📚 A-Level Further Mathematics: Calculating Significance Levels and Power of a Test | A-Level 进阶数学：计算检验的显著性水平与功效
In A-Level Further Mathematics, hypothesis testing is a core topic in statistics. Two closely related concepts often confuse students: the significance level of a test and the power of a test. This article explains how to calculate both, using step-by-step examples in the style of Edexcel exam questions.
在 A-Level 进阶数学中，假设检验是统计学的核心内容。其中两个密切相关的概念常常让学生感到困惑：检验的显著性水平和检验的功效。本文将通过 Edexcel 考试风格的逐步例题，解释如何计算这两者。
1. Hypothesis Testing Basics | 假设检验基础
A hypothesis test starts with a null hypothesis H₀ and an alternative hypothesis H₁. We collect data and decide whether the evidence is strong enough to reject H₀ in favour of H₁. The decision is based on a test statistic and a critical region.
假设检验始于原假设 H₀ 和备择假设 H₁。我们收集数据并判断证据是否足够充分，从而决定是否拒绝 H₀ 而接受 H₁。这一决策基于检验统计量和临界区域。
For example, we might test whether a coin is biased towards heads. Let X be the number of heads in 20 tosses. Under H₀, X ~ B(20, 0.5). We reject H₀ if X is too large.
例如，我们可能检验一枚硬币是否偏向正面。设 X 为 20 次抛掷中出现正面的次数。在 H₀ 下，X ~ B(20, 0.5)。如果 X 过大，我们就拒绝 H₀。
The significance level α is the probability of making a Type I error, that is, rejecting H₀ when it is actually true. The power of a test is the probability of correctly rejecting H₀ when H₁ is true, i.e. 1 − β, where β is the probability of a Type II error.
显著性水平 α 是犯第一类错误的概率，即当 H₀ 实际为真时错误地拒绝 H₀。检验的功效是当 H₁ 为真时正确拒绝 H₀ 的概率，即 1 − β，其中 β 是犯第二类错误的概率。
2. Significance Level and Critical Region | 显著性水平与临界区域
The significance level is chosen before the test, usually 5% or 1%. It defines how unlikely the observed result must be before we reject H₀. In practice, we find a critical region such that the probability of the test statistic falling in that region, given H₀, is as close as possible to α without exceeding it (for discrete distributions).
显著性水平在检验之前选定，通常为 5% 或 1%。它决定了观察结果必须有多不可能才使我们拒绝 H₀。实际操作中，我们找到一个临界区域，使得在 H₀ 成立时检验统计量落入该区域的概率尽可能接近 α 但不超过它（对于离散分布而言）。
For a one-tailed test with H₁: p > 0.5, the critical region is of the form X ≥ c. The actual significance level is P(X ≥ c | H₀). Note that this may be less than the nominal 5% because of discreteness.
对于单尾检验 H₁: p > 0.5，临界区域形式为 X ≥ c。实际显著性水平为 P(X ≥ c | H₀)。注意，由于离散性，这个值可能小于名义上的 5%。
To find c, we take the smallest integer c such that P(X ≥ c | H₀) ≤ 0.05. Equivalently, P(X ≤ c − 1) ≥ 0.95.
为了求 c，我们取满足 P(X ≥ c | H₀) ≤ 0.05 的最小整数 c。等价地，P(X ≤ c − 1) ≥ 0.95。
3. Calculating the Significance Level for a Binomial Test | 计算二项检验的显著性水平
Consider testing H₀: p = 0.5 against H₁: p > 0.5 using X ~ B(20, 0.5). Suppose we choose the critical region X ≥ 14. What is the significance level?
