📚 Pre-U WJEC Statistics: Case Study Practical Exercises | WJEC Pre-U 统计：案例分析实战演练
This article provides a practical, step-by-step walkthrough of a complete statistical case study tailored to the WJEC Pre-U Statistics specification. By engaging with a realistic dataset, you will reinforce your ability to plan an investigation, explore data, apply inference techniques, interpret outcomes, and produce a coherent statistical report.
本文针对WJEC Pre-U统计课程，带领你逐步完成一个完整的统计案例实战演练。通过分析一个基于真实情境的数据集，你将巩固如何设计调查、探索数据、应用推断方法、解读结果并撰写条理清晰的统计报告。
1. Understanding the Case Study Context | 理解案例背景
A secondary school plans to evaluate whether a new interactive teaching strategy (Method B) raises Year 12 Mathematics test scores compared with the traditional lecture style (Method A). Sixty students were randomly assigned to the two groups, 30 per method. For each student the following variables were recorded: TeachingMethod (A or B), Gender, StudyHours (independent study per week), TestScore (out of 100 marks), and FinalGrade (A*, A, B, C, D, E).
一所中学计划评估新的互动式教学策略（方法B）是否比传统讲授式（方法A）更能提高12年级数学考试成绩。60名学生被随机分配到两组，每组30人。每位学生记录以下变量：教学方法（A或B）、性别、每周自主学习小时数、考试成绩（百分制）以及最终等级（A*、A、B、C、D、E）。
The primary research questions are: Is there a significant difference in mean test scores between the two teaching methods? Does an association exist between teaching method and final grade? And can study hours be used to predict test score? These questions will guide the analysis.
主要研究问题是：两种教学方法下的平均考试成绩是否存在显著差异？教学方法与最终等级之间是否存在关联？以及学习小时数能否用于预测考试成绩？这些问题将引导后续分析。
2. Initial Data Exploration | 数据初探
Before formal testing, we examine the data using side-by-side box plots and histograms. A box plot of TestScore by TeachingMethod reveals the median, spread, and any outliers. From the simulated data, Method B shows a slightly higher median (72 vs 65), with overlapping interquartile ranges.
在进行正式检验前，我们通过并排箱线图和直方图观察数据。按教学方法分组的考试成绩箱线图显示了中位数、分布范围以及可能的异常值。模拟数据显示B组中位数略高（72 vs 65），四分位距有重叠。
Histograms for each group suggest roughly symmetric distributions, though Method A has a slight left skew. This visual check helps us assess whether parametric test assumptions might be tenable.
每组的直方图表明分布大致对称，但方法A略有左偏。这样的可视化检查有助于我们判断参数检验的假设是否可能成立。
Summary statistics are computed: for Method A the mean is 64.3 with standard deviation 11.5; for Method B the mean is 70.1 with standard deviation 10.8. The apparent difference of about 5.8 points prompts formal inference.
计算汇总统计量：方法A的均值为64.3，标准差11.5；方法B的均值为70.1，标准差10.8。大约5.8分的直观差异促使我们进行正式推断。
3. Descriptive Statistics and Assumptions | 描述性统计与假设条件
We report group means, medians, standard deviations and IQRs. The effect size (Cohen’s d) is approximately (70.1 – 64.3) / √[(11.5² + 10.8²)/2] ≈ 0.52, suggesting a medium effect. Normality is checked via Q‑Q plots and the Shapiro‑Wilk test; both groups give p‑values > 0.10, so the assumption of normality is not violated.
我们报告各组均值、中位数、标准差和四分位距。效应量（Cohen’s d）约为 (70.1 – 64.3) / √[(11.5² + 10.8²)/2] ≈ 0.52，提示中等效应。通过Q-Q图和Shapiro‑Wilk检验检查正态性；两组p值均大于0.10，因此正态性假设未被拒绝。
Homogeneity of variance is assessed with Levene’s test. The test yields p = 0.34, so we assume equal variances for the two‑sample t‑test. These checks are essential before trusting the inference.
方差齐性用Levene检验评估。检验得到p = 0.34，因此在双样本t检验中我们假定方差相等。这些检查是信任推断的必要前提。
4. Two‑Sample t‑Test | 双样本t检验
We test H₀: μ_A = μ_B against H₁: μ_A ≠ μ_B at the 5% significance level. With equal variances assumed, the test statistic is:
我们在5%显著性水平下检验H₀: μ_A = μ_B 与 H₁: μ_A ≠ μ_B。假定方差相等，检验统计量为：
t = (x̄_B – x̄_A) / (sₚ √(1/n_A + 1/n_B))
where pooled standard deviation sₚ = √[((n_A – 1)s_A² + (n_B – 1)s_B²) / (n_A + n_B – 2)]. Inserting the values yields t ≈ 2.04 with 58 degrees of freedom. The two‑tailed p‑value is approximately 0.046.
