尾检验。例如，一台灌装500毫升水的机器，需要检测平均容量是否发生了变化，而不特别关注是灌多还是灌少。此时备择假设的形式是“不等于”，写作 H₁: θ ≠ θ₀，其中 θ₀ 是原假设 H₀: θ = θ₀ 中指定的数值。

The total significance level α is split equally between the two tails of the sampling distribution. This means that the critical region lies at both extremes, and the test statistic must fall into either tail to reject H₀. Two‑tailed tests are generally more conservative than one‑tailed tests because the evidence must be strong enough to overcome the significance threshold on both sides.

总显著性水平 α 被平均分配到抽样分布的两个尾部。这意味着拒绝域位于两端，检验统计量必须落入任一侧尾部才能拒绝 H₀。双尾检验通常比单尾检验更保守，因为证据必须足够强，以克服分布在两侧的显著性门槛。

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2. Null and Alternative Hypotheses | 零假设和备择假设

For a two‑tailed test, the hypotheses are always structured as:

对于双尾检验，假设总是按如下结构设置：

H₀: θ = θ₀
H₁: θ ≠ θ₀

Here θ can represent a population proportion p, a population mean μ, or a population rate λ in a Poisson distribution. The null hypothesis assumes the parameter takes a specific value, often a standard or historical figure. The alternative hypothesis simply states that the parameter is different, without indicating whether it is larger or smaller.

这里的 θ 可以代表总体比例 p、总体均值 μ 或泊松分布的总体发生率 λ。原假设假定参数取某个特定值，通常是一个标准或历史数据。备择假设仅仅说明参数不同，不指明是更大还是更小。

Writing the hypotheses in the correct form is the first step in any test. Examiners often award marks for identifying that a test is two‑tailed based on wording such as “has changed”, “is different from”, or “there is evidence of a difference”.

在任何检验中，正确书写假设是关键的第一步。考官常会根据“发生了变化”“与……不同”“有差异的证据”等措辞，判断学生是否能识别出这是双尾检验，并给予分数。

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3. Significance Level and Rejection Regions | 显著性水平与拒绝域

In a two‑tailed test, if the significance level is α, the probability in each tail is α/2. The rejection region consists of the most extreme outcomes in both the lower and upper tails. For a continuous symmetric distribution like the normal distribution, the boundaries are symmetric around the population parameter under H₀. For example, with α = 0.05, the rejection regions for a standard normal test statistic are Z ≤ −1.96 and Z ≥ 1.96.

在双尾检验中，如果显著性水平为 α，每个尾部的概率为 α/2。拒绝域由分布下尾和上尾的最极端结果组成。对于正态分布这样连续对称的分布，边界线在 H₀ 下的总体参数两侧对称。例如，当 α = 0.05 时，标准正态检验统计量的拒绝域是 Z ≤ −1.96 和 Z ≥ 1.96。

For discrete distributions, such as the binomial, it may not be possible to split α exactly into two equal halves. Instead, we find the largest critical region in each tail such that the probability in each tail is as close as possible to α/2, while ensuring the total probability does not exceed α. This often leads to slight differences between the actual significance level and the nominal α.

对于离散分布，如二项分布，可能无法将 α 恰好均分成两半。我们转而寻找每个尾部最大的拒绝域，使每个尾部的概率尽可能接近 α/2，同时保证总概率不超过 α。这常导致实际显著性水平与名义上的 α 略有出入。

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4. Critical Values for Two-tailed Tests | 双尾检验的临界值

Critical values separate the rejection region from the acceptance region. For a two‑tailed normal test, the critical values are ±z(α/2), where z(α/2) is the z‑score that cuts off area α/2 in the upper tail. Commonly used z(α/2) values include:

临界值是将拒绝域与接受域分开的边界值。对于双尾正态检验，临界值为 ±z(α/2)，其中 z(α/2) 是截出上尾面积 α/2 的 z 分数。常用的 z(α/2) 值包括：

α	α/2	z(α/2)
0.10	0.05	1.645
0.05	0.025	1.960
0.01	0.005	2.576

For a binomial test, critical values are usually whole numbers found by listing cumulative probabilities. We look for the largest integer r such that P(X ≤ r) ≤ α/2 and the smallest integer s such that P(X ≥ s) ≤ α/2. The critical region is then X ≤ r and X ≥ s.

