Regression, Correlation and Hypothesis Testing | 回归、相关与假设检验

📚 Regression, Correlation and Hypothesis Testing | 回归、相关与假设检验

In Edexcel A-Level Mathematics, the topic of regression, correlation and hypothesis testing brings together bivariate data analysis, the product moment correlation coefficient (PMCC), least squares regression lines and formal hypothesis tests for correlation. This article covers the key concepts, formulas, interpretations and exam techniques you need to answer questions confidently.

在 Edexcel A-Level 数学中，回归、相关与假设检验这一主题将双变量数据分析、积矩相关系数（PMCC）、最小二乘回归线以及相关性的正式假设检验结合在一起。本文涵盖关键概念、公式、解释和考试技巧，帮助你自信地解答相关题目。

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1. Scatter Diagrams and Types of Correlation | 散点图与相关类型

A scatter diagram is used to display bivariate data, where each point represents a pair of values (x, y). Correlation describes the strength and direction of a linear relationship between the two variables. Positive correlation means that as x increases, y tends to increase. Negative correlation means that as x increases, y tends to decrease. Zero correlation means there is no linear pattern, although a non-linear relationship may still exist.

散点图用于展示双变量数据，每个点代表一对数值 (x, y)。相关描述两个变量之间线性关系的强度和方向。正相关意味着当 x 增大时，y 通常也增大。负相关意味着当 x 增大时，y 通常减小。零相关意味着没有线性模式，但仍可能存在非线性关系。

• Positive correlation: points rise from left to right.
• Negative correlation: points fall from left to right.
• Strong correlation: points lie close to a straight line.
• Weak correlation: points are widely scattered around a line.

正相关：点从左到右上升。负相关：点从左到右下降。强相关：点紧靠一条直线。弱相关：点围绕一条直线分散较宽。

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2. The Product Moment Correlation Coefficient (PMCC) | 积矩相关系数

ndicates a perfect negative linear correlation, and r = 0 indicates no linear correlation.

积矩相关系数通常记作 r，是线性相关的数值度量。它的取值范围始终在 -1 到 +1 之间。r = +1 表示完全正线性相关，r = -1 表示完全负线性相关，r = 0 表示没有线性相关。

r = Sxy ÷ √Sxx × Syy

Here Sxx, Syy and Sxy are summary statistics calculated from the data:

这里的 Sxx、Syy 和 Sxy 是由数据计算得到的汇总统计量：

Sxx = Σx² − (Σx)² ÷ n

Syy = Σy² − (Σy)² ÷ n

Sxy = Σxy − (Σx)(Σy) ÷ n

These quantities measure the total variation in x, the total variation in y, and the joint variation between x and y. Using them gives a standardised value of r that is not affected by the units of measurement.

这些量分别度量 x 的总变异、y 的总变异以及 x 与 y 的联合变异。使用它们可以得到标准化的 r 值，该值不受测量单位的影响。

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3. Calculating PMCC from Data | 从数据计算 PMCC

To calculate r from a set of bivariate data, first complete a table with columns for x, y, x², y² and xy. Then find the totals Σx, Σy, Σx², Σy² and Σxy. Substitute these totals into the formulas for Sxx, Syy and Sxy, and finally evaluate r.

要从一组双变量数据计算 r，首先制作一个包含 x、y、x²、y² 和 xy 列的表格。然后求出总和 Σx、Σy、Σx²、Σy² 和 Σxy。将这些总和代入 Sxx、Syy 和 Sxy 的公式，最后计算 r。

x	y	x²	y²	xy
...	...	...	...	...

Always check your arithmetic carefully, because a single error in Σxy or Σx² can change the value of r significantly. In the exam, you may be given some of these totals, so write down the formulas before substituting.