其中合并标准差 sₚ = √[((n_A – 1)s_A² + (n_B – 1)s_B²) / (n_A + n_B – 2)]。代入数值得到 t ≈ 2.04，自由度58。双侧p值约为0.046。
Since p < 0.05, we reject H₀ and conclude that there is a statistically significant difference in mean test scores. The evidence suggests that the interactive method is associated with higher scores.
由于p < 0.05，我们拒绝H₀，得出结论：平均考试成绩存在统计显著差异。证据表明互动教学法与更高成绩相关。
5. Confidence Interval for Difference in Means | 均值差的置信区间
A 95% confidence interval for μ_B – μ_A is computed as:
计算μ_B – μ_A的95%置信区间：
(x̄_B – x̄_A) ± t* × sₚ √(1/n_A + 1/n_B)
With t* ≈ 2.002, the interval is (0.12, 11.48). Since the interval does not contain zero, it reinforces the conclusion from the hypothesis test. The practical interpretation is that Method B improves scores, on average, by between about 0.1 and 11.5 marks.
t* ≈ 2.002，得到区间 (0.12, 11.48)。区间不包含零，这进一步支持了假设检验的结论。实际解释为：方法B平均提高约0.1到11.5分。
6. Correlation Analysis | 相关分析
We next explore the relationship between StudyHours and TestScore. The Pearson correlation coefficient r = 0.61 indicates a moderately strong positive linear association. A scatter plot confirms a linear trend with some scatter.
接着我们探索学习小时数与考试成绩的关系。Pearson相关系数 r = 0.61 表明中等偏强的正线性关联。散点图确认了带有一定离散的线性趋势。
To test whether the population correlation ρ is different from zero, we use:
为了检验总体相关系数ρ是否不为零，我们使用：
t = r √(n – 2) / √(1 – r²)
with n = 60. This gives t ≈ 5.91, with a p‑value < 0.001. There is very strong evidence that study hours and test score are linearly related.
n = 60，得到 t ≈ 5.91，p值 < 0.001。有极强的证据表明学习小时数与考试成绩存在线性关系。
7. Linear Regression | 线性回归
To predict TestScore from StudyHours, we fit a simple linear regression: Score = β₀ + β₁×StudyHours + ε. The estimated equation is Score = 38.2 + 4.1×StudyHours. The slope of 4.1 means that each additional study hour per week is associated with an increase of 4.1 marks, on average.
为用学习小时数预测考试成绩，我们拟合简单线性回归：Score = β₀ + β₁×StudyHours + ε。估计方程为 Score = 38.2 + 4.1×StudyHours。斜率4.1表示平均每周每多学习一小时，成绩提高4.1分。
The coefficient of determination R² = 0.372, so about 37% of the variability in test scores is explained by study hours. The t‑test for the slope gives p < 0.001, confirming a significant linear relationship. Residual plots show no obvious patterns, supporting model adequacy.
决定系数R² = 0.372，约37%的考试成绩变异性可被学习小时数解释。对斜率的t检验p < 0.001，证实线性关系显著。残差图无明显模式，支持模型适合性。
8. Chi‑Squared Test of Independence | 卡方独立性检验
We categorise FinalGrade as High (A*, A, B) and Low (C, D, E) to investigate an association with TeachingMethod. The observed frequencies are shown below:
我们将最终等级分为高（A*, A, B）和低（C, D, E），以研究其与教学方法的关联。观察频数如下表：
| High | Low | |
|---|---|---|
| Method A | 12 | 18 |
| Method B | 20 | 10 |
The chi‑squared test statistic is calculated as χ² = Σ (O – E)² / E. With 1 degree of freedom, we obtain χ² ≈ 4.80 and p ≈ 0.028. Rejecting the null hypothesis indicates a significant association between teaching method and grade.
卡方检验统计量 χ² = Σ (O – E)² / E。自由度为1，得到 χ² ≈ 4.80，p ≈ 0.028。拒绝原假设表明教学方法与等级之间存在显著关联。
We also compute Cramér’s V = √(χ² / (n × min(r‑1, c‑1))) = √(4.80/60) ≈ 0.28, suggesting a small‑to‑moderate effect. Thus the interactive method appears to yield a higher proportion of top grades.
我们还计算了Cramér’s V = √(χ² / (n × min(r‑1, c‑1))) = √(4.80/60) ≈ 0.28，提示小到中等效应。因此互动式教学法似乎带来了更高比例的高分等级。
9. Integrated Interpretation and Discussion | 综合分析解读与讨论
Bringing the results together, all three research questions receive affirmative answers: the t‑test detects a significant mean difference favouring Method B, the confidence interval excludes zero, the correlation and regression show that StudyHours is a meaningful predictor of TestScore, and the chi‑squared test reveals an association between teaching method and final grade categories.