对于二项检验，临界值通常是整数，通过列出累计概率找到。我们寻找最大的整数 r 使 P(X ≤ r) ≤ α/2，以及最小的整数 s 使 P(X ≥ s) ≤ α/2。拒绝域即为 X ≤ r 和 X ≥ s。

In a t‑test when the population variance is unknown, the critical values are taken from the t‑distribution with n−1 degrees of freedom. The two‑tailed critical value is t(α/2, ν), where ν = n−1. These values can be obtained from statistical tables provided in the Edexcel formula booklet.

在总体方差未知的 t 检验中，临界值取自自由度为 n−1 的 t 分布。双尾临界值为 t(α/2, ν)，其中 ν = n−1。这些值可从 Edexcel 公式手册的统计表中查到。

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5. Using the p-value Approach | 使用p值方法

The p‑value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming H₀ is true. For a two‑tailed test, this probability is doubled if the distribution is symmetric and the alternative is two‑sided. More generally, the p‑value equals the probability of results as far or further from the parameter in both directions.

p值是在 H₀ 成立的条件下，获得一个至少与观测值同样极端的检验统计量的概率。对于双尾检验，如果分布是对称的且备择假设是双侧的，这个概率要乘以2。更一般地，p值等于在参数两侧得到与观测结果同样远或更远的概率之和。

For a binomial test with test statistic X, the two‑tailed p‑value can be calculated as P(X ≤ observed) + P(X ≥ equally extreme value). If the observed count lies below the hypothesised proportion, we find the corresponding upper count that gives the same distance from the centre under symmetry. If the distribution is not perfectly symmetric, we still sum the probabilities of all values that are as or more extreme than the observed value in either direction.

对于检验统计量为 X 的二项检验，双尾 p 值可以计算为 P(X ≤ 观测值) + P(X ≥ 同等极端的另一端值)。如果观测频数低于假设比例，我们找到在对称情况下与中心距离相同的上侧计数。如果分布并不完全对称，我们仍对所有比观测值更极端的任一方向的取值概率求和。

Once the p‑value is found, we compare it with α. If p-value ≤ α, we reject H₀; otherwise, we do not reject H₀. This method is particularly useful when using technology, but in Edexcel A‑Level exams the critical region method is more frequently tested.

一旦找到 p 值，就将其与 α 比较。如果 p值 ≤ α，就拒绝 H₀；否则不拒绝 H₀。使用计算工具时该方法特别方便，但在 Edexcel A‑Level 考试中，更常考察临界域方法。

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6. Two-tailed Test for a Proportion (Binomial) | 比例的双尾检验（二项分布）

When testing a population proportion p, we define a random variable X ~ B(n, p₀) under H₀. The critical region is determined using the binomial distribution tables. For a two‑tailed test at significance level α, we find the lower critical value r and the upper critical value s as described earlier. The observed number of successes is then compared with these cut‑offs.

当检验总体比例 p 时，在 H₀ 下定义随机变量 X ~ B(n, p₀)。利用二项分布表确定拒绝域。对于显著性水平 α 的双尾检验，我们如前所述找到下临界值 r 和上临界值 s。然后比较观测到的成功次数与这些分界值。

For example, suppose H₀: p = 0.5, H₁: p ≠ 0.5, n = 20, α = 0.05. From binomial tables with p = 0.5, we find P(X ≤ 5) = 0.0207 and P(X ≤ 6) = 0.0577. To keep the lower tail probability ≤ 0.025, we take lower critical value r = 5. By symmetry, the upper critical value is s = 20 − 5 = 15. Thus the critical region is X ≤ 5 or X ≥ 15. The actual total significance level is P(X ≤ 5) + P(X ≥ 15) = 2 × 0.0207 = 0.0414.

例如，假设 H₀: p = 0.5，H₁: p ≠ 0.5，n = 20，α = 0.05。从 p = 0.5 的二项分布表，可得 P(X ≤ 5) = 0.0207，P(X ≤ 6) = 0.0577。为使下尾概率 ≤ 0.025，取下临界值 r = 5。由对称性，上临界值为 s = 20 − 5 = 15。因此拒绝域为 X ≤ 5 或 X ≥ 15。实际总显著性水平为 P(X ≤ 5) + P(X ≥ 15) = 2 × 0.0207 = 0.0414。

If the observed value falls inside {r+1, …, s−1} we do not reject H₀. The key to success in binomial two‑tailed questions is to always state the actual significance level and the explicit critical region.

