Class-Based Voting and Other Influences on Voting Patterns: A Statistical Analysis | 基于阶级的投票及其他投票模式影响因素：统计分析

📚 Class-Based Voting and Other Influences on Voting Patterns: A Statistical Analysis | 基于阶级的投票及其他投票模式影响因素：统计分析

Voting behaviour in the UK has often been summarised by the phrase "class is the basis of British party politics". From a mathematical perspective, this claim can be turned into a testable question: do different social classes vote in significantly different proportions?

英国投票行为常被概括为“阶级是英国政党政治的基础”。从数学角度看，这一说法可以转化为一个可检验的问题：不同社会阶级的投票比例是否存在显著差异？

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1. Voting Patterns as Data | 作为数据的投票模式

In A-Level Mathematics, we treat voting patterns as categorical data. Each voter is classified by social class and by party choice. We can summarise the relationship using a two-way table and calculate proportions, probabilities, and measures of association.

在A-Level数学中，我们将投票模式视为分类数据。每位选民按社会阶级和政党选择进行分类。我们可以用双向表总结这种关系，并计算比例、概率和关联度量。

The key question is whether the observed differences are large enough to be considered statistically significant, rather than the result of random sampling variation.

关键问题是，观察到的差异是否足够大，可以被认为具有统计显著性，而不是随机抽样变异的结果。

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2. Class-Based Voting: Operationalising Social Class | 基于阶级的投票：社会阶级的量化

Social class is often simplified into categories such as "working class" and "middle class". Suppose a survey records that 55% of working-class voters support Party A, while only 40% of middle-class voters do so.

社会阶级通常简化为“工人阶级”和“中产阶级”等类别。假设一项调查记录显示，55%的工人阶级选民支持A党，而中产阶级选民中只有40%支持A党。

The conditional probability P(A | WC) = 0.55 and P(A | given their class. The difference 0.55 - 0.40 = 0.15 is a simple measure of the class gap.

条件概率 P(A | WC) = 0.55 和 P(A | MC) = 0.40 描述了给定阶级时选民支持A党的概率。差值 0.55 - 0.40 = 0.15 是阶级差距的一个简单度量。

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3. Conditional Probability and Independence | 条件概率与独立性

If class and voting were independent, then P(A | WC) would equal P(A | MC) and both would equal the overall proportion P(A). Independence means knowing a voter's class gives no additional information about their party choice.

如果阶级与投票相互独立，那么 P(A | WC) 将等于 P(A | MC)，且两者都等于总体比例 P(A)。独立意味着知道选民的阶级不会提供关于其政党选择的额外信息。

We can test independence using a two-way table. For example, if the overall support for Party A is 48%, an independent model would predict about 48% support in every class category.

我们可以使用双向表检验独立性。例如，如果A党的总体支持率为48%，独立模型将预测每个阶级类别中的支持率都约为48%。

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4. Binomial Model for Voting Samples | 投票样本的二项分布模型

Suppose we take a random sample of n working-class voters and record the number X who support Party A. If the true proportion is p, then X follows a binomial distribution: X ~ B(n, p).

假设我们随机抽取 n 名工人阶级选民，记录支持A党的人数 X。如果真实比例为 p，则 X 服从二项分布：X ~ B(n, p)。

The mean of X is np and the variance is np(1 - p). These results allow us to calculate the probability of observing a particular number of Party A supporters under a given null hypothesis.

X 的均值是 np，方差是 np(1 - p)。这些结果使我们能够计算在给定原假设下观察到特定数量A党支持者的概率。

For large n, the binomial distribution can be approximated by a normal distribution with mean np and variance np(1 - p). This is useful for constructing hypothesis tests.

当 n 较大时，二项分布可用均值为 np、方差为 np(1 - p) 的正态分布来近似。这对于构造假设检验非常有用。

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5. Hypothesis Test for a Single Proportion | 单一比例假设检验

of working-class voters support Party A. The alternative hypothesis might be H₁: p ≠ 0.50.

为了检验关于阶级投票的说法，我们建立原假设。例如，H₀: p = 0.50，表示一半的工人阶级选民支持A党。备择假设可能是 H₁: p ≠ 0.50。

Given a sample proportion p̂ = X/n, the test statistic for a large sample is:

给定样本比例 p̂ = X/n，大样本检验统计量为：

z = (p̂ - p₀) ÷ √(p₀(1 - p₀) ÷ n)

z with critical values from the standard normal distribution. If z is beyond ±1.96 for a 5% we reject H₀ and conclude that the observed proportion is significantly different from p₀.

我们将 z 与标准正态分布的临界值进行比较。如果在5%显著性水平下 z 超出 ±1.96，就拒绝 H₀，并得出结论：观察到的比例与 p₀ 存在显著差异。

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6. Comparing Two Proportions: Class Gap Test | 比较两个比例：阶级差距检验

middle-class support for Party A differ significantly, we compare two sample proportions p̂₁ and p̂₂. The null hypothesis is H₀: p₁ = p₂.

为了检验工人阶级和中产阶级对A党的支持率是否存在显著差异，我们比较两个样本比例 p̂₁ 和 p̂₂。原假设为 H₀: p₁ = p₂。

Using a pooled estimate p̂ = (X₁ + X₂) ÷ (n₁ + n₂), the test statistic is:

使用合并估计 p̂ = (X₁ + X₂) ÷ (n₁ + n₂)，检验统计量为：

z = (p̂₁ - p̂₂) ÷ √(p̂(1 - p̂)(1/n₁ + 1/n₂))

If the test rejects H₀, we have statistical evidence of a class gap in voting behaviour. This does not prove causation but shows the difference is unlikely to have occurred by chance.

如果检验拒绝 H₀，我们就有了阶级差距在投票行为中存在的统计证据。这并不证明因果关系，但表明这种差异不太可能仅由偶然因素造成。

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7. Correlation Between Class Voting and Time | 阶级投票与时间的相关性

Class-based voting has declined in the UK since the 1960s. We can measure this trend by recording the percentage of working-class voters supporting the Labour Party each decade and calculating the product moment correlation coefficient (PMCC) with time.

自20世纪60年代以来，英国基于阶级的投票已经下降。我们可以记录每十年工人阶级选民支持工党的百分比，并计算其与时间的积矩相关系数（PMCC）来衡量这一趋势。

If the PMCC r is close to -1, there is a strong negative linear correlation between time and class voting. A hypothesis test for r can determine whether this correlation is statistically significant.

如果 PMCC r 接近 -1，则时间与阶级投票之间存在强负线性相关。对 r 的假设检验可以确定这种相关是否具有统计显著性。

A scatter diagram with a regression line would show the model:

带有回归线的散点图将显示模型：

ClassVoting = a + b × Time

where b is the average change in percentage points per year.

其中

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