Relationships between the presidency and Congress | 总统与国会之间的关系

📚 Relationships between the presidency and Congress | 总统与国会之间的关系

When studying the American political system, we often ask: is there a measurable link between presidential popularity and the legislative success of a president’s agenda in Congress? In Edexcel A‑Level Mathematics, you can use statistical tools – scatter diagrams, correlation coefficients, and linear regression – to quantify such real‑world relationships. This article treats the ‘relationship between the presidency and Congress’ as a case study for the Statistics content of your course, guiding you through data collection, correlation testing, and interpretation of results.

在研究美国政治体制时，我们常会问：总统的公众支持率与其在国会中推行议程的成功率之间是否存在可量化的联系？在Edexcel A‑Level数学中，你可以运用统计学工具——散点图、相关系数和线性回归——来量化这类现实世界的关系。本文将“总统与国会之间的关系”作为课程统计部分的一个案例进行研究，带你走完数据收集、相关性检验与结果解读的全过程。

1. Modelling Political Dynamics with Statistics | 用统计学建模政治动态

fine two numerical variables: X = annual average public approval rating of the president (% ons over several years, we can apply the Edexcel S1 techniques of bivariate data analysis.

政治学者长期争论总统公众支持在多大程度上影响国会行为。假设我们定义两个数值变量：X = 总统年度平均公众支持率（百分比），Y = 国会关键投票中与总统立场一致的百分比。将这些作为多年的配对观测值，我们就可以运用Edexcel S1的双变量数据分析技术。

ggests a significant positive correlation, we might conclude that higher approval ratings are associated with greater legislative success, even though correlation does not imply causation. Your task is to model the data, test for correlation, and construct a regression line for prediction.

在这类调查中，零假设通常是没有线性相关（ρ = 0）。若证据表明存在显著正相关，便可推断更高的支持率与更大的立法成功率相关联，尽管相关性并不意味着因果关系。你的任务是建立数据模型，检验相关性并构建用于预测的回归线。

---

2. Defining Variables and Collecting Data | 定义变量与收集数据

Before any calculation, you must be clear about the explanatory (independent) variable and the response (dependent) variable. Here, the president’s approval rating (X) is taken as the independent variable because it is likely to drive congressional support, rather than the other way around. The percentage of aligned congressional votes (Y) is the dependent variable.

任何计算开始前，必须明确解释变量（自变量）与响应变量（因变量）。此处将总统支持率（X）视为自变量，因为它很可能驱动国会支持，而非相反。国会投票一致性百分比（Y）则为因变量。

For an Edexcel style question, you might be given a small dataset. As an illustration, consider the following hypothetical table of ten observations.

对于Edexcel风格的考题，通常会给出一个小的数据集。以下虚构的十组观测值供作图解之用。

Year	X (Approval %)	Y (Congress Support %)
1	42	38
2	48	44
3	54	50
4	60	57
5	66	62
6	39	35
7	51	49
8	57	53
9	63	60
10	45	41

Check that the data appears to show X and Y moving in the same direction. A clear pattern suggests positive correlation, but we need a rigorous measure.

检查数据是否显示X与Y同向变动。明显的模式表明存在正相关，但我们需要严格的度量指标。

---

3. Scatter Diagrams and Visual Correlation | 散点图与视觉相关性

Plotting the points (xᵢ, yᵢ) on a scatter diagram is the first step of any bivariate analysis. Label the horizontal axis ‘Presidential Approval (%)’ and the vertical axis ‘Congressional Support (%)’. Each point represents one year.

绘制散点图 (xᵢ, yᵢ) 是任何双变量分析的第一步。横轴标注“总统支持率（%）”，纵轴标注“国会支持率（%）”。每个点代表一个年份。

From the scatter you can make an initial judgement: the points roughly follow an upward‑sloping pattern, suggesting a moderate to strong positive linear correlation. There are no obvious outliers that could unduly influence the correlation coefficient. You should also comment on scatter, i.e. the tightness of the cloud, which governs the strength of correlation.

从散点图可做出初步判断：各点大致沿一条向上倾斜的线分布，表明存在中等至较强的正线性相关。没有明显会过度影响相关系数的异常值。还应评论离散程度，即点云的集中度，它决定了相关性的强弱。

Edexcel exams often provide a pre‑drawn scatter diagram or ask you to sketch one. Always add a brief verbal description alongside your visual assessment.

Edexcel考试通常会提供现成的散点图或要求你绘制一幅。务必在视觉评估的同时给出简短的语言描述。

---

4. The Pearson Correlation Coefficient r | 皮尔逊相关系数 r

To measure the strength and direction of a linear relationship, we calculate Pearson’s product‑moment correlation coefficient r. The formula is:

为衡量线性关系的强度与方向，我们计算皮尔逊积矩相关系数 r。公式为：

r = Σ (xᵢ – x̄)(yᵢ – ȳ) / √[ Σ (xᵢ – x̄)² Σ (yᵢ – ȳ)² ]

Here x̄ and ȳ are the means of the X and Y values respectively. The numerator is often denoted S_xy, while the denominator involves √S_xx S_yy.