综合各项结果，三个研究问题均得到肯定答案：t检验检测到支持方法B的显著均值差异，置信区间不包含零，相关和回归表明学习小时数是考试成绩有意义的预测变量，卡方检验显示教学方法与最终等级分类之间存在关联。
However, the observational nature of the data prevents causal claims. Confounding factors such as student motivation, prior attainment, or teacher quality could influence both StudyHours and TestScore. In a more advanced analysis, one could control for such covariates using multiple regression or ANCOVA.
然而，数据的观察性质使我们不能断言因果关系。混淆因素如学生动力、先前水平或教师质量可能同时影响学习小时数和考试成绩。在更深入的分析中，可以通过多元回归或协方差分析控制这些协变量。
Effect sizes for all tests are moderate, so practical importance should be discussed alongside statistical significance. The school might consider the cost and feasibility of implementing Method B given these results.
所有检验的效应量均为中等，因此在讨论统计显著性的同时应兼顾实际重要性。学校可根据这些结果考虑实施方法B的成本和可行性。
10. Reporting and Conclusion | 报告撰写与结论
A structured statistical report should include: introduction with research questions, methodology (sampling, variables, tests chosen), results (descriptive statistics, test outcomes with p‑values and confidence intervals, effect sizes), and a conclusion that links back to the objectives. Avoid over‑selling the findings and note limitations clearly.
一份结构清晰的统计报告应包括：研究问题引言、方法（抽样、变量、所选检验）、结果（描述性统计、检验结果及p值和置信区间、效应量），以及联系研究目标的结论。避免过度夸大结果，并清晰说明局限性。
In our case, the report would state that there is evidence of a positive effect of the interactive method, that study hours are positively correlated with performance, and that the teaching method influences grade distributions. All conclusions are supported by appropriate statistical tests and confidence intervals.
在本案例中，报告将说明：有证据表明互动教学法有正面效果，学习小时数与成绩正相关，教学方法影响等级分布。所有结论均由适当的统计检验和置信区间支撑。
11. Common Pitfalls and How to Avoid Them | 常见陷阱与规避方法
Pitfall 1: Ignoring test assumptions. Always check normality and equal variance before using a t‑test. If assumptions are violated, consider a Mann‑Whitney U test or data transformation.
陷阱1：忽略检验假设。在使用t检验前必须检查正态性和方差齐性。若假设被违反，可考虑Mann‑Whitney U检验或数据转换。
Pitfall 2: Confusing statistical significance with practical importance. A tiny p‑value does not guarantee a large or meaningful effect. Report effect sizes such as Cohen’s d, r², or Cramér’s V.
陷阱2：混淆统计显著性与实际重要性。极小的p值并不保证效应量大或具有实际意义。应报告效应量，如Cohen’s d、r²或Cramér’s V。
Pitfall 3: Over‑interpreting regression without checking residuals. Always examine residual plots for non‑linear patterns, heteroscedasticity, or outliers. If necessary, try a non‑linear model or robust methods.
陷阱3：未检查残差就过度解释回归结果。务必检查残差图是否存在非线性模式、异方差或异常值。必要时尝试非线性模型或稳健方法。
Pitfall 4: Using chi‑squared with small expected frequencies. Ensure at least 80% of expected counts exceed 5. If not, merge categories or use Fisher’s exact test.
陷阱4：在期望频数过小时使用卡方检验。确保至少80%的期望频数大于5。否则应合并类别或使用Fisher精确检验。
Pitfall 5: Multiple testing without correction. If several hypotheses are tested, adjust significance levels (e.g., Bonferroni) to control the family‑wise error rate.
陷阱5：多重检验未进行校正。若检验多个假设，应调整显著性水平（如Bonferroni校正）以控制整体错误率。
12. Further Practice Suggestions | 拓展练习建议
To deepen your skills, try repeating the analysis with a different dataset: perhaps compare two exam revision techniques, or investigate the relationship between physical activity and pulse rate. Change one variable to a non‑normal distribution and explore non‑parametric alternatives.
为深化技能，请尝试用不同数据集重复分析：例如比较两种复习技巧，或探究体育活动与脉搏率的关系。将一个变量改为非正态分布并探索非参数替代方法。
You could also extend the case by adding a second predictor—such as attendance rate—to build a multiple regression model, or perform a two‑way ANOVA to examine the effect of both teaching method and gender. These extensions will solidify your understanding of Pre‑U statistical inference.
你也可以在案例中加入第二个预测变量——如出勤率——以构建多元回归模型，或执行双因素方差分析来考察教学方法和性别的共同影响。这些拓展将巩固你对Pre‑U统计推断的理解。
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