📚 Hypothesis Testing: Significance Testing for Zero Correlation | 假设检验：零相关的显著性检验方法
When we collect data from a sample, the sample correlation coefficient r summarises the strength and direction of a linear relationship between two quantitative variables. A nonzero r might reflect a real association, or it might occur purely by chance because of sampling variation. Hypothesis testing for zero correlation asks a focused question: if the population correlation coefficient ρ is actually 0, how likely would we be to observe a sample correlation as extreme as ours?
当我们从样本中收集数据时，样本相关系数 r 用于概括两个定量变量之间线性关系的强度和方向。非零的 r 可能反映真实的关联，也可能只是抽样波动造成的偶然结果。零相关的显著性检验提出了一个明确的问题：如果总体相关系数 ρ 实际上为 0，我们观察到像当前样本这样极端的样本相关系数的可能性有多大？
1. Why Test Zero Correlation? | 为什么要检验零相关？
ether two variables are associated. For example, does study time correlate with exam score? Does temperature correlate with ice-cream sales? decide whether the evidence against ρ = 0 is strong enough to reject the null hypothesis.
在许多实际研究中，我们想知道两个变量之间是否存在关联。例如，学习时间是否与考试成绩相关？气温是否与冰淇淋销量相关？如果仅仅观察 r，我们仍然无法判断结果是否有意义，因为每个样本都存在一定的随机波动。假设检验提供了一套正式的决策规则：它帮助我们判断反对 ρ = 0 的证据是否足够强，从而决定是否拒绝原假设。
2. Null and Alternative Hypotheses | 原假设与备择假设
The population correlation coefficient is denoted by ρ (the Greek letter rho). The null hypothesis always claims that there is no linear relationship in the population. The alternative hypothesis claims that there is a linear relationship, and it may be two-tailed or one-tailed depending on the research question.
总体相关系数用 ρ 表示。原假设始终声称总体中不存在线性关系。备择假设则声称存在线性关系，根据研究问题可以是非方向性的（双尾）或方向性的（单尾）。
H₀: ρ = 0 — the null hypothesis states that the population correlation is zero.
H₀: ρ = 0 — 原假设表明总体相关系数为零。
H₁: ρ ≠ 0 — a two-sided alternative: there is some linear correlation, either positive or negative.
H₁: ρ ≠ 0 — 双侧备择假设：存在某种线性相关，无论是正相关还是负相关。
H₁: ρ > 0 — a one-sided alternative: there is a positive linear correlation.
H₁: ρ > 0 — 单侧备择假设：存在正线性相关。
H₁: ρ < 0 — a one-sided alternative: there is a negative linear correlation.
H₁: ρ < 0 — 单侧备择假设：存在负线性相关。
In most introductory courses, the two-sided test H₁: ρ ≠ 0 is used unless the research question clearly predicts a direction.
在大多数基础课程中，除非研究问题明确预测了方向，否则通常使用双侧检验 H₁: ρ ≠ 0。
3. Conditions for the Test | 检验的前提条件
Before applying the standard Pearson correlation significance test, we need to check that the data satisfy several important conditions. If these conditions are badly violated, the test statistic may not have the assumed distribution, and the conclusion could be misleading.
在应用标准 Pearson 相关系数显著性检验之前，我们需要确认数据满足若干重要条件。如果这些条件被严重违反，检验统计量可能不具有假设的分布，结论也可能产生误导。
Random sampling: the observations should be a random sample from the population of interest.
随机抽样：观测值应来自目标总体的随机样本。
Bivariate normality: ideally the pair (X, Y) follows a bivariate normal distribution, or each variable is approximately normal.
二元正态性：理想情况下，数据对 (X, Y) 服从二元正态分布，或者每个变量近似正态。
Independent observations: each pair of measurements should be independent of the others.
观测独立性：每一对测量值应与其他测量值相互独立。
Linear relationship: the Pearson correlation only measures linear association, so a roughly straight-line pattern in the scatterplot is expected.