务必仔细检查计算，因为 Σxy 或 Σx² 中的一个错误就可能显著改变 r 的值。在考试中，你可能会被给出其中一些总和，因此代入前先写出公式。

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4. Interpreting PMCC and Its Limitations | 解释 PMCC 及其局限性

The value of r tells you the direction and strength of a linear relationship. Values close to +1 or -1 indicate a strong linear relationship, while values close to 0 indicate a weak linear relationship. However, a high value of r does not prove that one variable causes the other to change. Correlation does not imply causation.

r 的值告诉你线性关系的方向和强度。接近 +1 或 -1 的值表示强线性关系，而接近 0 的值表示弱线性关系。然而，较高的 r 值并不能证明一个变量的变化是由另一个变量引起的。相关并不意味因果。

• r close to 0 may hide a strong non-linear relationship, such as a quadratic or exponential curve.
• Outliers can inflate or deflate the value of r.
• Restricting the range of x or y can weaken the apparent correlation.
• A significant correlation does not prove causation.

接近 0 的 r 可能隐藏了强非线性关系，例如二次曲线或指数曲线。异常值可能夸大或缩小 r 的值。限制 x 或 y 的范围可能削弱表观相关性。显著的相关性并不能证明因果关系。

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5. Regression Lines: Least Squares Method | 回归线：最小二乘法

, where b is the gradient and a is the y-intercept. The least squares estimates are given by:

y 对 x 的回归线是根据最小化数据点到直线的垂直距离平方和这一原则得到的最佳拟合直线。其方程为 y = a + bx，其中 b 是斜率，a 是 y 轴截距。最小二乘估计由以下公式给出：

b = Sxy ÷ Sxx

a = ȳ − b x̄

Here x̄ is the mean of x and ȳ is the mean of y. The regression line always passes through the point (x̄, ȳ). This gives a useful check: after finding a and b, substitute x = x̄ and you should get y = ȳ.

这里 x̄ 是 x 的平均值，ȳ 是 y 的平均值。回归线总是通过点 (x̄, ȳ)。这提供了一个有用的检查：在求出 a 和 b 后，代入 x = x̄，你应得到 y = ȳ。

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6. Interpreting Regression Coefficients | 解释回归系数

x = 0, but this is only meaningful if x = 0 is within or close to the observed range of x.

斜率 b 表示 x 每增加一个单位时 y 的估计变化量。截距 a 表示当 x = 0 时 y 的估计值，但只有当 x = 0 位于或接近观测到的 x 范围内时，这个解释才有实际意义。

is age in years, then the gradient tells us that height increases by approximately 1.8 cm per year. The intercept of 3.2 cm would not be meaningful for age 0 unless the data set includes very young children.

例如，如果回归线为 y = 3.2 + 1.8x，其中 y 是身高（cm），x 是年龄（年），那么斜率告诉我们身高每年大约增加 1.8 cm。截距 3.2 cm 在年龄为 0 时没有实际意义，除非数据集包括非常小的儿童。

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7. Making Predictions and Extrapolation Risks | 做出预测与外推风险

Once a regression line has been found, it can be used to predict y for a given value of x. Predictions made within the range of the original data are called interpolations and are usually reliable, provided the linear model is appropriate. Predictions made outside the range of the original data are called extrapolations and can be unreliable because the linear relationship may not continue beyond the observed range.

一旦求出回归线，就可以用它来预测给定 x 值时 y 的值。在原始数据范围内做出的预测称为内插，通常在模型合适时较为可靠。在原始数据范围之外做出的预测称为外推，可能不可靠，因为线性关系在观测范围之外可能不再成立。

• Interpolation: predict within the observed x-range.
• Extrapolation: predict outside the observed x-range.
• Extrapolation is dangerous because the trend may change.

内插：在观测到的 x 范围内进行预测。外推：在观测到的 x 范围之外进行预测。外推具有风险，因为趋势可能发生变化。

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8. Hypothesis Testing for Correlation | 相关性的假设检验

s can be two-tailed, H1: ρ ≠ 0, or one-tailed, H1: ρ > 0 or H1: ρ < 0, depending on the wording of the question.

