📚 A-Level Maths: Hypothesis Testing for Zero Correlation | A-Level 数学：零相关系数的假设检验
In A-Level Mathematics and Further Mathematics, the product moment correlation coefficient r is used to summarise how strongly two variables are linearly related in a sample. But a sample value of r is only an estimate: even when two variables are completely unrelated in the population, a random sample will almost never give exactly r = 0. Hypothesis testing for zero correlation turns that raw sample value into a defensible statistical decision about the population correlation coefficient ρ.
在 A-Level 数学与进阶数学中，积矩相关系数 r 用来概括样本中两个变量之间线性关系的强弱。但样本中的 r 只是一个估计值：即使总体中两个变量毫无关系，随机抽样也几乎不会得到恰好 r = 0。零相关系数的假设检验，正是要把这个样本值转化为关于总体相关系数 ρ 的、有依据的统计判断。
1. Why Test for Zero Correlation? | 为什么要检验零相关
If you measure the height of ten random students and the price of a cup of coffee in ten random cities, you will still get some value of r, perhaps 0.3. That does not mean a real relationship exists. Sampling variation alone can produce noticeable correlation by chance, especially when the sample is small. The hypothesis test asks a single question: is the observed correlation large enough, relative to the size of the sample, to be unlikely to have arisen from pure chance?
如果你随机测量十个学生的身高、以及十个随机城市中一杯咖啡的价格，你仍然会得到一个 r 值，也许是 0.3。这并不意味着真的存在关系。仅凭抽样波动，尤其是样本量较小时，就足以偶然产生可观的相关系数。假设检验要回答的核心问题是：观察到的相关程度相对于样本量是否足够大，以致于不太可能纯属偶然？
The test has three practical purposes:
这个检验有三个实际目的：
- To decide whether an apparent linear relationship is statistically significant. | 判断表面上的线性关系是否具有统计显著性。
- To control the risk of claiming a relationship that does not exist (a Type I error). | 控制错误宣称存在关系的风险（第一类错误）。
- To replace vague words such as ‘strong’ or ‘weak’ with a clear, examinable decision rule. | 用清晰、可评分的决策规则取代 ‘强’ 或 ‘弱’ 这类模糊措辞。
2. The Product Moment Correlation Coefficient (PMCC) | 积矩相关系数（PMCC）
The test statistic is Pearson’s product moment correlation coefficient, usually calculated from summary statistics rather than from scratch. In the A-Level formula booklet it appears as follows:
检验统计量是皮尔逊积矩相关系数，通常由汇总统计量计算而非逐项计算。在 A-Level 公式表中它写作：
r = Sₓᵧ ÷ √(Sₓₓ × Sᵧᵧ)
where the three summary quantities are:
其中三个汇总量为：
Sₓᵧ = Σxᵢyᵢ − (Σxᵢ)(Σyᵢ) ÷ n
Sₓₓ = Σxᵢ² − (Σxᵢ)² ÷ n
Sᵧᵧ = Σyᵢ² − (Σyᵢ)² ÷ n
Key properties you must know: r always lies between −1 and 1, so −1 ≤ r ≤ 1; r measures only linear association, so a perfect curve can give r close to 0; and r is unchanged by any linear coding of the form x → a + bx with b > 0 (the sign of r flips if b < 0).
必须掌握的关键性质：r 的取值范围是 −1 到 1，即 −1 ≤ r ≤ 1；r 只衡量线性关联，因此一条完美的曲线也可能给出接近 0 的 r；对于形如 x → a + bx 的线性编码，当 b > 0 时 r 不变（若 b < 0 则 r 的符号反转）。
3. The Sampling Distribution Idea Behind the Test | 检验背后的抽样分布思想
The logic is the same as any other hypothesis test in A-Level: assume the null hypothesis is true, then ask how unusual the observed statistic would be. If the population correlation coefficient ρ is really 0, then r is a random variable centred on 0 with a spread that depends on n. Roughly speaking, the standard error behaves like 1 ÷ √(n − 1), so larger samples give values of r that cluster more tightly around 0.