线性关系：Pearson 相关只衡量线性关联，因此散点图应大致呈现直线模式。
No strong outliers: extreme outliers can inflate or deflate r and change the conclusion.
无强异常值：极端异常值可能夸大或缩小 r，从而改变检验结论。
4. The Test Statistic | 检验统计量
ion coefficient r can be transformed into a t statistic. This t statistic follows a t-distribution with n − 2 degrees of freedom, where n is the sample size.
当条件满足且 H₀: ρ = 0 成立时，样本相关系数 r 可以转化为 t 统计量。该 t 统计量服从自由度为 n − 2 的 t 分布，其中 n 是样本容量。
t = r√n − 2 / √1 − r²
Here r is the sample correlation coefficient, n is the number of data pairs, and the degrees of freedom are:
其中 r 是样本相关系数，n 是数据对数量，自由度为：
df = n − 2
Notice that if r = 0, the test statistic is 0. As |r| ks and the absolute value of t becomes very large, providing stronger evidence against H₀.
注意，如果 r = 0，则检验统计量为 0。当 |r| 越接近 1 时，分母变小，|t| 变得很大，从而提供更强的拒绝 H₀ 的证据。
5. One-Tailed and Two-Tailed Tests | 单尾与双尾检验
The choice between a one-tailed and a two-tailed test affects the critical value and the p-value. In a two-tailed test, we reject H₀ when the observed t is sufficiently far from 0 in either direction. In a one-tailed test, we only reject H₀ when the t value falls in the predicted direction.
选择单尾检验还是双尾检验会影响临界值与 p 值。在双尾检验中，当观测到的 t 值在任一方向上距离 0 足够远时，我们拒绝 H₀。在单尾检验中，只有当 t 值落入预测的方向时，我们才拒绝 H₀。
For H₁: ρ ≠ 0, use a two-tailed test. The significance level α is split between the two tails.
对于 H₁: ρ ≠ 0，使用双尾检验。显著性水平 α 被分配到两个尾端。
For H₁: ρ > 0, use an upper-tailed test. Only large positive t values provide evidence against H₀.
对于 H₁: ρ > 0，使用上尾检验。只有大的正 t 值才提供反对 H₀ 的证据。
For H₁: ρ < 0, use a lower-tailed test. Only large negative t values provide evidence against H₀.
对于 H₁: ρ < 0，使用下尾检验。只有大的负 t 值才提供反对 H₀ 的证据。
6. Decision Rules: Critical Values and p-Values | 决策规则：临界值与 p 值
2 and the chosen significance level α. Alternatively, we can compute a p-value and compare it with α.
计算出 t 统计量后，我们需要将其与 t 分布表中的临界值进行比较，使用自由度 df = n − 2 以及选定的显著性水平 α。另一种方法是计算 p 值，并将其与 α 比较。
If |t| > t* (the critical value), reject H₀.
如果 |t| > t*（临界值），则拒绝 H₀。
If p-value < α, reject H₀.
如果 p 值 < α，则拒绝 H₀。
ail to reject H₀. We conclude that there is insufficient evidence of a linear correlation.
如果 p 值 ≥ α，则不能拒绝 H₀。我们得出结论：没有足够证据表明存在线性相关。
tself. For H₀: ρ = 0, the critical value of r can be found from t* using the relationship:
有些考试局会直接提供相关系数的临界值表。对于 H₀: ρ = 0，相关系数的临界值可以通过 t* 用以下关系求得：
r* = t* / √t*² + n − 2
The table below shows selected t critical values for two-tailed tests.
下表给出了双尾检验中部分 t 临界值。
| Degrees of freedom | α = 0.05 | α = 0.01 |
| 5 | 2.571 | 4.032 |
| 8 | 2.306 | 3.355 |
| 10 | 2.228 | 3.169 | Published by TutorHao | Mathematics Revision Series | aleveler.com
