📚 Year 12 WJEC Statistics Unit Test Mock Paper Walkthrough | Year 12 WJEC 统计单元测试模拟卷解析
This article provides a detailed walkthrough of a typical Year 12 WJEC Statistics unit test mock paper, covering both Unit 1 (Data, Probability and Distributions) and Unit 2 (Normal Distribution, Correlation and Regression). Each question is broken down with step-by-step solutions, key points, and common pitfalls. The explanations are designed to reinforce your understanding of statistical concepts and boost your confidence for the actual examination.
本文详细解析了一份典型的 Year 12 WJEC 统计单元测试模拟卷，涵盖 Unit 1（数据、概率与分布）和 Unit 2（正态分布、相关与回归）。每道题目均配有逐步求解过程、关键知识点和常见易错点。这些解析旨在巩固你对统计概念的理解，并提升你应对真实考试的信心。
1. Overview of the Mock Paper | 模拟卷概述
The mock paper is structured to mirror the style and weighting of the WJEC AS Statistics papers. It contains 10 compulsory questions, mixing short calculations with longer, structured problem-solving tasks. Questions cover sampling methods, data presentation and interpretation, probability, discrete random variables, the binomial distribution, hypothesis testing, the normal distribution, and correlation and regression. The total marks for the paper are 80, with a recommended time of 1 hour 45 minutes. Understanding how marks are allocated helps you manage your time effectively during the real exam.
模拟卷的结构模拟了 WJEC AS 统计试卷的风格和权重。试卷包含 10 道必答题，混合了简短计算和较长的结构化问题解决任务。题目涵盖抽样方法、数据展示与解读、概率、离散随机变量、二项分布、假设检验、正态分布以及相关与回归。试卷总分 80 分，建议用时 1 小时 45 分钟。了解各题分值分配有助于你在真实考试中有效管理时间。
2. Question 1: Sampling Techniques | 第1题：抽样技术
ol survey of 800 students across four year groups. A clear description of how to implement each method is required, along with advantages and disadvantages. For stratified sampling, you must also calculate the number of students to be sampled from each stratum if a sample of 80 is required. The calculation involves finding the proportion of each year group and multiplying by 80. For example, if Year 10 has 240 students, the sample size for that stratum is (240/800)×80 = 24. The mark scheme awards points for mentioning random number generators, sampling frames, and proportional allocation.
本题考察简单随机抽样、系统抽样和分层抽样的知识。题目给出一个关于对一所学校四个年级共 800 名学生进行调查的场景。你需要清晰描述每种方法的实施步骤，并说明其优缺点。对于分层抽样，还需计算在抽取样本量为 80 的情况下，每个分层应抽取的学生人数。计算方法是求出每个年级的比例，再乘以 80。例如，若 Year 10 有 240 名学生，则该分层的样本量应为 (240/800)×80 = 24。评分方案会就提及随机数生成器、抽样框架和按比例分配等要点给分。
3. Question 2: Histograms and Data Presentation | 第2题：直方图与数据展示
lass widths. First, calculate frequency density for each class using the formula frequency density = frequency ÷ class width. Then plot bars with the class interval on the horizontal axis and frequency density on the vertical axis. Ensure the area of each bar is proportional to the frequency. The question also asks you to estimate the median and interquartile range from the histogram, using linear interpolation within the relevant class intervals. These estimates demonstrate your ability to extract summary measures from graphical displays. Remember that the area under the histogram represents total frequency, and careful reading of scales is essential.
