Pre-U CAIE Statistics: Hands-On Case Study Practice | Pre-U CAIE 统计：案例分析实战演练

📚 Pre-U CAIE Statistics: Hands-On Case Study Practice | Pre-U CAIE 统计：案例分析实战演练

This article presents a fully worked case study in Pre-U CAIE Statistics, guiding you through the complete analytical process from formulating research questions to reporting conclusions. We explore a genuine scenario comparing three teaching methods for an examination, using real statistical techniques such as descriptive analysis, ANOVA, post hoc tests, effect sizes, regression modelling, ANCOVA and non‑parametric alternatives.

本文为您呈现一个完整的 Pre-U CAIE 统计案例分析，带您亲历从研究问题提出到结论报告的全分析流程。我们基于一个真实的教学方法比较场景，综合运用描述性统计、方差分析、事后检验、效应量、回归建模、协方差分析及非参数替代方法，步步为营，筑牢实战能力。

1. Introduction and Research Question | 引言与研究问题

An international school aims to improve performance in its Pre-U Statistics module. Three teaching strategies were trialled: Group A received traditional lectures, Group B engaged in structured group discussions, and Group C used an interactive online platform. The outcome variable is the score (out of 100) on a common end‑of‑module test. The driving question is whether the mean scores differ significantly across the three methods.

一所国际学校希望提升其 Pre-U 统计模块的成绩，试行了三种教学策略：A 组采用传统讲授，B 组开展结构化小组讨论，C 组使用互动式在线平台。结果变量是模块结束时统一测试的成绩（满分 100）。核心研究问题是，三种教学方法的平均成绩是否存在显著差异。

The null hypothesis states H₀: μₐ = μₑ = μₒ, where μₐ, μₑ and μₒ are the population mean scores for groups A, B and C respectively. The alternative is H₁: at least one mean differs. A significance level of α = 0.05 is adopted throughout.

原假设为 H₀: μₐ = μₑ = μₒ，其中 μₐ、μₑ 和 μₒ 分别为 A、B、C 三组的总体均值。备择假设 H₁：至少有一组均值不同。全文统一采用显著性水平 α = 0.05。

2. Data Description and Summary Statistics | 数据描述与汇总统计

Thirty students were randomly assigned, ten per group. A snippet of the raw data appears below.

三十名学生被随机分配，每组 10 人。原始数据节选如下表所示。

The full dataset yields the following descriptive statistics.

完整数据集汇总得到以下描述统计量。

Group C displays the highest average (89.3), while Group A records the lowest (72.2). Standard deviations are small and similar, suggesting consistent within‑group performance.

C 组平均成绩最高（89.3），A 组最低（72.2）。各组标准差均较小且相近，表明组内表现较为一致。

3. Visual Exploration: Boxplots and Beyond | 可视化探索：箱线图及其他

Side‑by‑side boxplots (not displayed here) would reveal that Group A’s distribution sits noticeably lower, Group B’s is intermediate, and Group C’s is highest with almost no overlap. The interquartile ranges are compact, and no outliers are present. This purely graphical check already hints at substantial differences among the three teaching strategies.

并列箱线图（本文未展示）可清晰看出，A 组分布明显偏低，B 组居中，C 组最高且几乎无重叠。四分位距紧凑，未发现离群值。这一纯图形的初步审视已暗示三种教学策略间存在实质性差异。

Exploring the shape of each distribution is important before formal testing. All three groups appear roughly symmetric, supporting the use of parametric procedures, though formal assumption checks remain essential.

正式检验前审视各分布形态十分重要。三组数据均大致对称，这支持使用参数方法，但正式的前提条件检验仍不可省略。

4. Assumption Checking for ANOVA | 方差分析的前提条件检验

One‑way ANOVA assumes independence, normality of residuals (or normality within each group for modest samples) and homogeneity of variances. Random assignment ensures independence.

单因素方差分析要求独立性、残差正态性（或在中等样本下各组内正态性）以及方差齐性。随机分组保证了独立性。

Normality was assessed with Shapiro‑Wilk tests. For Group A, W = 0.962, p = 0.812; Group B, W = 0.945, p = 0.613; Group C, W = 0.971, p = 0.898. All p‑values exceed 0.05, so we do not reject the null hypothesis of normality. A normal Q‑Q plot (not shown) confirms that points lie near the diagonal.

采用 Shapiro‑Wilk 检验评估正态性。A 组 W = 0.962，p = 0.812；B 组 W = 0.945，p = 0.613；C 组 W = 0.971，p = 0.898。p 值均大于 0.05，故不拒绝正态性原假设。正态 Q‑Q 图（未展示）亦印证散点紧贴对角线。

Levene’s test for equality of variances gave F(2, 27) = 0.21, p = 0.815, indicating no significant departure from variance homogeneity. Consequently, the classical one‑way ANOVA is appropriate.

Levene 方差齐性检验得 F(2, 27) = 0.21，p = 0.815，表明未能拒绝方差齐性的原假设。因此经典的单因素方差分析是合适的。

5. One‑Way ANOVA: Testing Differences in Means | 单因素方差分析：检验均值差异

We partition total variability into between‑group and within‑group components. The ANOVA table is shown

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