📚 The Principle of Hypothesis Testing | 假设检验的基本原理
Hypothesis testing is a fundamental procedure in inferential statistics that allows us to assess claims about a population parameter by examining evidence provided by a sample. It formalises the process of deciding whether an observed effect is genuinely present or simply due to random chance.
假设检验是推断统计中的一个基本方法，它使我们能够通过样本提供的证据来评判关于总体参数的主张。它将判断观察到的效应是真实存在还是仅由随机因素引起的过程形式化。
1. Introduction to Hypothesis Testing | 假设检验引言
In any scientific investigation, we often wish to test a statement about a population, such as "the mean lifetime of a new battery is 500 hours" or "a new teaching method improves test scores". Hypothesis testing provides a structured framework to evaluate such claims using sample data, helping us avoid drawing conclusions based on insufficient evidence.
在任何科学调查中，我们常常希望检验某个关于总体的说法，例如“一种新电池的平均寿命为500小时”或“一种新教学方法能提高考试成绩”。假设检验提供了一个结构化的框架，利用样本数据来评估这些主张，帮助我们避免依据不充分证据得出结论。
The logic resembles a court trial: the null hypothesis represents "innocence" (no effect), and we require strong evidence to reject it in favour of the alternative hypothesis. The process involves measures of uncertainty, such as the significance level and the p-value, to control the risk of making an incorrect decision.
其逻辑类似于法庭审判：零假设代表“无罪”（即无效应），我们需要强有力的证据才能拒绝零假设而支持备择假设。这个过程中涉及不确定性度量，如显著性水平和p值，以控制做出错误决策的风险。
2. Null and Alternative Hypotheses | 零假设与备择假设
meter equals a specific value. It is always expressed with an equality (e.g., H₀: μ = μ₀).
零假设，记作 H₀，是声称没有效应、没有差异或总体参数等于某个特定值的陈述。它总是用等式表达（例如 H₀: μ = μ₀）。
The alternative hypothesis, denoted H₁ or Hₐ, is the statement we wish to find evidence for. It contradicts the null hypothesis and can take one of three forms: two-tailed (μ ≠ μ₀), upper-tailed (μ > μ₀), or lower-tailed (μ < μ₀).
备择假设，记作 H₁ 或 Hₐ，是我们希望找到证据支持的陈述。它与零假设相对，可以有三种形式：双尾（μ ≠ μ₀）、上尾（μ > μ₀）或下尾（μ < μ₀）。
or wishing to detect any deviation would use H₁: μ ≠ 10. If the concern is only that bolts are too thin, the alternative would be H₁: μ < 10.
例如，如果制造商声称螺栓的平均直径为10 mm，我们设 H₀: μ = 10。质量检验员若要检测任何偏差，会使用 H₁: μ ≠ 10。若只关心螺栓过细，备择假设则为 H₁: μ < 10。
3. Significance Level and Critical Region | 显著性水平与拒绝域
lues are α = 0.05, 0.01, or 0.10. In IB examinations, α is usually stated in the question.
显著性水平，用 α 表示，是当零假设实际为真时拒绝零假设的概率。它由研究人员在收集数据之前选定；常见值为 α = 0.05、0.01 或 0.10。在 IB 考试中，α 通常会在题目中给出。
= −1.96 and z = 1.96: if the test statistic falls below −1.96 or above 1.96, we reject H₀.
拒绝域（临界区域）由那些导致拒绝 H₀ 的检验统计量的值组成。边界值称为临界值。对于 α = 0.05 的双尾 z 检验，临界值为 z = −1.96 和 z = 1.96：如果检验统计量低于 −1.96 或高于 1.96，我们就拒绝 H₀。
In a diagram of the sampling distribution, the critical region corresponds to the tail(s) area that sums to α. The non-rejection region is the complement, representing the range of test statistic values for which the sample evidence is not strong enough to reject H₀.
在抽样分布的图示中，拒绝域对应于总和为 α 的尾部面积。不拒绝域是其补集，代表样本证据尚不足以拒绝 H₀ 的检验统计量取值范围。
4. Test Statistic and Its Distribution | 检验统计量及其分布
A test statistic is a summary measure calculated from sample data, used to decide whether to reject H₀. It measures how far the sample statistic deviates from the hypothesised parameter, scaled by the standard error.
检验统计量是由样本数据计算出的汇总度量，用于决定是否拒绝 H₀。它度量了样本统计量与假设的参数值之间的偏差有多大，并以标准误作为尺度。
For a test about a population mean μ with known population standard deviation σ, the test statistic Z follows a standard normal distribution under H₀:
Z = (x̄ − μ₀) / (σ / √n)
对于一个已知总体标准差 σ 的总体均值 μ 的检验，在 H₀ 下检验统计量 Z 服从标准正态分布：
Z = (x̄ − μ₀) / (σ / √n)
When σ is unknown, the sample standard deviation s is used, and the test statistic becomes a t-statistic with n − 1 degrees of freedom, provided the population is approximately normal:
t = (x̄ − μ₀) / (s / √n) , df = n − 1
当 σ 未知时，使用样本标准差 s，检验统计量成为自由度为 n − 1 的 t 统计量，且要求总体近似正态：
t = (x̄ − μ₀) / (s / √n) , 自由度 = n − 1
In IB Mathematics, both z-tests and t-tests appear depending on the context; the choice hinges on whether σ is known and whether the sample size is large enough for the Central Limit Theorem to apply.
在 IB 数学中，z 检验和 t 检验会根据上下文出现；选择取决于 σ 是否已知以及样本量是否足够大以应用中心极限定理。
5. P-value: Definition and Interpretation | P值：定义与解释
The p-value is the probability, assuming H₀ is true, of obtaining a test statistic as extreme as, or more extreme than, the one observed from the sample. It quantifies the strength of evidence against H₀: the smaller the p-value, the stronger the evidence.