题目给出了一张包含 50 名自行车手完成试骑所用时间（精确到分钟）的分组频数表。你需要绘制一个具有不等组距的直方图。首先，使用频数密度 = 频数 ÷ 组距的公式计算每组的频数密度。然后，以组区间为横轴、频数密度为纵轴绘制矩形条。务必确保每个矩形的面积与频数成正比。该题还要求你通过直方图估算中位数和四分位距，这需要在相关组区间内使用线性插值法。这些估算值体现了你从图形展示中提取汇总度量的能力。请记住，直方图下的面积代表总频数，准确读取刻度至关重要。
4. Question 4: Measures of Central Tendency and Dispersion | 第3题：集中趋势与离散度量
e, and standard deviation. The WJEC specification expects you to use the formula for sample standard deviation: s = √[Σ(x - x̄ squared deviations. A comparison of the mean and median can indicate skewness in the data – iate when outliers are present; the interquartile range is resistant to outliers, while the range and standard deviation are not.
题目给出了一个包含 12 天日降雨量（单位：毫米）的小型数据集，要求你计算平均数、中位数、众数、全距、四分位距和标准差。WJEC 考试大纲要求使用样本标准差公式：s = √[Σ(x - x̄)²/(n-1)]。需清晰展示所有步骤，包括离差平方的计算。通过比较平均数和中位数可以判断数据的偏态——如果平均数大于中位数，则分布呈正偏态。你还需要说明当存在异常值时，哪种离散度量更为合适；四分位距对异常值具有抗性，而全距和标准差则不具备。
5. Question 5: Probability Rules | 第4题：概率法则
ay be used to represent events A and B from a sample space. You are asked to calculate P(A ∪ B), P(A ∩ B' rmine whether events are mutually exclusive or independent. For independence, check if P(A ∩ × P(B). The question also includes a conditional probability part, where you must use P(A|B) = P(A ∩ n and the final answer. Justifying independence with a numerical comparison is essential for full credit.
本题要求应用概率的加法和乘法法则。可以利用维恩图或列联表来表示样本空间中的事件 A 和 B。题目要求计算 P(A ∪ B)、P(A ∩ B')，并判断事件是否互斥或独立。对于独立性，需检验是否满足 P(A ∩ B) = P(A) × P(B)。题目还包含一个条件概率部分，此时你必须使用 P(A|B) = P(A ∩ B)/P(B)。应尽可能化简分数。评分方案通常要求明确写出所用公式，然后代入数值并给出最终答案。通过数值比较来证明独立性是获得满分的关键。
6. Question 6: Tree Diagrams and Conditional Probability | 第5题：树图与条件概率
A typical problem involves bags containing coloured counters, with two successive selections described. You are asked to draw a fully labelled probability tree diagram, showing probabilities on each branch. Remember that probabilities on the second set of branches are conditional on the outcome of the first selection. The question then requires you to calculate the probability of specific combined events, such as both counters being the same colour, using the multiplication and addition rules along the branches. Finally, you are usually asked to find a reverse conditional probability like P(first counter was red | second counter is blue). This is solved by applying the formula for conditional probability or using Bayes' theorem, dividing the probability of the relevant path by the total probability of the given condition. Show your method clearly, as method marks are generous.
典型题目涉及装有彩色筹码的袋子，并描述了两次连续的抽取。你需要绘制一个完整标注的概率树图，在每根树枝上标出概率。请记住，第二层树枝上的概率是在第一次抽取结果的条件下成立的条件概率。题目随后要求你运用树枝上的乘法和加法法则，计算特定组合事件的概率，例如两个筹码颜色相同的概率。最后，通常会要求你求一个反向条件概率，如 P(第一个筹码是红色 | 第二个筹码是蓝色)。这需要应用条件概率公式或贝叶斯定理来求解，即用相关路径的概率除以给定条件的总概率。清晰展示你的解法，因为方法分通常给得很大方。
7. Question 7: Discrete Random Variables | 第6题：离散随机变量
terms of constant k. First, use the fact that ΣP(X = x) = 1 to find k. Then calculate the expected value E(X) = Σ [x · P(X = x)] and the variance Var(X) = E(X²) - [E(X)]². The question may also ask for the standard deviation and the expected value of a linear transformation, such as E(5X - 3) = 5E(X) - 3. Showing all intermediate columns (x·P(X=x) and x²·P(X=x)) in a table is a safe way to minimise arithmetic errors and gain method marks. Units are not required for probability, but numerical answers should be given as exact fractions or decimals to three significant figures.
