📚 AQA A Level Maths Unit 5 January 2021 Question Paper | AQA A Level 数学 单元5 2021年1月试卷
The AQA A Level Maths Unit 5 paper from January 2021 is a Statistics 1 unit covering core statistical ideas that appear frequently in the International A Level Mathematics specification. This article reviews the main topics, formulas, and exam techniques linked to that paper, so you can approach similar questions with confidence.
AQA A Level 数学 2021 年 1 月单元 5 试卷属于统计学 1 单元，覆盖国际 A Level 数学考试大纲中频繁出现的核心统计概念。本文回顾该试卷涉及的主要主题、公式与解题技巧，帮助你更有信心地应对类似题目。
1. Paper Structure and Assessment Focus | 试卷结构与考查重点
Unit 5 is normally assessed through a written paper lasting 1 hour 30 minutes, with a mix of short structured questions and longer problem-solving tasks. The January 2021 paper followed the standard pattern of testing statistical calculation, interpretation, and use of probability models.
单元 5 通常通过一份 1 小时 30 分钟的笔试进行评估，题型包括简短的结构题和较长的应用问题。2021 年 1 月的试卷延续了标准模式，重点考查统计计算、结果解释以及概率模型的应用。
Examiners expect clear method statements, correct rounding to 3 significant figures unless stated otherwise, and full working for hypothesis tests. You should also label diagrams and state conclusions in context rather than simply writing 'reject H₀'.
阅卷人要求写出清晰的方法说明，除非题目另有要求，否则结果通常保留 3 位有效数字，并且假设检验需要写出完整步骤。结论必须结合题目背景进行陈述，而不是只写“拒绝 H₀”。
2. Essential Formula Sheet Revision | 必备公式复习
Many Unit 5 questions depend on accurate recall and rearrangement of the following formulas. The formula booklet provides some of these, but using them fluently saves time in the exam.
单元 5 的许多题目都依赖于准确记忆和变形以下公式。公式手册会提供其中一部分，但熟练使用它们能帮助你在考试中节省时间。
| Statistic | Formula | 中文说明 |
|---|---|---|
| Mean | x̄ = (Σxᵢ)/n | 平均数 |
| Sample variance | s² = Σ(xᵢ - x̄)²/(n - 1) | 样本方差 |
| Standard deviation | s = √s² | 标准差 |
| Addition law | P(A∪B) = P(A) + P(B) - P(A∩B) | 加法法则 |
| Conditional probability | P(A|B) = P(A∩B)/P(B) | 条件概率 |
| Expectation of X | E(X) = Σ xᵢ·P(X=xᵢ) | X 的期望 |
| Variance of X | Var(X) = E(X²) - [E(X)]² | X 的方差 |
| Binomial probability | P(X=r) = ⁿCᵣ pʳ qⁿ⁻ʳ | 二项分布概率 |
| Binomial mean and variance | μ = np, σ² = npq | 二项分布期望与方差 |
| Normal standardisation | z = (x - μ)/σ | 正态标准化 |
| Product-moment correlation | r = Sxy / √Sxx·Syy | 积矩相关系数 |
| Regression line | y = a + bx, b = Sxy / Sxx, a = ȳ - b x̄ | 回归直线 |
In the exam, write the formula first, substitute values carefully, and then give the final answer to the requested degree of accuracy. Avoid rounding intermediate values too early.
考试时应先写出公式，仔细代入数值，然后按题目要求的精确度给出最终答案。避免过早对中间结果进行舍入。
3. Measures of Location and Spread | 位置与离散程度的度量
Questions on location require you to calculate or interpret the mean, median, and mode from raw data, frequency tables, or grouped data. The mean is the most commonly used average, but outliers can pull it away from the centre of the data.
关于数据位置的题目要求你根据原始数据、频数表或分组数据计算或解释平均数、中位数和众数。平均数是最常用的集中量数，但异常值可能使它远离数据中心。
Measures of spread include the range, interquartile range, variance, and standard deviation. The interquartile range is preferred when outliers are present because it focuses on the middle 50% of the data.
离散程度的度量包括极差、四分位距、方差和标准差。当数据中存在异常值时，四分位距更适用，因为它只关注中间 50% 的数据。
When using grouped data, treat each value as the midpoint of its class interval. The formula for variance then uses the midpoint values, and you may be asked to estimate the standard deviation.
处理分组数据时，将每个值视为其组的中点。方差公式中使用这些中点值，题目可能要求你估算标准差。
4. Probability Laws and Tree Diagrams | 概率法则与树状图
multiplication law, and conditional probability. Two events A and B are independent if P(A∩B) = P(A) × P(B), or equivalently if P(A|B) = P(A).
单元 5 的概率题通常结合加法法则、乘法法则和条件概率。若两个事件 A 和 B 满足 P(A∩B) = P(A) × P(B)，或者等价地 P(A|B) = P(A)，则它们相互独立。
Tree diagrams are useful for sequential events without replacement. Multiply along branches for joint probabilities, and add those joint probabilities when more than one path satisfies the required condition.