这里 x̄ 和 ȳ 分别为 X 和 Y 的均值。分子常记为 S_xy，分母涉及 √S_xx S_yy。下面将使用 Unicode 符号代替下标。

Using the dataset above, first compute the sums Σx, Σy, Σx², Σy², and Σxy. Then find x̄ = Σx / n, ȳ = Σy / n. With n = 10, we obtain:

使用上述数据集，先计算 Σx、Σy、Σx²、Σy² 和 Σxy。然后求 x̄ = Σx/n, ȳ = Σy/n。在 n = 10 的情况下，我们得到：

x̄ = 52.5, ȳ = 48.9

Now compute S_xy = Σ (xᵢ – x̄)(yᵢ – ȳ) = 556.5, S_xx = Σ (xᵢ – x̄)² = 744.5, S_yy = Σ (yᵢ – ȳ)² = 620.9.

接着计算 S_xy = Σ (xᵢ – x̄)(yᵢ – ȳ) = 556.5, S_xx = Σ (xᵢ – x̄)² = 744.5, S_yy = Σ (yᵢ – ȳ)² = 620.9。

Finally, r = 556.5 / √744.5 × 620.9 trong positive linear correlation between presidential approval and congressional support.

最后，r = 556.5 / √744.5 × 620.9 ≈ 0.818。该值表明总统支持率与国会支持之间存在强正线性相关。

---

广告 / Sponsored

5. Hypothesis Testing for Correlation | 相关系数的假设检验

Once r is calculated, we need to test whether it is significantly different from zero, i.e. whether the population correlation coefficient ρ is non‑zero. State the hypotheses:

计算出 r 后，需检验它是否显著异于零，即总体相关系数 ρ 是否非零。设立假设：

H₀: ρ = 0
H₁: ρ > 0 (one‑tailed) or ρ ≠ 0 (two‑tailed)

Given that political theory predicts a positive link, a one‑tailed test is often appropriate. The test statistic follows a t‑distribution:

鉴于政治理论预期正相关，常采用单尾检验。检验统计量服从 t 分布：

t = r √n – 2 / √1 – r²

Plugging in r = 0.818 and n = 10 gives t ≈ 4.03. The degrees of freedom are n – 2 = 8. Looking up the 5% critical value for a one‑ . 1.860), we see that 4.03 > 1.860, so we reject H₀. There is sufficient evidence at the 5% significance level to support a positive correlation.

代入 r = 0.818, n = 10 得 t ≈ 4.03。自由度为 n – 2 = 8。查 Edexcel 表格中 8 自由度单尾 5% 临界值（约为 1.860），可见 4.03 > 1.860，故拒绝 H₀。在 5% 的显著性水平上有充分证据支持正相关。

You may also be asked to use the critical values for r directly from the product‑moment correlation coefficient table. For 8 df, the 5% one‑iled critical value is 0.5494. Our r = 0.818 exceeds this, confirming the same conclusion.

考试中也可能要求直接使用积矩相关系数表中的 r 临界值。对于 8 自由度，5% 单尾临界值为 0.5494。我们的 r = 0.818 大于此值，同样证实结论。

---

6. Coefficient of Determination r² | 决定系数 r²

ficient, r², is called the coefficient of determination. For our data, r² ≈ 0.669 (or 66.9% that can be explained by the linear relationship with X (presidential approval). About 67% s votes with the president is associated with changes in approval rating. The remaining 33% is due to other factors.

相关系数的平方 r² 称为决定系数。本例中 r² ≈ 0.669（即 66.9%）。它表示 Y（国会支持率）的变异中可由与 X（总统支持率）的线性关系解释的比例。国会投票与总统一致程度的变异约有 67% 与支持率变化相关联，其余 33% 归因于其他因素。

This gives a more intuitive feel for the strength of the relationship than r alone. An r² above 0.5 is generally considered strong in social‑science contexts.

相比单独的 r，r² 能更直观地体现关系强弱。在社会科学背景下，r² 高于 0.5 通常被视为强相关。

---

7. Linear Regression: Predicting Congressional Support | 线性回归：预测国会支持

If we want to predict Y for a given X, we use the least‑squares regression line of Y on X. The equation is y = a + bx, where:

若要基于给定的 X 预测 Y，我们使用 Y 对 X 的最小二乘回归线。方程为 y = a + bx，其中：

b = S_xy / S_xx
a = ȳ – b x̄

Using our previously computed sums, b = 556.5 / 744.5 ≈ 0.747, and a = 48.9 – 0.747 × 52.5 ≈ 9.69. Hence the regression equation is:

利用此前算出的和，b = 556.5 / 744.5 ≈ 0.747，a = 48.9 – 0.747 × 52.5 ≈ 9.69。因此回归方程为：

y = 9.69 + 0.747x

This line can be drawn on the scatter diagram by choosing two x‑values, finding the corresponding y, and joining the points.

