📚 Hypothesis Testing for a Zero Correlation Coefficient | 相关系数为零的假设检验
In A-Level Further Mathematics, we often need to decide whether an observed linear relationship in a sample is strong enough to prove that a real relationship exists in the population. A hypothesis test for a zero correlation coefficient uses the sample product moment correlation coefficient r to test whether the population correlation coefficient ρ is zero.
在进阶数学中，我们常常需要判断样本中观察到的线性关系是否强到足以证明总体中真实存在联系。相关系数为零的假设检验，就是利用样本积矩相关系数 r 来检验总体相关系数 ρ 是否为零。
1. The Population and Sample Correlation Coefficients | 总体与样本相关系数
The Greek letter ρ represents the population product moment correlation coefficient. It is a fixed unknown number between −1 and 1. The sample coefficient r is an estimate of ρ, calculated from n paired observations (x₁, y₁), (x₂, y₂), ..., (xₙ, yₙ).
希腊字母 ρ 表示总体积矩相关系数，它是一个固定的未知数，取值范围在 −1 到 1 之间。样本相关系数 r 是 ρ 的估计值，由 n 组成对数据 (x₁,y₁), (x₂,y₂),…,(xₙ,yₙ) 计算得到。
r = Sxy / √SxxSyy = Σ(xᵢ − x̄)(yᵢ − ȳ) / √[Σ(xᵢ − x̄)² Σ(yᵢ − ȳ)²]
A value of r close to 1 suggests a strong positive linear relationship; a value close to −1 suggests a strong negative linear relationship; a value close to 0 suggests little or no linear relationship.
当 r 接近 1 时，说明存在很强的正线性关系；当 r 接近 −1 时，说明存在很强的负线性关系；当 r 接近 0 时，说明几乎没有线性关系。
2. Setting Up the Hypotheses | 建立原假设与备择假设
The null hypothesis always states that the population correlation coefficient is zero. The alternative hypothesis depends on the direction of the claim being tested.
原假设始终说明总体相关系数为零。备择假设则取决于题目要检验的方向。
H₀: ρ = 0, H₁: ρ > 0 (one-tailed positive test)
H₀: ρ = 0, H₁: ρ < 0 (one-tailed negative test)
H₀: ρ = 0, H₁: ρ ≠ 0 (two-tailed test)
Use a one-tailed test when the question predicts a positive or a negative correlation. Use a two-tailed test when the question only asks whether there is any linear correlation.
如果题目预测正相关或负相关，就使用单尾检验。如果题目只问是否存在线性相关，就使用双尾检验。
3. Conditions for the Test | 检验的前提条件
The Pearson product moment correlation coefficient test is valid only when certain assumptions are satisfied. These conditions are important in exam questions, especially when data are collected from a sample.
皮尔逊积矩相关系数检验只有在满足特定前提条件时才是有效的。处理从样本中收集的数据时，这些条件尤其重要。
First, the n paired observations must be a random sample of independent pairs from the population. Second, for the Pearson test, the data are assumed to come from a bivariate normal distribution, or at least that the underlying relationship is linear and there are no extreme outliers.
第一，n 组成对数据必须是来自总体的随机样本，并且各对数据相互独立。第二，对于皮尔逊检验，通常假设数据来自二元正态分布，或者至少总体关系是线性的，且没有极端异常值。
If the normality assumption is doubtful, or if the data are given in ranks, Spearman's rank correlation coefficient can be used instead.
如果正态性假设受到怀疑，或者数据以排名形式给出，可以改用斯皮尔曼秩
Published by TutorHao | A-Level 进阶数学 Revision Series | aleveler.com
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