题目给出离散随机变量 X 的概率分布表，其取值 x = 0, 1, 2, 3，对应的概率 P(X = x) 用常数 k 表示。首先，利用 ΣP(X = x) = 1 求出 k。然后计算期望值 E(X) = Σ [x · P(X = x)] 和方差 Var(X) = E(X²) - [E(X)]²。题目可能还会要求计算标准差以及线性变换的期望值，如 E(5X - 3) = 5E(X) - 3。在表格中列出所有中间计算列（x·P(X=x) 和 x²·P(X=x)）是一种稳妥的方法，可以减少算术错误并获得方法分。概率问题不需要单位，但数值答案应以精确分数或保留三位有效数字的小数形式给出。
8. Question 8: Binomial Distribution | 第7题：二项分布
the same probability of success, for example, a manufacturer producing components with a 5% defect rate, and a random sample of 20 is taken. You must define the random variable: 'Let X be the number of defective components in the sample. X ~ B(20, 0.05)' > 5). For WJEC, you are expected to use the binomial probability formula P(X = r) = ⁿCᵣ pʳ (1-p)ⁿ⁻ʳ s or a calculator that is permitted in the exam. The questions often include a phrase like 'at least' or 'more than' ble to find the mean and variance of a binomial distribution: E(X) = np, Var(X) = np(1-p).
题目描述了一个具有固定独立试验次数且每次成功概率相同的场景，例如，某制造商生产的组件缺陷率为 5%，随机抽取 20 件。你必须定义随机变量：“设 X 为样本中缺陷组件的数量。X ~ B(20, 0.05)”。然后计算诸如 P(X = 2)、P(X ≤ 3) 和 P(X > 5) 等概率。对于 WJEC 考试，要求你使用二项概率公式 P(X = r) = ⁿCᵣ pʳ (1-p)ⁿ⁻ʳ，但也可以使用考试允许的统计表格或计算器。题目中常出现“至少”或“多于”等措辞，这检验你处理累积分布的能力。你还应能求出二项分布的均值和方差：E(X) = np，Var(X) = np(1-p)。
9. Question 9: Hypothesis Testing with Binomial | 第8题：基于二项分布的假设检验
ing a claim about a population proportion. For example, a student claims that more than 30% ded. A sample of 15 pupils is taken, and 7 are left-handed. You are asked to test at the 5% pothesis H₁: p > 0.30. Find the probability of getting the observed result or more extreme, assuming H₀ is true: P(X ≥ 7). Using B(15, 0.30), calculate this probability; if it is less than 0.05, reject H₀. The conclusion must be written in the context of the problem. The WJEC mark scheme penalises omitting the context phrase and not linking the p-value to the significance level. Also remember to state the critical value and critical region if required; for a 5% upper-tail test, find the smallest r such that P(X ≥ r) ≤ 0.05.
本题涉及对总体比例进行假设检验。例如，某学生声称超过 30% 的学生是左撇子。抽取了 15 名学生作为样本，其中有 7 名是左撇子。要求在 5% 的显著性水平下进行检验。步骤如下：定义检验统计量 X 及其在原假设 H₀: p = 0.30 下的分布；陈述备择假设 H₁: p > 0.30。在假定 H₀ 为真的条件下，计算得到观察结果或更极端结果的概率：P(X ≥ 7)。利用 B(15, 0.30) 计算该概率；如果小于 0.05，则拒绝 H₀。结论必须结合问题背景来写。WJEC 评分方案会因遗漏背景语句以及未将 p 值与显著性水平联系起来而扣分。还要记住，如果需要，应说明临界值和拒绝域；对于 5% 的上尾检验，找出最小的 r 使得 P(X ≥ r) ≤ 0.05。
10. Question 10: Normal Distribution | 第9题：正态分布
d the probability that a randomly selected bag weighs less than 1485 g. Standardise to Z = (1485 - 1500)/12 = -1.25, then use the standard normal table to find P(Z < -1.25). (b) Find the percentage of bags with weight between 1490 g and 1515 g. (c) Determine the weight exceeded by only 10% of bags – l curve and shade the required area; it helps avoid sign errors. For the WJEC exam, you are expected to use the percentage points table or statistical tables provided. Express final probabilities as decimals to four decimal places.