考虑使用 X ~ B(20, 0.5) 检验 H₀: p = 0.5 对 H₁: p > 0.5。假设我们选择临界区域 X ≥ 14。那么显著性水平是多少？
We calculate P(X ≥ 14) = P(X = 14) + P(X = 15) + … + P(X = 20). Using binomial probabilities or a calculator, we get:
我们计算 P(X ≥ 14) = P(X = 14) + P(X = 15) + … + P(X = 20)。利用二项概率公式或计算器，可得：
P(X ≥ 14) = 1 − P(X ≤ 13) = 1 − 0.9423 = 0.0577
Thus the actual significance level is 5.77%, which is above 5%. So this critical region gives a significance level larger than 5%. To keep α ≤ 0.05, we would need X ≥ 15, giving P(X ≥ 15) = 1 − P(X ≤ 14) = 1 − 0.9793 = 0.0207.
因此实际显著性水平为 5.77%，高于 5%。所以这个临界区域给出的显著性水平大于 5%。为了确保 α ≤ 0.05，我们需要 X ≥ 15，此时 P(X ≥ 15) = 1 − P(X ≤ 14) = 1 − 0.9793 = 0.0207。
In exams, you must state the actual significance level, not just the nominal level, because the discrete nature of X makes the true probability different.
在考试中，你必须写出实际显著性水平，而不只是名义水平，因为 X 的离散性使得真实概率有所不同。
4. Type I and Type II Errors | 第一类错误与第二类错误
A Type I error occurs when we reject H₀ when H₀ is true. The probability of a Type I error is exactly the significance level α.
第一类错误是当 H₀ 实际上为真时我们拒绝了 H₀。犯第一类错误的概率恰好就是显著性水平 α。
A Type II error occurs when we fail to reject H₀ when H₁ is true. Its probability is denoted β. The value of β depends on the true value of the population parameter under H₁.
第二类错误是当 H₁ 实际上为真时我们没有拒绝 H₀。其概率记为 β。β 的值取决于 H₁ 下总体参数的真值。
For a given critical region, β is calculated using the distribution under H₁. For example, if H₁: p = 0.6 and critical region X ≥ 15 with X ~ B(20, p), then β = P(X ≤ 14 | p = 0.6).
对于给定的临界区域，β 使用 H₁ 下的分布来计算。例如，若 H₁: p = 0.6，临界区域 X ≥ 15，X ~ B(20, p)，则 β = P(X ≤ 14 | p = 0.6)。
5. Power of a Test: Definition | 检验功效的定义
The power of a test is the probability of rejecting H₀ when H₁ is true. It is a measure of how well the test detects the effect specified by H₁.
检验功效是当 H₁ 为真时拒绝 H₀ 的概率。它衡量检验能够检测到 H₁ 所指定效应的能力。
Power = 1 − β = P(X ∈ critical region | H₁ true).
功效 = 1 − β = P(X ∈ 临界区域 | H₁ 为真)。
High power means the test is sensitive to the alternative hypothesis. In practice, we want tests with high power and low significance level, but these two goals often conflict.
高效功意味着检验对备择假设很敏感。在实际中，我们希望检验具有高效功和低显著性水平，但这两个目标常常互相冲突。
6. Calculating Power for a Binomial Test | 计算二项检验的功效
Let us continue the previous example. Suppose we use X ~ B(20, p) and critical region X ≥ 15. We want to find the power when H₁: p = 0.6.
我们继续前面的例子。假设使用 X ~ B(20, p)，临界区域 X ≥ 15。我们想求当 H₁: p = 0.6 时检验的功效。
Under H₁, X ~ B(20, 0.6). The power is P(X ≥ 15 | p = 0.6). Using binomial tables or a calculator:
在 H₁ 下，X ~ B(20, 0.6)。功效为 P(X ≥ 15 | p = 0.6)。使用二项分布表或计算器：
P(X ≥ 15) = 1 − P(X ≤ 14) = 1 − 0.8744 = 0.1256
So the power is only 12.56%. This means that if the true probability of heads is 0.6, we have only a 12.6% chance of correctly rejecting H₀. The test is not very powerful for this particular alternative.
