The Labour Party | 工党民意调查中的 A-Level 数学统计应用

📚 The Labour Party | 工党民意调查中的 A-Level 数学统计应用

In Edexcel A-Level Mathematics, statistics questions often use real-world contexts such as opinion polls. A common exercise is to model the proportion of voters supporting a political party, for example the Labour Party, using the binomial distribution, hypothesis testing, and confidence intervals. This article uses a fictional survey about the Labour Party to revise the core statistical techniques required by the Edexcel specification.

在 Edexcel A-Level 数学中，统计题常以民意调查等真实情境为背景。常见练习是将支持某政党（例如工党）的选民比例用二项分布、假设检验和置信区间来建模。本文使用一项关于工党的虚构调查，复习 Edexcel 考试大纲要求的核心统计方法。

1. Why Opinion Polls Appear in A-Level Statistics | 为什么民意调查会出现在 A-Level 统计中

Opinion polls are ideal for testing statistical ideas because they involve a population proportion p, a random sample of size n, and observable counts of ‘successes’. The Labour Party support rate in a poll can be treated as an unknown parameter p that we estimate or test using sample data.

民意调查非常适合考查统计概念，因为它涉及总体比例 p、容量为 n 的随机样本，以及可观测的“成功”次数。民调中工党的支持率可视为未知参数 p，我们利用样本数据对其进行估计或假设检验。

Edexcel frequently sets questions around polling data because it allows examiners to assess binomial distributions, normal approximations, confidence intervals, and hypothesis tests all within one theme.

Edexcel 常以民调数据为背景出题，因为这样可以在同一主题下综合考查二项分布、正态近似、置信区间和假设检验。

2. Defining the Random Variable and Binomial Conditions | 定义随机变量与二项分布条件

Let X be the number of Labour supporters in a random sample of n voters. X follows a binomial distribution, X ~ B(n, p), provided that each voter is independent, there are only two outcomes (Labour or not Labour), the probability p remains constant, and the sample size is fixed.

设 X 为 n 名选民的随机样本中支持工党的人数。若每位选民相互独立、只有两种结果（支持工党或不支持）、概率 p 保持不变且样本量固定，则 X 服从二项分布，记为 X ~ B(n, p)。

Fixed number of trials n | 固定试验次数 n
Independent trials | 每次试验相互独立
Two outcomes: Labour / not Labour | 两种结果：支持工党 / 不支持工党
Constant probability p | 概率 p 保持不变

In a real poll, independence may be violated if voters in the same household influence each other, so the binomial model is a simplification that still works well for large, well-designed samples.

在实际民调中，如果同一家庭的选民相互影响，独立性可能不成立，因此二项模型是一种简化，但对于设计良好、样本量较大的调查仍然适用。

3. Calculating Probabilities Using the Binomial Formula | 使用二项公式计算概率

To find the probability of exactly k Labour supporters in a sample of 10 voters when p = 0.35, use:

当 p = 0.35，要在 10 名选民中恰好有 k 名工党支持者，可使用：

P(X = k) = ₁₀Cₖ · 0.35ᵏ · 0.65¹⁰⁻ᵏ

For example, P(X = 4) = ₁₀C₄ · 0.35⁴ · 0.65⁶ ≈ 0.2377. This means there is about a 23.8% chance that exactly 4 out of 10 voters support Labour.

例如 P(X = 4) = ₁₀C₄ · 0.35⁴ · 0.65⁶ ≈ 0.2377。这意味着 10 名选民中恰好有 4 人支持工党的概率约为 23.8%。

Always show the formula in your working, but use the calculator’s Binomial PD function to obtain the numerical value quickly.

解题时应写出公式，但可用计算器的二项概率密度函数快速得到数值。

4. Cumulative Probabilities and Inequalities | 累积概率与不等式

Exam questions often ask for P(X ≤ 3), P(X < 5), P(X ≥ 6) or P(2 ≤ X ≤ 7). Use the binomial cumulative distribution function (Binomial CD) on your calculator. Note that P(X < 5) = P(X ≤ 4) for a discrete variable.

考试常要求计算 P(X ≤ 3)、P(X < 5)、P(X ≥ 6) 或 P(2 ≤ X ≤ 7)。此时使用计算器的二项累积分布函数（Binomial CD）。注意对离散变量而言，P(X < 5) = P(X ≤ 4)。

If 100 voters are sampled and p = 0.35, then P(X ≤ 30) represents the probability that no more than 30 Labour supporters appear, which can be found directly from the calculator.

若抽样 100 名选民且 p = 0.35，则 P(X ≤ 30) 表示工党支持者不超过 30 人的概率，可直接由计算器求得。

For P(X ≥ a), use 1 − P(X ≤ a − 1). This avoids the common mistake of including the boundary value incorrectly.

计算 P(X ≥ a) 时，使用 1 − P(X ≤ a − 1)。这样可以避免边界值处理不当的常见错误。

5. Hypothesis Testing for a Proportion | 比例假设检验

A hypothesis test for p usually has the structure:

关于 p 的假设检验通常形式为：

H₀: p = 0.35 H₁: p ≠ 0.35 or H₁: p > 0.35 / p < 0.35

Suppose a political analyst claims Labour support has fallen below 35%. In a sample of 80 voters, 22 support Labour. Test at the 5% significance level whether the proportion is less than 0.35.

假设一位政治分析人士声称工党支持率已降至 35% 以下。在 80 名选民样本中，有 22 人支持工党。在 5% 显著性水平下检验该比例是否小于 0.35。

Under H₀ we define X ~ B(80, 0.35). The observed count is x = 22. We find P(X ≤ 22). If this p-value is less than 0.05, we reject H₀ and conclude there is evidence support is below 35%.

在原假设下定义 X ~ B(80, 0.35)，观测频数为 x = 22。求 P(X ≤ 22)。若该 p 值小于 0.05，则拒绝 H₀，得出结论：有证据表明支持率低于 35%。

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

When n is large and p is not too close to 0 or 1, X ~ B(n, p) can be approximated by a normal distribution with mean μ = np and variance σ² = np(1 − p). The usual condition is np > 5 and n(1 − p) > 5.

当 n 较大且 p 不太接近 0 或 1 时，X ~ B(n, p) 可用正态分布近似，其均值 μ = np，方差 σ² = np(1 − p)。通常要求 np > 5 且 n(1 − p) > 5。

To calculate P(X ≤ 22) for X ~ B(80, 0.35), we apply a continuity correction: P(X ≤ 22) ≈ P(Y ≤ 22.5) where Y ~ N(28, 18.2), since np = 28 and np(1 − p) = 18.2.

对于 X ~ B(80, 0.35)，计算 P(X ≤ 22) 时使用连续性校正：P(X ≤ 22) ≈ P(Y ≤

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The Labour Party | 工党民意调查中的 A-Level 数学统计应用