如果观测值落在 {r+1, …, s−1} 内，我们不拒绝 H₀。在二项分布双尾检验题中成功的关键是，一定要说明实际显著性水平并明确写出拒绝域。

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7. Two-tailed Test for a Mean (Normal Distribution) | 均值的双尾检验（正态分布）

When the population is normally distributed with known variance σ², or when the sample size is large enough for the Central Limit Theorem to apply, we use a z‑test. The test statistic is:

当总体服从正态分布且方差 σ² 已知，或样本量足够大使中心极限定理适用时，我们使用 z 检验。检验统计量为：

Z = (x̄ − μ₀) / (σ / √n)

For a two‑tailed test, we reject H₀ if |Z| > z(α/2). Equivalently, the critical region for the sample mean x̄ is x̄ < μ₀ − z(α/2) × (σ/√n) or x̄ > μ₀ + z(α/2) × (σ/√n).

对于双尾检验，若 |Z| > z(α/2) 则拒绝 H₀。等价地，样本均值 x̄ 的拒绝域为 x̄ < μ₀ − z(α/2) × (σ/√n) 或 x̄ > μ₀ + z(α/2) × (σ/√n)。

Suppose we test H₀: μ = 500 against H₁: μ ≠ 500, with σ = 10, n = 25, x̄ = 495, and α = 0.05. Then z = (495 − 500) / (10/5) = −2.5. The critical value is z(0.025) = 1.96. Since |−2.5| > 1.96, we reject H₀ and conclude there is evidence that the mean differs from 500.

假设检验 H₀: μ = 500，H₁: μ ≠ 500，已知 σ = 10，n = 25，x̄ = 495，α = 0.05。则 z = (495 − 500) / (10/5) = −2.5。临界值为 z(0.025) = 1.96。因为 |−2.5| > 1.96，我们拒绝 H₀，得出结论有证据表明均值与500不同。

If the test statistic falls between −z(α/2) and z(α/2), we do not reject H₀. The conclusion should always be stated in the context of the problem, not simply “reject H₀” or “accept H₀”.

如果检验统计量落在 −z(α/2) 与 z(α/2) 之间，我们不拒绝 H₀。结论始终要结合问题背景来表述，而不只是说“拒绝 H₀”或“接受 H₀”。

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8. Two-tailed Test for a Mean with Unknown Variance (t-test) | 方差未知时均值的双尾检验（t检验）

When the population variance is unknown and the sample size is small (n < 30), or when the population is normally distributed but σ is estimated by the sample standard deviation s, we use a t‑test. The test statistic is:

当总体方差未知且样本量较小（n < 30），或总体服从正态分布但 σ 由样本标准差 s 估计时，我们使用 t 检验。检验统计量为：

t = (x̄ − μ₀) / (s / √n)

The critical values are ±t(α/2, n−1). The number of degrees of freedom is ν = n−1. If the absolute value of the calculated t exceeds the critical value, we reject H₀.

临界值为 ±t(α/2, n−1)。自由度为 ν = n−1。如果计算出的 t 的绝对值超过临界值，则拒绝 H₀。

For instance, a sample of 10 measurements gives x̄ = 24.3, s = 1.8. Testing H₀: μ = 25 vs H₁: μ ≠ 25 at α = 0.05, t = (24.3 − 25) / (1.8/√10) ≈ −1.23. With 9 degrees of freedom, t(0.025,9) = 2.262. Since |−1.23| < 2.262, we do not reject H₀. There is insufficient evidence to say the mean differs from 25.

例如，一个容量为10的样本给出 x̄ = 24.3，s = 1.8。检验 H₀: μ = 25 对 H₁: μ ≠ 25，α = 0.05，t = (24.3 − 25) / (1.8/√10) ≈ −1.23。自由度为9时，t(0.025,9) = 2.262。因为 |−1.23| < 2.262，我们不拒绝 H₀。没有足够证据表明均值不等于25。

In Edexcel exams, it is crucial to check the conditions: the population must be normal or the sample size large enough that the t‑test is robust. The t‑test is widely used for small samples when the data are roughly symmetric.

