Presidential Elections: Modelling Polls and Significance Tests | 总统选举：民意调查建模与显著性检验

📚 Presidential Elections: Modelling Polls and Significance Tests | 总统选举：民意调查建模与显著性检验

Presidential elections generate a flood of polling data. In Edexcel A-Level Mathematics, topics such as the binomial distribution, normal approximation, confidence intervals and hypothesis tests give us the tools to judge whether an apparent lead is statistically meaningful or just sampling noise.

总统选举会产生大量民意调查数据。在 Edexcel A-Level 数学中，二项分布、正态近似、置信区间和假设检验等主题为我们提供了判断某一表面领先是否具有统计意义、还是仅仅抽样噪声的工具。

1. Binomial Model for Voter Support | 选民支持率的二项分布模型

Suppose a candidate has true support proportion p in the population. If we randomly sample n voters, the number X who support the candidate follows X ~ B(n, p). The probability of exactly x supporters is P(X = x) = C(n, x) pˣ (1-p)ⁿ⁻ˣ.

假设某候选人在总体中的真实支持率为 p。如果我们随机抽取 n 名选民，支持该候选人的人数 X 服从 X ~ B(n, p)。恰好有 x 名支持者的概率为 P(X = x) = C(n, x) pˣ (1-p)ⁿ⁻ˣ。

For example, if p = 0.52 and n = 10, the probability that exactly 6 voters support the candidate is P(X = 6) = C(10, 6) (0.52)⁶ (0.48)⁴ ≈ 0.220. This shows that even with a true majority, a single small sample can vary considerably.

例如，若 p = 0.52 且 n = 10，恰好有 6 名选民支持的概率为 P(X = 6) = C(10, 6) (0.52)⁶ (0.48)⁴ ≈ 0.220。这表明即使真实支持率过半，单次小样本的结果也可能有很大波动。

In Edexcel exams you may use a calculator function such as Binomial PD to compute these probabilities quickly, but you must still show the distribution and the values used.

在 Edexcel 考试中，你可以使用计算器的二项分布概率功能快速计算这些概率，但仍须写出分布和所用数值。

2. Normal Approximation to the Binomial | 二项分布的正态近似

When n is large and both np and n(1-p) are greater than 5, X is approximately normal with mean μ = np and variance σ² = np(1-p). The sample proportion p̂ = X/n then has approximately N(p, p(1-p)/n).

当 n 较大且 np 与 n(1-p) 均大于 5 时，X 近似服从均值为 μ = np、方差为 σ² = np(1-p) 的正态分布。此时样本比例 p̂ = X/n 近似服从 N(p, p(1-p)/n)。

p̂ ~ N(p, p(1-p)/n)

This approximation is the backbone of election polling because polls typically use n = 1000 or more, making calculations with the normal distribution much quicker than summing many binomial probabilities.

这一近似是选举民调的支柱，因为民调通常使用 n = 1000 或更大的样本，用正态分布计算比逐项加和二项概率要快得多。

We often standardise the sample proportion to obtain a z-score: Z = (p̂ – p) / √(p(1-p)/n) ~ N(0, 1). This allows us to use standard normal tables or calculator functions.

我们经常将样本比例标准化，得到 z 分数

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

更多咨询请联系16621398022（同微信）