📚 AS AQA Mathematics Unit 3 June 2019: Statistics 1B Revision and Worked Solutions | AQA AS 数学 Unit 3 2019年6月统计1B复习与答案解析
The June 2019 AQA AS Mathematics Unit 3 paper is a key assessment for students who have studied the applied statistics option. It tests your ability to interpret data, use probability models, and communicate statistical reasoning clearly under timed conditions.
2019 年 6 月的 AQA AS 数学 Unit 3 试卷是学习应用统计选项的学生面临的一项关键评估。它考查你在限时条件下解读数据、运用概率模型以及清晰表达统计推理的能力。
This article gives you a complete revision framework for the Statistics 1B route, including topic breakdowns, worked examples, common examiner comments, and a last-minute checklist. All practice questions are written in the style of the June 2019 paper, but they are original and do not reproduce AQA copyrighted material.
本文为 Statistics 1B 方向提供完整的复习框架，包括主题拆解、例题讲解、常见考官点评以及考前最终清单。所有练习题均按照 2019 年 6 月试卷的风格编写，但均为原创，不复制 AQA 的受版权保护内容。
1. Why Unit 3 Matters in AS AQA Mathematics | 为什么 Unit 3 对 AQA AS 数学如此重要
Unit 3 is the applied paper in your AS Mathematics qualification. If your school follows the statistics route, this paper is usually Statistics 1B, and it carries substantial weight in your final AS grade.
Unit 3 是 AS 数学资格中的应用卷。如果你的学校选择统计方向，这份试卷通常是 Statistics 1B，它在你的最终 AS 成绩中占有相当大的比重。
The June 2019 paper required students to move between real-world contexts, probability calculations, and statistical models. Success depends not only on knowing formulas but also on choosing the correct model and interpreting results in context.
2019 年 6 月的试卷要求学生能够在真实情境、概率计算和统计模型之间灵活切换。成功不仅取决于你是否记住了公式，还取决于你能否选择正确的模型并结合情境解释结果。
Many students lose marks because they give numerical answers without a concise interpretation. Examiners reward precise wording such as ‘there is evidence to reject the null hypothesis’ or ‘the expected number is 12’ rather than a bare number.
许多学生因为没有对数值结果给出简洁的解释而失分。考官会给精确的表述加分，例如“有证据拒绝原假设”或“期望数量为 12”，而不是只给出一个数字。
2. Paper Structure and Command Words | 试卷结构与指令词
The June 2019 Unit 3 Statistics 1B paper typically lasts 1 hour 30 minutes and carries 75 marks. You are allowed to use a calculator, and a formula booklet is normally provided by AQA.
2019 年 6 月的 Unit 3 Statistics 1B 试卷通常用时 1 小时 30 分钟，满分 75 分。你可以使用计算器，AQA 通常会提供公式手册。
| Section | Typical Content | Marks Range |
|---|---|---|
| A | Data presentation and summary | 15-20 |
| B | Probability and Venn diagrams | 15-20 |
| C | Binomial and normal distributions | 25-30 |
| D | Correlation and regression | 10-15 |
Command words such as ‘state’, ‘calculate’, ‘interpret’, and ‘suggest’ tell you exactly how much working is needed. ‘State’ requires a short answer, while ‘interpret’ requires a sentence linked to the original context.
像 state、calculate、interpret 和 suggest 这样的指令词会告诉你需要写出多少解题步骤。state 要求简短回答，而 interpret 则要求结合原始情境写出完整的句子。
In the June 2019 paper, questions often combined two or three topics. For example, a question might ask you to draw a box plot, then calculate a probability from grouped data, then comment on the shape of the distribution.
在 2019 年 6 月的试卷中，题目常常结合两到三个主题。例如，一道题可能要求你绘制箱线图，然后根据分组数据计算概率，再对分布形状作出评论。
3. Topic 1: Numerical Measures and Data Representation | 主题一：数值度量与数据表示
You must be confident finding the mean, median, mode, range, interquartile range, variance, and standard deviation from both raw and grouped data.
你必须能够熟练地从原始数据和分组数据中求平均数、中位数、众数、极差、四分位距、方差和标准差。
For raw data, the mean is x̄ = Σx ÷ n. For grouped data, use midpoints: x̄ = Σfx ÷ Σf. The variance is s² = Σ(x − x̄)² ÷ (n − 1) for a sample, or s² = Σx² ÷ n − x̄² for raw data.
对于原始数据，平均数公式为 x̄ = Σx ÷ n。对于分组数据，使用组中值：x̄ = Σfx ÷ Σf。样本方差为 s² = Σ(x − x̄)² ÷ (n − 1)，原始数据的方差也可写为 s² = Σx² ÷ n − x̄²。
s = √[ Σ(x − x̄)² ÷ (n − 1) ]
When a box plot is required, remember to use the five-number summary: minimum, lower quartile Q₁, median Q₂, upper quartile Q₃, and maximum. Outliers are usually values below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR.