在Edexcel考试中，检查条件是至关重要的：总体必须正态，或样本量足够大使得t检验稳健。当数据大致对称时，t检验广泛用于小样本情形。

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9. Comparing One-tailed and Two-tailed Tests | 单尾与双尾检验比较

A one‑tailed test allocates the entire significance level α to one tail, making it easier to reject H₀ if the effect is in the predicted direction. A two‑tailed test splits α between two tails, requiring stronger evidence in either direction. Table shows the differences:

单尾检验将整个显著性水平 α 分配到一个尾部，如果效应在预测方向上，就更容易拒绝 H₀。双尾检验将 α 分配到两个尾部，要求在任一方向都有更强的证据。下表展示区别：

Feature	One‑tail	Two‑tail
H₁	θ > θ₀ or θ < θ₀	θ ≠ θ₀
Rejection region	one tail	both tails
Critical value (normal)	z(α) only	±z(α/2)
p‑value	one‑sided probability	doubled or sum of two tails
Power for a specific direction	higher if direction correct	lower for a given α

It is vital to decide which test to use before seeing the data, and this decision should be based on the wording of the research question. Edexcel questions often prompt you to state whether a two‑tailed test is appropriate.

在查看数据之前就决定使用哪种检验至关重要，该决定应基于研究问题的措辞。Edexcel 的题目常要求你说明双尾检验是否合适。

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10. Common Mistakes in Two-tailed Tests | 双尾检验的常见错误

One frequent mistake is using the one‑tailed critical value in a two‑tailed test, for instance using z = 1.645 when α = 0.05 instead of z = 1.96. This leads to an incorrect rejection region and invalid conclusion.

一个常见错误是在双尾检验中使用单尾临界值，例如当 α = 0.05 时使用 z = 1.645 而非 z = 1.96。这会导致拒绝域错误，结论无效。

Another common error is failing to double the p‑value for symmetric two‑tailed tests. Students often report only the probability in one tail and compare it directly with α, effectively performing a one‑tailed test. In Edexcel mark schemes, stating the two‑tailed p‑value correctly is essential.

另一个普遍错误是未将对称双尾检验的 p 值翻倍。学生常只报告一个尾部的概率并直接与 α 比较，实际上进行的是单尾检验。在Edexcel评分方案中，正确给出双尾 p 值至关重要。

For discrete distributions, simply dividing α by 2 and reading the critical value for α/2 from one tail can produce a total error probability noticeably less than α. The correct approach is to find the critical region so that the total probability in both tails is as large as possible without exceeding α. Students must state the actual significance level explicitly.

对于离散分布，简单地将 α 除以2并从单尾表中读取 α/2 对应的临界值，可能会使总错误概率显著小于 α。正确的方法是找到拒绝域，使两个尾部的概率之和尽可能大但不超 α。学生必须明确说明实际显著性水平。

Finally, some learners misinterpret the conclusion. Failing to reject H₀ does not prove H₀ is true; it simply means the evidence is not strong enough to support a difference. Always phrase conclusions in terms of the evidence.

最后，一些学习者会错误解读结论。无法拒绝 H₀ 并不证明 H₀ 为真；它仅仅意味着证据不足以来支持存在差异。始终用证据的措辞来表达结论。

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11. Worked Example: Two-tailed Binomial Test | 例题：二项分布双尾检验

Question: A coin is tossed 20 times yielding 6 heads. Test at the 5% significance level whether the coin is fair.

问题： 抛一枚硬币20次得到6次正面。在5%显著性水平下检验硬币是否公平。

Hypotheses: H₀: p = 0.5, H₁: p ≠ 0.5. Let X ~ B(20, 0.5) be the number of heads.

假设： H₀: p = 0.5, H₁: p ≠ 0.5。设 X ~ B(20, 0.5) 为正面次数。

Significance level: α = 0.05, so each tail should have probability as close as possible to 0.025 without exceeding it.

显著性水平： α = 0.05，因此每个尾部概率应尽可能接近但不超0.025。

Critical values: For p = 0.5, P(X ≤ 5) = 0.0207 ≤ 0.025, but P(X ≤ 6) = 0.0577 > 0.025. So lower critical value r = 5. By symmetry, upper critical value is s = 20 − 5 = 15. Rejection region: X ≤ 5 or X ≥ 15. Actual significance level = 2 × 0.0207 = 0.0414.

临界值： 对于 p = 0.5，P(X ≤ 5) = 0.0207 ≤ 0.025，但 P(X ≤ 6) = 0.0577 > 0.025。因此下临界值 r = 5。由对称性，上临界值 s = 20 − 5 = 15。拒绝域：X ≤ 5 或 X ≥ 15。实际显著性水平 = 2 × 0.0207 = 0.0414。

Test statistic: Observed X = 6. Since 6 is not ≤ 5 and not ≥ 15, it does not fall in the rejection region. Therefore we do not reject H₀. There is insufficient evidence at the 5% level to suggest the coin is biased.

检验统计量： 观测值 X = 6。由于6既不≤5也不≥15，未落入

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