Democratic States: Statistical Modelling of Election Turnout | 民主国家：选举投票率的统计建模

📚 Democratic States: Statistical Modelling of Election Turnout | 民主国家：选举投票率的统计建模

In democratic states, election results and opinion polls are usually reported as proportions of voters supporting a particular party, candidate, or policy. Edexcel A-Level Mathematics gives you a rigorous framework for testing whether an observed sample proportion is consistent with a claimed population proportion. This article walks through the binomial model, normal approximation, hypothesis tests, confidence intervals, and sample size calculations that appear in the statistics papers.

在民主国家，选举结果和民意调查通常以支持某一政党、候选人或政策的选民比例来报告。Edexcel A-Level 数学为你提供了一个严谨的框架，用来检验观察到的样本比例是否与声称的总体比例一致。本文梳理了统计试卷中出现的二项模型、正态近似、假设检验、置信区间和样本量计算。

1. Why Model Voting in Democratic States? | 为什么对民主国家的投票行为建模？

In a democratic state, every voter can be thought of as a random individual with a probability p of supporting a specific candidate. When a polling agency samples n voters at random, the number of supporters becomes a binomial random variable. This simple model allows us to quantify uncertainty and to test claims about the whole electorate.

在一个民主国家，每一位选民都可以被看作一个随机个体，其支持某位候选人的概率为 p。当民调机构随机抽取 n 名选民时，支持者的人数就成为一个二项随机变量。这个简单的模型使我们能够量化不确定性，并检验关于全体选民的各类说法。

The key assumption is that each sampled voter is independent of the others and has the same probability p of success. In practice, this means the sample must be random and should be no more than 10% of the population. Edexcel exam questions will normally state that these assumptions are reasonable.

关键假设是每位被抽样的选民相互独立，并且每位选民支持成功的概率 p 相同。在实践中，这意味着样本必须是随机的，并且不应超过总体的 10%。Edexcel 考题通常会说明这些假设是合理的。

2. The Binomial Model for Voter Preferences | 选民偏好的二项模型

Let X be the number of voters in a random sample of size n who support candidate A. If the true population proportion is p, then X follows a binomial distribution with parameters n and p. We write this as X ~ B(n, p).

设 X 为容量为 n 的随机样本中支持候选人 A 的选民人数。如果真实的总体比例为 p，则 X 服从参数为 n 和 p 的二项分布。我们将其写作 X ~ B(n, p)。

The probability of observing exactly k supporters is given by the binomial probability mass function. In polls, this formula helps calculate the exact chance of a particular sample result under a given claimed value of p.

恰好观察到 k 名支持者的概率由二项概率质量函数给出。在民调中，该公式有助于在给定的声称 p 值下计算特定样本结果的确切概率。

P(X = k) = ₙCₖ pᵏ (1 − p)ⁿ⁻ᵏ

Here ₙCₖ is the binomial coefficient, calculated as n! / [k!(n – k)!]. On a calculator, this is usually the nCr function. You should always write down the formula before substituting values, because Edexcel awards method marks for the correct structure.

这里 ₙCₖ 是二项系数，计算公式为 n! / [k!(n – k)!]。在计算器上，这通常是 nCr 函数。你应该在代入数值之前先写出公式，

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