AQA International A-level Unit S2 (Statistics) Revision Guide | AQA 国际 A-level 统计 S2 单元复习指南

📚 AQA International A-level Unit S2 (Statistics) Revision Guide | AQA 国际 A-level 统计 S2 单元复习指南

The AQA International A-level Statistics 2 (S2) unit builds on S1 and focuses on continuous probability models, the normal distribution, statistical inference, and chi-squared tests. You will be expected to use integration to handle continuous random variables, to carry out z-tests and confidence intervals, and to interpret chi-squared results in context.

AQA 国际 A-level 统计第二单元（S2）在 S1 基础上深入学习连续概率模型、正态分布、统计推断和卡方检验。考试要求能用积分处理连续随机变量，能实施 z 检验与置信区间，并能结合背景解释卡方检验结果。

The examination is a written paper that emphasises accurate calculation, complete hypothesis-testing structure, and clear interpretation of statistical conclusions in real-world language. A solid grasp of notation and calculator use is essential, but final answers must be supported by method.

考试为笔试，重点考查计算准确性、假设检验的完整步骤和统计结论的文字解释。扎实掌握符号和计算器使用至关重要，但最终答案必须展示完整方法。

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1. Continuous Random Variables and PDFs | 连续随机变量与概率密度函数

c) = 0 for any single value c, so inclusive and exclusive inequalities give the same area.

连续随机变量 X 用概率密度函数 f(x) 描述。pdf 必须满足对所有 x 有 f(x) ≥ 0，且曲线下总面积为 1：∫₋∞^∞ f(x) dx = 1。概率就是面积：P(a < X < b) = ∫ₐᵇ f(x) dx。注意连续变量中任意单点概率 P(X = c) = 0，因此 ≤ 和 < 给出相同面积。

For example, if f(x) = kx² for 0 ≤ x ≤ 2, find k by setting the total area to 1.

例如，若 f(x) = kx²，0 ≤ x ≤ 2，通过令总面积为 1 求 k。

1 = ∫₀² kx² dx = k[x³/3]₀² = 8k/3 → k = 3/8

Always check that f(x) is non-negative on its support and that the calculated total area is exactly 1 before moving on.

务必先检查 f(x) 在支撑集上非负且计算总面积为 1，再继续后续计算。

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2. Cumulative Distribution Function, Median and Mode | 累积分布函数、中位数与众数

= 0.75. The mode is the value or values of x where f(x) is greatest, if a maximum exists.

累积分布函数 F(x) = P(X ≤ x) = ∫₋∞^x f(t) dt，从 0 递增到 1。中位数 m 满足 F(m) = 0.5；下四分位数满足 F(Q₁) = 0.25，上四分位数满足 F(Q₃) = 0.75。众数是使 f(x) 最大的 x 值（若最大值存在）。

ivative relationship F′(x) = f(x) is useful when the pdf is given as a piecewise function.

不要混淆 F(x) 与 f(x)：F(x) 是从左侧累积的概率，而 f(x) 是密度，其值可以大于 1。当 pdf 以分段函数给出时，导数关系 F′(x) = f(x) 很有用。

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3. Expectation and Variance for Continuous Variables | 连续变量的期望与方差

ar(X). The same linear properties apply: E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X).

连续随机变量 X 的期望 E(X) = μ = 在支撑集上 ∫ x f(x) dx，且 E(X²) = ∫ x² f(x) dx。于是 Var(X) = E(X²) − [E(X)]²，标准差 σ = √Var(X)。线性性质同样适用：E(aX + b) = aE(X) + b，Var(aX + b) = a²Var(X)。

d Var(X) = (b − a)²/12. These results are often quoted but can be verified by integration.

对 [a, b] 上的连续均匀分布，pdf 为 f(x) = 1/(b − a)，因此 E(X) = (a + b)/2，Var(X) = (b − a)²/12。这些结果常被引用，但也可通过积分验证。

Practise integrating piecewise pdfs: split the support at each piece, compute the expectation and second moment separately, then combine the results.

练习对分段 pdf 积分：在每一段分别计算期望和二阶矩，再合并结果。

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4. Normal Distribution Basics and Standardisation | 正态分布基础与标准化

The normal distribution N(μ, σ²) has pdf f(x) = (1/(σ√2π)) e^[−(x−μ)²/(2σ²)]. To find probabilities, standardise: Z = (X − μ)/σ. Then Z ~ < z). For P(Z > z) use 1 − Φ(z); for P(a < X < b) use Φ((b−μ)/σ) − Φ((a−μ)/σ).

