Statistical Report Writing Framework and Model Answers for Year 13 CAIE | CAIE 13年级统计报告写作框架与范文

📚 Statistical Report Writing Framework and Model Answers for Year 13 CAIE | CAIE 13年级统计报告写作框架与范文

Writing a coherent statistical report is a crucial skill for Year 13 CAIE Statistics students. Whether you are answering a long-form exam question or completing an investigative project, you must present your analysis in a structured, logical way that demonstrates deep understanding of the statistical enquiry cycle.

撰写条理清晰的统计报告是CAIE 13年级统计学学生的一项重要技能。无论是回答考试中的长篇问题还是完成调查项目，你都必须以一种结构清晰、逻辑严密的方式呈现分析，体现出对统计探究周期的深刻理解。

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1. Understanding the Assessment Context | 理解评估背景

In CAIE A Level Mathematics (9709), Paper 6 (Probability & Statistics 2) often includes questions that require you to carry out a hypothesis test from start to finish, and to write a conclusion in context. This mirrors the structure of a short statistical report, and communication marks are awarded for clear statistical reasoning.

在CAIE A Level Mathematics (9709) 试卷6（概率与统计2）中，常有要求从头到尾进行假设检验并在情境下写出结论的题目。这相当于一篇小型统计报告的结构，而且清晰的统计推理能力可以获得交流分。

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2. The Statistical Enquiry Cycle as a Blueprint | 以统计探究周期为蓝图

The statistical enquiry cycle—Problem, Plan, Data, Analysis, Conclusion—provides a natural framework for any statistical report. Following this cycle ensures that your writing flows logically from the initial question to the final decision, with every step justified.

统计探究周期——问题、计划、数据、分析、结论——为任何统计报告提供了天然的框架。遵循这个周期可确保你的行文从初始问题到最终决定逻辑通顺，每一步都有理有据。

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3. Overall Structure of a Statistical Report | 统计报告的整体结构

Your report should be divided into clear sections matching the enquiry cycle. The table below summarises the key components and gives brief guidance in both English and Chinese.

你的报告应划分为与探究周期相匹配的清晰节段。下表概括了关键组成部分，并给出中英双语简要指导。

Section	English Guidance	中文指导
1. Introduction	State the problem, the population, and the parameter of interest.	陈述问题、总体以及感兴趣的参数。
2. Hypotheses	Define H₀ and H₁ precisely, noting the tail direction and significance level α.	精确定义H₀和H₁，注明尾部方向和显著性水平α。
3. Data & Assumptions	Describe sample size, sampling method, and check assumptions (normality, independence).	描述样本量、抽样方法，并检验假设（正态性、独立性）。
4. Summary Statistics	Present x̄, s, n, and any relevant graph. Comment on shape and outliers.	呈现x̄、s、n以及相关图形。评论分布形状和异常值。
5. Inferential Method	Justify the chosen test (e.g., one-sample t-test, χ² test).	论证所选检验（如单样本t检验、χ²检验）的合理性。
6. Test Statistic & p-value	Calculate t, χ², or z; determine the p-value or compare with critical value.	计算t、χ²或z值；确定p值或与临界值比较。
7. Decision	Reject H₀ if p < α, otherwise do not reject H₀.	若p < α则拒绝H₀，否则不拒绝H₀。
8. Conclusion in Context	Interpret the decision using the original problem wording.	用原始问题措辞解释该决定。
9. Evaluation	Discuss limitations, possible errors, and suggestions for improvement.	讨论局限性、可能的错误以及改进建议。

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4. Crafting the Problem Statement and Hypotheses | 撰写问题陈述与假设

Always begin with a clear statement of the problem and the population parameter of interest. If performing a hypothesis test, define the null hypothesis H₀ and the alternative hypothesis H₁ precisely, using proper notation, and state whether the test is one-tailed or two-tailed.