一台机器装填面粉袋，其重量服从正态分布，均值 μ = 1500 g，标准差 σ = 12 g。常见问题有：(a) 求随机抽取一袋面粉重量小于 1485 g 的概率。将其标准化：Z = (1485 - 1500)/12 = -1.25，然后查标准正态表得 P(Z < -1.25)。(b) 求重量介于 1490 g 到 1515 g 之间的袋子所占的百分比。(c) 确定只有 10% 的袋子超过的重量值——这是逆正态分布问题。找出对应于尾部概率为 0.10 的 Z 值，然后求解 X = μ + Zσ。务必画出正态分布曲线草图并给所需区域涂上阴影；这有助于避免符号错误。在 WJEC 考试中，要求你使用所给的正态分布表或统计表。最终概率以小数形式给出，保留四位小数。
11. Question 11: Correlation and Regression | 第10题：相关与回归
ent on the correlation. (b) Calculate the product moment correlation coefficient (PMCC) using the formula r = Sxy / √Sxx Syy, where Sxx = Σx² - (Σx)²/n, and similarly for Syy and Sxy rry out a hypothesis test for correlation, with H₀: ρ = 0, H₁: ρ > 0 (one-tailed) at the 5% e. (d) Calculate the equation of the least squares regression line y = a + bx, where b = Sxy / Sxx and a = ȳ - b x̄ n time, commenting on the reliability of the prediction (interpolation vs. extrapolation).
题目给出了八名学生复习时间（x）与测验成绩（y）的双变量数据。你需要：(a) 绘制散点图，并对相关性加以评述。(b) 使用公式 r = Sxy / √Sxx Syy 计算积矩相关系数 (PMCC)，其中 Sxx = Σx² - (Σx)²/n，Syy 和 Sxy 类似计算。题目可能会提供 Σx、Σy、Σx²、Σy²、Σxy 等汇总统计量以减轻计算负担。(c) 对相关性进行假设检验，H₀: ρ = 0，H₁: ρ > 0（单尾），在 5% 水平下使用表中的临界值进行判断。(d) 计算最小二乘回归直线方程 y = a + bx，其中 b = Sxy / Sxx，a = ȳ - b x̄，并解释斜率的含义。(e) 利用该回归直线预测给定复习时间下的测验成绩，并评述预测的可靠性（内插与外推）。
12. Exam Tips and Final Advice | 考试技巧与最终建议
Success in WJEC Statistics comes from a blend of conceptual understanding and careful arithmetic. Always show your working – even if the final answer is wrong, you can gain marks for correct method. Label diagrams fully, and always write your hypotheses with parameters (μ, p, ρ), not sample statistics. When using statistical tables, double-check that you are reading the correct tail or percentage points. Manage your time: spend no more than 2 minutes per mark on average, and leave a few minutes to check your calculations at the end. Replicate the style of these walkthroughs in your own revision by working through past papers under timed conditions and comparing your solutions to the mark schemes. The more you practise, the more confident you will become.
要在 WJEC 统计考试中取得成功，需要将概念理解与细致计算相结合。务必展示你的解题过程——即使最终答案错误，你仍可因正确的方法而得分。完整标注图表，并始终使用参数（μ, p, ρ）而非样本统计量来书写假设。使用统计表时，要反复确认你查阅的是正确的尾部或百分位点。合理管理时间：平均每题每分不超过 2 分钟，并留出几分钟在最后检查计算。在你自主复习时，请模仿本文解析的风格，在规定时间内演练历年真题，并将你的解答与评分方案进行比对。练习越多，你就会越自信。
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