你可以对总体相关系数（通常记作 ρ）进行假设检验。零假设是总体中没有线性相关：H0: ρ = 0。备择假设可以是双尾的，H1: ρ ≠ 0，也可以是单尾的，H1: ρ > 0 或 H1: ρ < 0，具体取决于题目的表述。

The test statistic is the sample product moment correlation coefficient r. You compare the absolute value of r with a critical value obtained from the PMCC critical value table, using the sample size n and the chosen significance level. If |r| is greater than the critical value, you reject H0 and conclude that there is evidence of correlation.

检验统计量是样本积矩相关系数 r。你将 r 的绝对值与从 PMCC 临界值表中查得的临界值进行比较，该表使用样本容量 n 和选定的显著性水平。如果 |r| 大于临界值，则拒绝 H0，并得出存在相关性的结论。

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9. Critical Values and Conclusions | 临界值与结论

For a two-tailed test at the 5% ize n = 12, the critical value is approximately 0.576. This means that if the sample r has |r| > 0.576, the result is significant at the 5% level and H0 is rejected. If |r| ≤ 0.576, there is insufficient evidence to reject H0.

对于样本容量 n = 12、显著性水平 5% 的双尾检验，临界值约为 0.576。这意味着如果样本 r 满足 |r| > 0.576，则结果在 5% 水平下显著，拒绝 H0。如果 |r| ≤ 0.576，则没有足够证据拒绝 H0。

Sample size n	5% two-tailed critical value	1% two-tailed critical value
5	0.878	0.959
12	0.576	0.708

For a one-tailed test, use the one-tailed critical values from the same table, which are smaller than the two-tailed values for a given significance level and sample size. Always state your conclusion in context, for example: 'There is evidence at the 5% level of a positive correlation between height and weight.'

对于单尾检验，使用同一表格中的单尾临界值，在给定显著性水平和样本容量下，单尾临界值小于双尾临界值。始终结合上下文陈述结论，例如：“在 5% 显著性水平下，有证据表明身高与体重之间存在正相关。”

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10. Worked Example: PMCC and Hypothesis Test | 例题：PMCC 与假设检验

Five pairs of data are recorded: (1, 2), (2, 4), (3, 5), (4, 4), (5, 7). Calculate the PMCC and test at the 5% level whether there is evidence of linear correlation.

记录五对数据：(1, 2)、(2, 4)、(3, 5)、(4, 4)、(5, 7)。计算 PMCC，并在 5% 显著性水平下检验是否有线性相关的证据。

x	y	x²	y²	xy
1	2	1	4	2
2	4	4	16	8
3	5	9	25	15
4	4	16	16	16
5	7	25	49	35
Σ = 15	Σ = 22	Σ = 55	Σ = 110	Σ = 76

Sxx = 55 − 15² ÷ 5 = 55 − 45 = 10, and Syy = 110 − 22² ÷ 5 = 110 − 96.8 = 13.2. Therefore:

使用 n = 5，我们得到 Sxy = 76 − (15 × 22) ÷ 5 = 76 − 66 = 10。同时 Sxx = 55 − 15² ÷ 5 = 55 − 45 = 10，Syy = 110 − 22² ÷ 5 = 110 − 96.8 = 13.2。因此：

r = 10 ÷ √10 × 13.2 = 10 ÷ √132 ≈ 0.870

For a two-tailed test at the 5% level with n = 5, the critical value is 0.878. Since |r| = 0.870 < 0.878, we do not reject H0. There is insufficient evidence at the 5% level to conclude that the population correlation is non-zero.

对于 n = 5 的 5% 双尾检验，临界值为 0.878。由于 |r| = 0.870 < 0.878，我们不拒绝 H0。在 5% 显著性水平下，没有足够证据断定总体相关不为零。

The regression line of y on x is also found from b = Sxy ÷ Sxx = 10 ÷ 10 = 1 and a = ȳ − b x̄ = 4.4 − 1 × 3

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