始终以清晰陈述问题和感兴趣的总体参数开始。如果进行假设检验，要精确定义原假设H₀和备择假设H₁，使用正确的符号，并说明是单尾还是双尾检验。

For example: ‘We are testing whether the mean weight μ of packets filled by a machine is 500 g. H₀: μ = 500, H₁: μ ≠ 500, a two-tailed test at the 5% significance level.’

例如：“我们检验机器包装的平均重量μ是否为500克。H₀: μ = 500，H₁: μ ≠ 500，双尾检验，显著性水平5%。”

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5. Data Collection and Pre-processing | 数据收集与预处理

Describe the sample clearly: size n, how the data were collected, and whether any observations have been removed as outliers. If the data are given in the exam, state the source and any necessary background.

清晰地描述样本：样本量n、数据的收集方式、是否有任何观测值作为异常值被移除。如果数据是考试给定，说明来源和任何必要背景。

Check the assumptions required for your test. For a t-test, verify that the sample is random, the data are approximately normal (with no extreme skew), and observations are independent.

检查检验所需的假设。对于t检验，要验证样本是随机的、数据近似正态（无极端偏斜）且观测值独立。

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6. Descriptive Statistics and Visualisation | 描述性统计与可视化

Report the sample mean x̄, sample standard deviation s, and the sample size n. A well-chosen graph, such as a box plot or histogram, strengthens your report. Always comment on the shape, centre, spread, and any unusual features.

报告样本均值x̄、样本标准差s和样本量n。一个精心选择的图形，如箱形图或直方图，能增强报告说服力。务必就分布形状、中心、散布和任何异常特征作出评论。

For example: ‘The histogram of the 25 data points appears roughly symmetric with no obvious outliers, supporting the assumption of approximate normality.’

例如：“25个数据点的直方图大致对称，无明显异常值，支持近似正态性的假设。”

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7. Choosing the Right Inferential Method | 选择合适的推断方法

Link the choice of test to the data type and assumptions. For a normal population with unknown population standard deviation σ, use a one-sample t-test. For categorical data arranged in a contingency table, use the χ² test of independence.

将检验的选择与数据类型和假设联系起来。对于总体标准差σ未知的正态总体，使用单样本t检验。对于列联表中的分类数据，使用χ²独立性检验。

Justify your choice by mentioning the parameter, the sample size, and whether the population variance is known. Your reasoning shows the examiner that you understand why a specific test is appropriate.

通过提及参数、样本量以及总体方差是否已知来论证选择。你的推理向考官表明你理解为何特定检验是合适的。

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8. Conducting the Hypothesis Test Step by Step | 逐步进行假设检验

Present the calculation clearly:

清晰地展示计算过程：

Step 1: Set up H₀ and H₁. Step 2: Calculate the test statistic, for example t = (x̄ – μ₀) / (s / √n). Step 3: Determine the degrees of freedom (n – 1). Step 4: Find the p-value using tables or calculator, or find the critical value. Step 5: Compare p with α, or compare the test statistic with the critical value.

第1步：设立H₀和H₁。第2步：计算检验统计量，例如 t = (x̄ – μ₀) / (s / √n)。第3步：确定自由度 (n – 1)。第4步：使用表格或计算器求得p值，或求出临界值。第5步：将p与α比较，或将检验统计量与临界值比较。

Always show the values you substituted. If using a calculator, note the function used (e.g., ‘using the t-test function on a CASIO fx-991EX’).

始终展示代入的数值。若使用计算器，注明所用功能（例如“使用CASIO fx-991EX的t-test功能”）。

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9. Interpreting p-values and Making Decisions | 解读p值与做出决策

Compare the p-value to the significance level α. If p < α, reject H₀; otherwise, do not reject H₀. Remember that a small p-value indicates the observed result is unlikely if H₀ were true.

将p值与显著性水平α比较。如果p < α，拒绝H₀；否则不拒绝H₀。记住，小的p值表示如果H₀为真，观察到的结果不太可能发生。

Always write the decision in the context of the problem, not just ‘reject H₀’. For example: ‘There is sufficient evidence at the 5% level to conclude that the mean packet weight has changed from 500 g.’