树状图适用于不放回的连续事件。沿分支相乘得到联合概率，当多个路径都能满足所需条件时，将这些联合概率相加。
For conditional probability, clearly identify the reduced sample space. For example, P(A|B) uses only the part of the sample space where B has already occurred.
处理条件概率时，要明确缩小后的样本空间。例如，P(A|B) 只使用 B 已经发生的那部分样本空间。
5. Discrete Random Variables | 离散随机变量
bilities must sum to 1, and each individual probability must be between 0 and 1 inclusive.
离散随机变量 X 取一组可列的值，每个值对应概率 P(X=x)。所有概率之和必须为 1，且每个概率必须在 0 到 1 之间。
cut is Var(X) = E(X²) - [E(X)]², which is often faster than using the definition directly.
期望值 E(X) 是加权平均值，而 Var(X) 衡量分布的离散程度。常用的计算捷径是 Var(X) = E(X²) - [E(X)]²，这通常比直接使用定义更快。
) + b and Var(aX+b) = a²Var(X). Adding a constant changes the location but not the spread.
对于线性变换，需要记住 E(aX+b) = aE(X) + b 以及 Var(aX+b) = a²Var(X)。加上一个常数会改变位置，但不会改变离散程度。
6. Binomial Distribution | 二项分布
The binomial distribution is used when there is a fixed number of independent trials n, each with the same probability of success p. The number of successes X then follows the distribution X ~ B(n, p).
二项分布适用于固定次数的独立试验 n，且每次试验成功的概率 p 相同。成功次数 X 服从分布 X ~ B(n, p)。
The probability of exactly r successes is given by P(X=r) = ⁿCᵣ pʳ qⁿ⁻ʳ ive probabilities such as P(X ≤ r), use the tables or calculator with the correct n and p.
恰好取得 r 次成功的概率为 P(X=r) = ⁿCᵣ pʳ qⁿ⁻ʳ，其中 q = 1 - p。对于 P(X ≤ r) 之类的累积概率，需要根据正确的 n 和 p 使用表格或计算器。
often appear in hypothesis tests or as a check after calculating individual probabilities.
二项随机变量的期望和方差分别为 E(X) = np 和 Var(X) = npq。它们经常出现在假设检验中，或作为计算单个概率后的检验。
7. Normal Distribution | 正态分布
The normal distribution is symmetric and bell-shaped, with mean μ and standard deviation σ. Total area under the curve is 1, and areas represent probabilities.
正态分布是对称的钟形曲线，具有均值 μ 和标准差 σ。曲线下的总面积为 1，面积代表概率。
To find probabilities for X ~ /σ. Then use the standard normal table or calculator to find Φ(z). For inverse problems, work backwards from the probability to the z-value and then solve for x.
对于 X ~ N(μ, σ²)，需要先标准化，即 z = (x - μ)/σ。然后使用标准正态表或计算器求 Φ(z)。对于逆问题，要从概率反推 z 值，再解出 x。
e, P(X ≥ 10) becomes P(Y ≥ 9.5) after the correction, where Y is the normal approximation.
用正态分布近似二项分布时，必须进行连续性修正。例如，P(X ≥ 10) 修正后变为 P(Y ≥ 9.5)，其中 Y 是正态近似变量。
8. Hypothesis Testing | 假设检验
A hypothesis test begins with a null hypothesis H₀ and an alternative hypothesis H₁. The alternative can be one-tailed, such as p > 0.3, or two-tailed, such as p ≠ 0.3. The significance level is usually 5% or 1%.
假设检验从原假设 H₀ 和备择假设 H₁ 开始。备择假设可以是单边的，例如 p > 0.3，也可以是双边的，例如 p ≠ 0.3。显著性水平通常为 5% 或 1%。
For a binomial test, calculate the probability of obtaining the observed result or a more extreme result under H₀. If this p-value is less than the significance level, reject H₀ in favour of H₁.
对于二项检验，需要计算在原假设成立时得到观测结果或更极端结果的概率。如果该 p 值小于显著性水平，则拒绝 H₀，接受 H₁。
When the normal approximation is used, compute the test statistic z and compare it with the critical value from the normal distribution. Always state your conclusion in the context of the original problem.
使用正态近似时，计算检验统计量 z，并与正态分布的临界值进行比较。结论必须结合原始问题的具体情境进行陈述。
9. Correlation and Regression | 相关与回归
ient r lies between -1 and 1, where values close to ±1 indicate strong linear association.
相关分析用于衡量两个变量之间线性关系的强度和方向。积矩相关系数 r 的取值范围在 -1 到 1 之间，接近 ±1 表示线性关系较强。
least squares regression line has the form y = a + bx. The gradient b is calculated from Sxy / SxxPublished by TutorHao | Mathematics Revision Series | aleveler.com
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