当需要绘制箱线图时，请记住使用五数综合法：最小值、下四分位数 Q₁、中位数 Q₂、上四分位数 Q₃ 和最大值。异常值通常定义为小于 Q₁ − 1.5 × IQR 或大于 Q₃ + 1.5 × IQR 的数值。
- Mean is affected by extreme values, but the median is resistant.
- Use the median and IQR for skewed data; use the mean and standard deviation for roughly symmetric data.
- Label box plots fully and give a scale.
- 平均数受极端值影响，而中位数具有抗性。
- 对于偏斜数据使用中位数和四分位距；对于大致对称的数据使用平均数和标准差。
- 箱线图要完整标注并给出刻度。
4. Topic 2: Probability Rules and Venn Diagrams | 主题二：概率规则与韦恩图
Probability questions in Unit 3 often involve tree diagrams, Venn diagrams, conditional probability, and the addition rule. You must be precise with overlap and complement events.
Unit 3 中的概率题常涉及树状图、韦恩图、条件概率和加法规则。你必须准确处理重叠事件和补事件。
The addition rule is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events, the intersection is zero. Conditional probability is P(A | B) = P(A ∩ B) ÷ P(B).
加法规则为 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。对于互斥事件，交集为零。条件概率公式为 P(A | B) = P(A ∩ B) ÷ P(B)。
P(A | B) = P(A ∩ B) / P(B)
In June 2019 style questions, the context may be something like ‘a student is selected at random from a group studying Biology, Chemistry, or both’. Draw the Venn diagram first, fill in the known values, and then use the totals to find missing entries.
在 2019 年 6 月风格的题目中，情境可能是“从学习生物、化学或两者都学的学生中随机抽取一名学生”。先画出韦恩图，填入已知数值，然后利用总数找到缺失项。
Always check that your Venn diagram totals match the sample space. If the probabilities do not sum to 1, you have either missed a region or double-counted an intersection.
务必检查韦恩图中的总数是否与样本空间一致。如果概率之和不为 1，你要么遗漏了某个区域，要么重复计算了某个交集。
5. Topic 3: Discrete Random Variables and Expectation | 主题三：离散随机变量与期望
A discrete random variable takes distinct values, each with a probability. The sum of all probabilities must equal 1, and each probability must be between 0 and 1 inclusive.
离散随机变量取不连续的值，每个值都有对应的概率。所有概率之和必须等于 1，且每个概率必须在 0 到 1 之间（含端点）。
The expected value E(X) is the long-run average, calculated as ΣxP(X = x). The variance Var(X) = E(X²) − [E(X)]², where E(X²) = Σx²P(X = x).
期望值 E(X) 是长期平均值，计算公式为 ΣxP(X = x)。方差 Var(X) = E(X²) − [E(X)]²，其中 E(X²) = Σx²P(X = x)。
E(X) = Σ x P(X = x)
Exam questions often ask you to find an unknown probability k, then calculate the mean and standard deviation, and finally write a sentence interpreting the expected value. Do not just give a number; say something like ‘on average, the number of defective items per box is 1.4’.
题目经常要求你先求出未知概率 k，然后计算平均数和标准差，最后写一句话解释期望值。不要只给出数字；要写出类似“平均而言，每箱缺陷品数量为 1.4”这样的句子。
Be careful with units and context. If the variable is money in pounds, then E(X) and standard deviation are also in pounds. Mixing units or omitting units can cost a mark.
注意单位和情境。如果变量是英镑金额，那么 E(X) 和标准差的单位也是英镑。混淆单位或遗漏单位都可能扣分。
6. Topic 4: Binomial Distribution | 主题四：二项分布
The binomial distribution applies when there are a fixed number of independent trials, each with the same probability of success. If X ~ B(n, p), then
当试验次数固定、各次试验独立且每次试验成功概率相同时，适用二项分布。如果 X ~ B(n, p)，那么
P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ
Use your calculator to find binomial probabilities quickly. For example, if X ~ B(20, 0.4), then P(X = 8) can be found using the binomial probability function, and P(X ≥ 15) = 1 − P(X ≤ 14).
使用计算器可以快速求出二项分布概率。例如，若 X ~ B(20, 0.4)，则 P(X = 8) 可通过二项概率函数求出，P(X ≥ 15) = 1 − P(X ≤ 14)。
You must be able to identify a binomial distribution from a word problem. Look for phrases like ‘a random sample of n items’ and ‘the probability that an item is defective is p’.
你必须能够从文字题中识别出二项分布。注意像“随机抽取 n 件物品”和“一件物品有缺陷的概率为 p”这样的短语。
Always state the distribution clearly, for example X ~ B(12, 0.25),
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