正态分布 N(μ, σ²) 的 pdf 为 f(x) = (1/(σ√2π)) e^[−(x−μ)²/(2σ²)]。求概率时先标准化：Z = (X − μ)/σ，则 Z ~ N(0, 1)。查表得 Φ(z) = P(Z < z)。P(Z > z) 用 1 − Φ(z)；P(a < X < b) 用 Φ((b−μ)/σ) − Φ((a−μ)/σ)。

Z = (X − μ) / σ

A sketch of the normal curve is essential. Label the mean, the boundary values, and the relevant area to avoid sign and table errors. Remember that the total area under the curve is 1 and the distribution is symmetric about the mean.

正态曲线草图必不可少。标出均值、边界值和相关面积，以避免符号和查表错误。记住曲线下总面积为 1，且分布关于均值对称。

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5. Reverse Normal Calculations | 正态分布反向计算

zσ. This appears in questions about limiting values, pass marks, or guaranteed lifetimes.

反向正态问题中，已知概率求对应值 x。先查表找到满足 Φ(z) = p 的 z，再用 x = μ + zσ 反代。这类题常涉及界限值、及格线或保证寿命。

For example, if X ~ N(50, 4²) and 5% × 4 = 56.58. Always state whether the given probability is a left-tail or right-tail area.

例如，若 X ~ N(50, 4²)，且 5% 的数值超过界限，则 P(Z > z) = 0.05 得 z = 1.6449，所以 x = 50 + 1.6449 × 4 = 56.58。务必说明已知概率是左尾面积还是右尾面积。

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6. Normal Approximation to Binomial and Poisson | 二项分布与泊松分布的正态近似

bution can approximate binomial and Poisson probabilities when parameters are large. For X ~ B(n, p), use N(np, np(1−p)) when np > 5 and n(1−p) > 5. For X ~ tinuity correction: P(X ≤ 10) becomes P(Y < 10.5), and P(X ≥ 10) becomes P(Y > 9.5).

当参数较大时，正态分布可近似二项和泊松分布。对 X ~ B(n, p)，在 np > 5 且 n(1−p) > 5 时使用 N(np, np(1−p))。对 X ~ Po(λ)，在 λ > 15 时使用 N(λ, λ)。必须做连续性校正：如 P(X ≤ 10) 变为 P(Y < 10.5)，P(X ≥ 10) 变为 P(Y > 9.5)。

Distribution | 分布	Normal approximation | 正态近似	Condition | 条件
B(n, p)	N(np, np(1−p))	np > 5, n(1−p) > 5
Po(λ)	N(λ, λ)	λ > 15

The continuity correction is the most common source of lost marks. Remember to adjust the boundary by 0.5 in the direction that includes the discrete value when converting to the continuous scale.

连续性校正是最常见的失分点。记住在转换到连续尺度时，要将边界向包含离散值的方向调整 0.5。

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7. Hypothesis Testing for a Normal Mean | 正态均值的假设检验

opulation mean from a normal distribution with known variance uses the z-statistic: z = (x̄ − μ₀) / (σ/√n ≠ μ₀. Compare z with the critical value or values from N(0,1). For a two-tailed test at 5% significance, the critical values are ±1.96.

已知方差时，正态总体均值的假设检验使用 z 统计量：z = (x̄ − μ₀) / (σ/√n)。写出 H₀: μ = μ₀ 和 H₁: μ < μ₀、μ > μ₀ 或 μ ≠ μ₀。将 z 与 N(0,1) 临界值比较。5% 显著性水平双侧检验临界值为 ±1.96。

z = (x̄ − μ₀) / (σ/√n)

Always present a five-step structure: define hypotheses, state significance level, calculate the test statistic, find the critical value or p-value, and write a contextual conclusion. Do not accept or reject without reference to the problem context.

始终使用五步结构：定义假设、说明显著性水平、计算检验统计量、求临界值或 p 值、写出结合背景的结论。不要在脱离问题背景的情况下简单接受或拒绝。

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8. Confidence Intervals for Means and Proportions | 均值与比例的置信区间

A confidence interval for the mean of a normal population with known σ is x̄ ± z × σ/√n. For a 95% confidence level,

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