其逻辑与 A-Level 中其他假设检验完全相同：先假定原假设成立，再看观察到的统计量有多不寻常。若总体相关系数 ρ 真的为 0，则 r 是一个以 0 为中心的随机变量，其离散程度取决于 n。粗略地说，标准误的行为类似 1 ÷ √(n − 1)，因此样本越大，r 的取值就越集中在 0 附近。
This explains the central exam fact: the critical value gets smaller as n gets larger. A correlation of r = 0.6 is significant for n = 12 but not significant for n = 5, because with only five pairs of data such a value is entirely plausible by chance.
这就解释了考试中的核心事实：n 越大，临界值越小。r = 0.6 在 n = 12 时显著，但在 n = 5 时不显著，因为只有五对数据时，这样的数值完全可能是偶然产生的。
4. Setting Up the Hypotheses | 建立原假设与备择假设
Always write the hypotheses in both words and symbols, and always use ρ (rho) for the population and r for the sample. Mixing these up loses marks. The null hypothesis is virtually always ‘no linear correlation in the population’, written H₀: ρ = 0. The alternative hypothesis depends on the direction of the claim being tested.
假设必须同时用文字和符号写出，并且总体用 ρ（rho）、样本用 r，混淆二者会丢分。原假设几乎总是 ‘总体中不存在线性相关’，写作 H₀: ρ = 0。备择假设则取决于被检验的主张是否有方向性。
| Type of test | Alternative hypothesis | English wording | 中文表述 |
| Two-tailed | H₁: ρ ≠ 0 | There is some correlation | 存在某种相关 |
| One-tailed (positive) | H₁: ρ > 0 | There is positive correlation | 存在正相关 |
| One-tailed (negative) | H₁: ρ < 0 | There is negative correlation | 存在负相关 |
Use a one-tailed test only when the question itself states a direction, for example ‘the researcher believes that more revision leads to higher marks’. Words such as ‘leads to’, ‘increases’ or ‘decreases’ signal a one-tailed test; words such as ‘is related to’ or ‘is associated with’ signal a two-tailed test.
只有当题目本身指明方向时才使用单尾检验，例如 ‘研究者认为复习时间越长分数越高’。’导致’、’提高’、’降低’ 等词暗示单尾检验；’与……有关’、’与……相关联’ 等词则暗示双尾检验。
5. Choosing the Significance Level and Tail Type | 选择显著性水平与单双尾
The significance level is normally given in the question, most often 5%, sometimes 1% or 10%. It is the probability of rejecting H₀ when it is actually true. You must never choose the level after looking at your value of r, because that is data snooping and invalidates the test. If the question gives a choice, choose a sensible level in advance and state it explicitly.
显著性水平通常由题目给定，最常见的是 5%，有时为 1% 或 10%。它是在 H₀ 实际为真时错误拒绝 H₀ 的概率。绝不能先算出 r 再回头选择水平，因为那属于数据操纵，会使检验失效。若题目允许选择，应事先选定合理的水平并明确写出。
Choosing a 1% level makes it harder to reject H₀ (the critical value is larger), so it protects against false positive claims. Choosing a 10% level makes rejection easier but increases the risk of a Type I error. In an exam you simply follow the question, but you should be able to explain this trade-off in a written answer.
选择 1% 水平会使拒绝 H₀ 更困难（临界值更大），从而降低误报风险；选择 10% 水平更容易拒绝 H₀，但第一类错误的风险更高。考试中一般直接按题目要求操作，但你应能在文字题中解释这一权衡关系。
6. Reading the Critical Values Table | 读懂临界值表
Statistical tables for the PMCC are indexed by n, the number of pairs of data, not by degrees of freedom. This is the single most common error in this topic. Find your value of n in the left-hand column, then read across to the column for your significance level and tail type.
PMCC 的统计临界值表以 n（数据对的个数）为索引，而不是自由度。这是本专题中最常见的错误。先在左列找到你的 n，再横读到对应的显著性水平与尾型所在列。
| n | Two-tailed 5% | Two-tailed 1% | One-tailed 5% |
| 8 | 0.7067 | 0.8343 | 0.6215 |
| 10 | 0.6319 | 0.7646 | 0.5494 |
| 12 | 0.5760 | 0.7079 | 0.4973 |
| 15 | 0.5140 | 0.6411 | 0.4413 |
| 20 | 0.4438 | 0.5614 |
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