始终在问题情境下写出决策，而不仅仅是“拒绝H₀”。例如：“在5%水平下有足够证据表明包装平均重量已从500克发生改变。”

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10. Writing Conclusions and Discussing Limitations | 撰写结论与讨论局限性

A good conclusion links back to the original research question, states what the data suggest, and acknowledges the significance level. It also reflects on the quality of the evidence.

一个好的结论会回扣原始研究问题，陈述数据所表明的结果，并承认显著性水平。它还会反思证据的质量。

Discuss any assumptions that may not be perfectly met, such as non-random sampling or a small sample size, and how this could affect the validity of the conclusion. Mention Type I and Type II errors where relevant.

讨论任何可能未被完全满足的假设，例如非随机抽样或小样本量，以及这如何影响结论的效度。在相关处提及第I类和第II类错误。

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11. Model Answer 1: One-Sample t-Test Report | 范文1：单样本t检验报告

Scenario: A sample of 25 students has a mean score of 72 with a standard deviation of 8. Test the claim that the population mean is less than 75 at the 5% significance level.

情境: 一个包含25名学生的样本，平均成绩为72，标准差为8。在5%显著性水平下检验总体均值小于75的假设。

Report Introduction: We investigate whether the true mean score μ of all students is below 75. Let μ be the population mean score. H₀: μ = 75, H₁: μ < 75; this is a left-tailed test with α = 0.05.

报告引言：我们研究所有学生的真实平均成绩μ是否低于75。令μ为总体平均成绩。H₀: μ = 75, H₁: μ < 75；这是一个左尾检验，α = 0.05。

Assumptions and Data: A random sample of n = 25 is assumed. The sample mean x̄ = 72, sample standard deviation s = 8. A box plot shows a roughly symmetric distribution with no outliers, justifying the use of a t-test.

假设与数据：假设为n = 25的随机样本。样本均值x̄ = 72，样本标准差s = 8。箱形图显示出大致对称的分布且无异常值，这为使用t检验提供了依据。

Test Statistic: t = (72 – 75) / (8 / √25) = -3 / 1.6 = -1.875. Degrees of freedom = 24.

检验统计量：t = (72 – 75) / (8 / √25) = -3 / 1.6 = -1.875。自由度 = 24。

p-value and Decision: Using a t-table or calculator, the one-tailed p-value for t = -1.875 with df = 24 is approximately 0.036. Since 0.036 < 0.05, we reject H₀. The test statistic also lies in the critical region (t < -1.711).

p值与决策：使用t表或计算器，对于df=24，t = -1.875的单尾p值约为0.036。由于0.036 < 0.05，我们拒绝H₀。检验统计量也落在拒绝域内（t < -1.711）。

Conclusion in Context: There is sufficient evidence at the 5% significance level to support the claim that the true mean score is less than 75. However, the sample size is moderate, so we should be cautious about generalisation.

情境结论：在5%显著性水平下，有足够证据支持真实平均成绩小于75的说法。但样本量不大，因此我们在推广时需谨慎。

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12. Model Answer 2: Chi-Squared Test of Independence Report | 范文2：卡方独立性检验报告

Scenario: A survey categorises 200 people by gender (Male/Female) and preference for a new product (Like/Dislike/Neutral). The observed frequencies are given. Test whether gender and preference are independent at the 1% significance level.

情境: 一项调查将200人按性别（男/女）和对新产品的偏好（喜欢/不喜欢/中立）分类。给出了观测频数。在1%显著性水平下检验性别与偏好是否独立。

Report Introduction: The aim is to test if there is an association between gender and product preference. H₀: Gender and preference are independent, H₁: They are not independent. Significance level α = 0.01.

报告引言：目的是检验性别与产品偏好之间是否存在关联。H₀：性别与偏好独立，H₁：它们不独立。显著性水平α = 0.01。

Observed and Expected Frequencies: We compute the expected frequencies using (row total × column total)/grand total. All expected values are greater than 5, satisfying the test assumption. The degrees of freedom are (2‑1)(3‑1) = 2.

观测与期望频数：我们使用（行合计 ×

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