该直线可通过选取两个 x 值、求对应的 y 并连接两点而画在散点图上。

Always check you have labelled the regression line correctly as ‘y on x’ and remember that the line passes through the mean point (x̄, ȳ).

务必正确标注回归线为“y 对 x”，并记住该直线经过均值点 (x̄, ȳ)。

---

8. Interpreting Regression Coefficients in Context | 结合语境解释回归系数

The slope b = 0.747 means that, on average, for every 1‑percentage‑ dicted congressional support when approval is zero; this has little practical meaning because approval ratings never reach zero, but it is mathematically necessary for the line.

斜率 b = 0.747 意味着，平均而言总统支持率每增加 1 个百分点，国会支持总统立场的百分比预期将增加约 0.75 个百分点。截距 a = 9.69 是支持率为零时的预测国会支持率；由于支持率永远不会降到零，该数值实际意义不大，但在数学上对直线必不可少。

Be cautious about extrapolation: using the model to predict Y for X values far outside the observed range (e.g., an approval of 90%) may be unreliable because the relationship may not stay linear.

外推需谨慎：用模型对远超出观测范围的 X 值（例如支持率 90%）预测 Y 可能不可靠，因为该关系可能并不保持线性。

---

9. Assessing the Fit: Residuals | 评估拟合：残差

s between observed Y values and the values predicted by the regression line: residual = yᵢ – onstant variance. A random scatter of residuals around zero supports the model’s validity.

残差是观测 Y 值与回归线预测值之间的差值：残差 = yᵢ – (a + bxᵢ)。将残差与 X 作图有助于检验线性与等方差性假设。残差围绕零值随机散布即支持模型的有效性。

A patterned residual plot (e.g., a curve) would suggest a non‑linear relationship, meaning a linear regression might not be the best model. In our political case study, if a president’s approval rating interacts with party control of Congress, the relationship could become more complex, requiring further investigation.

若残差图呈现规律性（如曲线），则表明存在非线性关系，意味着线性回归可能并非最佳模型。在我们的政治案例中，若总统支持率与国会党派控制发生交互，关系可能变得更为复杂，需进一步探究。

---

10. Assumptions and Limitations of the Model | 模型的假设与局限

The bivariate analysis behind r and regression relies on several assumptions: the relationship is linear, the data are a random sample from a bivariate normal distribution, observations are independent, and variability of Y around the regression line is roughly constant. In reality, political data are seldom perfectly normal, and independence can be questionable because events in one year may influence successive years.

r 和回归背后的双变量分析依赖于若干假设：关系是线性的，数据来自二元正态分布的随机样本，观测值独立，以及 Y 围绕回归线的变异大致恒定。现实中政治数据很少完全正态，且独立性存疑，因为某年发生的事件可能影响后续年份。

Moreover, correlation does not imply causation. A high r might reflect the influence of a third variable, such as the state of the economy, which simultaneously boosts both public approval and legislative cooperation. Always discuss such caveats in exam interpretations.

此外，相关并不意味着因果。高 r 值可能反映第三个变量的影响，例如经济状况可同时提升公众支持率与立法合作度。在考试中解读时务必讨论此类注意事项。

---

11. Coding the Data for Simplicity | 数据变换（编码）以简化计算

dexcel S1 frequently uses coding to make manual calculations easier. Suppose we let u = (x – m)/p and v = (y – , p, q are constants. The correlation coefficient is unchanged by linear coding, meaning r_uv = r_xy. This property is extremely useful: you can simplify the numbers without affecting the measure of correlation.

Edexcel S1 常使用编码以简化手工计算。设 u = (x – m)/p，v = (y – n)/q，其中 m, n, p, q 为常数。线性编码不改变相关系数，即 r_uv = r_xy。这一性质极为有用：你可简化数字而不影响相关性的度量。

However, be careful: the regression line changes under coding. If you code the variables, you must back‑transform the regression equation to the original units when interpreting the results.

但需注意：编码后回归线会改变。若对变量进行编码，在解读结果时必须将回归方程逆变换回原始单位。

---

12. Exam Techniques and Edexcel Practice | 考试技巧与 Edexcel 实练

Typical exam questions will give you a small table, ask for a scatter plot, the calculation of r, a hypothesis test, and the regression line. They may also ask you to interpret the gradient and intercept in context. Ensure you show all steps clearly, including the formulas with substituted values. Use your calculator efficiently to check sums, but write key

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）