P值是假定 H₀ 为真时，获得与样本观测值一样极端或更极端检验统计量的概率。它量化了反对 H₀ 的证据强度：p值越小，证据越强。
lue > α, do not reject H₀. For a two-tailed z-test, the p-value is calculated as 2 × P(Z > |z|); for an upper-tailed test it is P(Z > z), and for a lower-tailed test it is P(Z < z).
使用p值的决策规则是：如果 p值 ≤ α，拒绝 H₀；如果 p值 > α，不拒绝 H₀。对于双尾 z 检验，p值计算公式为 2 × P(Z > |z|)；对于上尾检验为 P(Z > z)，对于下尾检验为 P(Z < z)。
A common misinterpretation is that the p-value is the probability that H₀ is true. This is incorrect; it is the probability of the data given H₀, not the reverse. Always remember that a high p-value does not prove H₀, it merely indicates insufficient evidence to reject it.
一个常见的误解是认为 p值 是 H₀ 为真的概率。这是错误的；它是给定 H₀ 下数据出现的概率，而非其逆。务必记住，较高的 p值 并不能证明 H₀，它仅仅表示缺乏足够的证据来拒绝 H₀。
6. One-tailed versus Two-tailed Tests | 单尾检验与双尾检验
The choice between a one-tailed and a two-tailed test is guided by the research question. A two-tailed test is used when the interest lies in any deviation from the hypothesised value (H₁: μ ≠ μ₀). A one-tailed test is used when the deviation is specified in one direction only (H₁: μ > μ₀ or H₁: μ < μ₀).
选择单尾检验还是双尾检验取决于研究问题。当关注点在于偏离假设值的任何偏差时，使用双尾检验（H₁: μ ≠ μ₀）。当偏差仅指定一个方向时，使用单尾检验（H₁: μ > μ₀ 或 H₁: μ < μ₀）。
The table below summarises the rejection region and p-value computation for each type. One-tailed tests place the entire α in one tail, giving a higher chance of detecting an effect in that direction, but they cannot detect an effect in the opposite direction.
下表总结了每种类型的拒绝域和p值计算方式。单尾检验将全部 α 置于一侧尾部，从而有更大机会检测该方向的效应，但无法检测相反方向的效应。
| Test Type / 检验类型 | H₁ / 备择假设 | Rejection Region / 拒绝域 | P-value / P值 |
|---|---|---|---|
| Two-tailed / 双尾 | μ ≠ μ₀ | Both tails / 两侧尾部 | 2 × P(Z > |z|) / 2 × P(Z > |z|) |
| Upper-tailed / 上尾 | μ > μ₀ | Right tail / 右侧尾部 | P(Z > z) / P(Z > z) |
| Lower-tailed / 下尾 | μ < μ₀ | Left tail / 左侧尾部 | P(Z < z) / P(Z < z) |
In IB problems, it is crucial to decide the nature of the test before reading the data; the hypotheses must be stated before the analysis is performed. Switching to a one-tailed test after seeing the data inflates the Type I error rate.
在 IB 问题中，务必在查看数据之前确定检验的性质；假设必须在分析之前陈述。看到数据后再改为单尾检验会增大第一类错误率。
7. Rejection Rules and Conclusions | 拒绝规则与结论
There are two equivalent approaches to reaching a conclusion: the critical value method and the p-value method. Both lead to the same decision for a given α. The conclusion must be stated clearly in the context of the problem.
有两种等价的得出结论的方法：临界值法和 p值法。对于给定的 α，两者会得到相同的决策。结论必须在问题情境下清晰陈述。
ect H₀ and state that there is sufficient evidence at the α level to support the alternative hypothesis. For example, " . There is sufficient evidence to conclude that the mean lifetime differs from 1000 hours."
如果检验统计量落入拒绝域，或者 p值 ≤ α，我们拒绝 H₀，并声明在 α 水平上有足够证据支持备择假设。例如，“我们拒绝 H₀。有足够证据推断平均寿命不等于1000小时。”
If the test statistic does not fall in the critical region, or if p-value > α, we do not reject H₀. We say that the evidence is insufficient to support H₁. A common phrasing is: "We do not reject H₀. There is insufficient evidence to suggest that the mean lifetime is different from 1000 hours." It is incorrect to say "we accept H₀" because non-rejection does not prove H₀ is true.
如果检验统计量未落入拒绝域，或者 p值 > α，我们不拒绝 H₀。我们说证据不足以支持 H₁。常见的表述是：“我们不拒绝 H₀。没有足够证据表明平均寿命不同于1000小时。” 说“我们接受 H₀”是不正确的，因为不拒绝并不能证明 H₀ 为真。
8. Type I and Type II Errors | 第一类错误与第二类错误
ting a Type I error is exactly the significance level α. If α = 0.05, we are accepting a 5% risk of rejecting H₀ when it is actually correct.
第一类错误发生在我们拒绝了真实的零假设时。犯第一类错误的概率恰好是显著性水平 α。如果 α = 0.05，我们接受了当 H₀ 实际正确时拒绝它的 5% 风险。
A Type II error occurs when we fail to reject a false null hypothesis. The probability of a Type II error is denoted by β (beta). The value of β depends on the true value of the parameter, the sample size, and the chosen α.
第二类错误发生在我们未能拒绝错误的零假设时。第二类错误的概率用 β 表示。β 的值取决于参数的真实值、样本量和所选的 α。
The relationship between the two errors involves a trade-off:
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