因此功效仅为 12.56%。这意味着如果真实的正面概率是 0.6，我们只有 12.6% 的机会正确拒绝 H₀。对于这个特定的备择假设，检验的功效不高。
7. Power Depends on the Alternative Parameter | 功效依赖于备择参数
The power of a test changes as the true parameter moves away from H₀. The further the true value is from the null value, the larger the power tends to be.
检验功效随真实参数偏离 H₀ 的程度而变化。真实值与原假设值相差越远，功效往往越大。
For instance, in the above example with critical region X ≥ 15, the power when p = 0.7 is:
例如，在上述临界区域 X ≥ 15 的例子中，当 p = 0.7 时的功效为：
P(X ≥ 15 | p = 0.7) = 1 − P(X ≤ 14) = 1 − 0.5836 = 0.4164
And when p = 0.8, power = 1 − P(X ≤ 14 | p = 0.8) = 1 − 0.1958 = 0.8042. As expected, power increases as p increases.
当 p = 0.8 时，功效 = 1 − P(X ≤ 14 | p = 0.8) = 1 − 0.1958 = 0.8042。正如预期，功效随 p 增大而增大。
You may be asked to calculate power for a given value of the parameter, or to comment on how power changes. The key idea is that power is a conditional probability based on the true distribution under H₁.
你可能会被要求计算给定参数值下的功效，或评述功效如何变化。关键思想是功效是基于 H₁ 下真实分布的条件概率。
8. Using Normal Approximation for Power | 使用正态近似计算功效
When n is large, we can approximate the binomial distribution by a normal distribution. This is often needed in Edexcel A-Level Further Statistics when calculating power for large samples.
当 n 很大时，我们可以用正态分布近似二项分布。在 Edexcel A-Level 进阶统计中，计算大样本功效时经常需要这样做。
Suppose X ~ B(n, p). For large n, X ≈ N(np, np(1−p)). We apply a continuity correction when using the normal approximation to a discrete distribution.
设 X ~ B(n, p)。当 n 较大时，X ≈ N(np, np(1−p))。当使用正态近似离散分布时，需要应用连续性修正。
For example, let X ~ B(100, p). We test H₀: p = 0.5 against H₁: p > 0.5 at the 5% significance level. We first find the critical region. Using the normal approximation with continuity correction:
例如，设 X ~ B(100, p)。我们检验 H₀: p = 0.5 对 H₁: p > 0.5，显著性水平为 5%。首先求临界区域。使用带有连续性修正的正态近似：
Under H₀, X ≈ N(50, 25). We need the value c such that P(X ≥ c) ≤ 0.05. Using the normal approximation, we find the 95th percentile of N(50, 25) is 50 + 1.645 × 5 = 58.225. With continuity correction, the critical region starts at X ≥ 59 (since P(X ≥ 59) ≈ P(Y > 58.5) where Y ~ N(50, 25)).
在 H₀ 下，X ≈ N(50, 25)。我们需要求满足 P(X ≥ c) ≤ 0.05 的值 c。使用正态近似，N(50, 25) 的第 95 百分位数为 50 + 1.645 × 5 = 58.225。加上连续性修正，临界区域从 X ≥ 59 开始（因为 P(X ≥ 59) ≈ P(Y > 58.5)，其中 Y ~ N(50, 25)）。
9. Worked Example: Power with Normal Approximation | 例题：使用正态近似计算功效
Continuing the above test, find the power when the true value is p = 0.55.
继续上述检验，求当真实值 p = 0.55 时检验的功效。
Under H₁, X ~ B(100, 0.55), so approximately X ≈ N(55, 24.75) since np = 55 and np(1−p) = 24.75. The standard deviation is √24.75 ≈ 4.975.
在 H₁ 下，X ~ B(100, 0.55)，因此近似 X ≈ N(55, 24.75)，因为 np = 55，np(1−p) = 24.75。标准差为 √24.75 ≈ 4.975。
Power = P(X ≥ 59 | p = 0.55). Applying continuity correction, we approximate this by P(Y > 58.5) where Y ~ N(55, 24.75).
功效 = P(X ≥ 59 | p = 0.55)。应用连续性修正，我们将其近似为 P(Y > 58.5)，其中 Y ~ N(55, 24.75)。
P(Y > 58.5) = P(Z > (58.5 − 55)/4.975) = P(Z > 0.704) ≈ 0.2407
So the power is about 24.1%. This illustrates that even a moderate deviation from p = 0.5 is hard to detect with n = 100 at the 5% significance level.
所以功效约为 24.1%。这说明即使与 p = 0.5 存在中等偏差，在 n = 100 和 5% 显著性水平下也难以检测出来。
10. Relationship Between Significance Level and Power | 显著性水平与功效的关系
If we increase the significance level, the critical region becomes larger, so the power increases. However, this also increases the probability of making a Type I error. There is a trade-off between α and power.
如果增大显著性水平，临界区域会变大，因此功效增大。但与此同时，犯第一类错误的概率也会增大。α 与功效之间存在权衡。
For a fixed sample size, you cannot simultaneously make α very small and power very high. To improve power without increasing α, you need a larger sample size.
在样本量固定时，你不能同时使 α 非常小且功效非常高。为了在不增大 α 的情况下提高功效，你需要更大的样本量。
In Edexcel questions, you might be asked to compare two critical regions and state which gives greater power. The one with the larger rejection region always has more power, but also a higher significance level.
在 Edexcel 考题中，你可能会被要求比较两个临界区域，并指出哪个具有更大的功效。拒绝区域较大的那个总是具有更高的功效，但显著性水平也更高。
11. Common Pitfalls and Exam Tips | 常见错误与考试提示
One common mistake is using the nominal significance level (e.g. 5%) as the actual significance level in discrete tests. Always calculate the actual probability using the binomial distribution under H₀.
一个常见错误是在离散检验中使用名义显著性水平（例如 5%）作为实际显著性水平。务必使用 H₀ 下的二项分布计算实际概率。
Another mistake is forgetting to use the correct distribution when calculating power. Power must be computed under the alternative hypothesis H₁, not under H₀.
另一个错误是计算功效时忘记使用正确的分布。功效必须在备择假设 H₁ 下计算，而不是在 H₀ 下计算。
Always state whether the test is one-tailed or two-tailed. The critical region and the power calculation depend on this. In a two-tailed test, the critical region is split between both tails, and power calculations are slightly more involved.
始终说明检验是单尾还是双尾。临界区域和功效计算依赖于这一点。在双尾检验中，临界区域分布在两个尾部，功效计算略微复杂。
When using the normal approximation, remember to apply the continuity correction. Examiners penalise missing continuity corrections in approximations.
使用正态近似时，请记得应用连续性修正。考官会对近似中遗漏连续性修正扣分。
12. Summary | 总结
The significance level of a test is the probability of a Type I error, calculated under H₀. The power of a test is 1 − β, the probability of correctly rejecting H₀ when H₁ is true, calculated under H₁. For binomial distributions, both are found by summing probabilities from the relevant distribution.
检验的显著性水平是犯第一类错误的概率，在 H₀ 下计算。检验的功效是 1 − β，即在 H₁ 为真时正确拒绝 H₀ 的概率，在 H₁ 下计算。对于二项分布，两者都通过对相应分布的概率求和得到。
To calculate the significance level, identify the critical region and find P(reject H₀ | H₀). To calculate power, use the same critical region but find P(reject H₀ | H₁). Remember that power depends on the specific value of the parameter under H₁.
计算显著性水平时，确定临界区域并求 P(拒绝 H₀ | H₀)。计算功效时，使用相同临界区域但求 P(拒绝 H₀ | H₁)。记住功效依赖于 H₁ 下参数的具体值。
Mastering these calculations is essential for Edexcel A-Level Further Mathematics Paper 3 (Further Statistics). Practice with a range of binomial and normal examples to build confidence.
掌握这些计算对 Edexcel A-Level 进阶数学 Paper 3（进阶统计）至关重要。通过练习各种二项和正态的例子来建立信心。
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