📚 Mastering Case Studies in Statistics | 统计学案例分析实战演练
In the Cambridge Pre-U Statistics syllabus, case studies represent the culmination of your statistical training — they require you to integrate descriptive techniques, probability models, inferential reasoning, and regression analysis into a coherent narrative that answers a real-world question. This article walks through a complete case study from problem definition to final recommendations, demonstrating how to apply the statistical toolkit in a structured, exam-ready manner.
在剑桥Pre-U统计学课程中，案例分析是统计训练的综合体现——它要求你将描述性技术、概率模型、推断推理和回归分析整合成一个连贯的叙述，以回答现实世界的问题。本文从头到尾演示一个完整的案例研究，从问题定义到最终建议，展现如何以结构化、适合考试的方式应用统计工具箱。
1. Understanding the Case Context | 理解案例背景
A mid-sized supermarket chain, FreshMart, has recently launched a digital loyalty app that offers personalised discounts. Management wants to know whether customers who use the app spend significantly more per visit than non-app users, and whether the effect varies by store location. The dataset consists of 500 transactions sampled across 10 stores over one month, with variables: app_user (yes/no), spend_per_visit (GBP), store_location (urban/suburban), age, and basket_size (number of items).
一家中型连锁超市 FreshMart 最近推出了一款提供个性化折扣的数字会员应用。管理层想知道使用该应用的顾客每次消费金额是否显著高于非应用用户，以及这种效应是否因门店位置而异。数据集包含一个月内从10家门店抽取的500笔交易，变量包括：是否应用用户、每次消费金额（英镑）、门店位置（市区/郊区）、年龄和购物篮大小（件数）。
2. Defining the Problem and Objectives | 明确问题与目标
The primary research question is: ‘Is the mean spend per visit of app users greater than that of non-app users?’ This translates into a one-tailed hypothesis test. A secondary objective is to model spend_per_visit using app_user status and store_location, adjusting for age and basket_size, to quantify the app’s impact after controlling for confounders. The pre-analysis plan specifies significance level α = 0.05 for all tests and a 95% confidence level for intervals.
首要研究问题是：“应用用户每次消费的平均金额是否高于非应用用户？”这转化为一个单尾假设检验。次要目标是以是否应用用户和门店位置为自变量，并控制年龄和购物篮大小，对每次消费金额进行建模，从而在控制混杂因素后量化应用的影响。分析计划规定所有检验的显著性水平 α = 0.05，置信区间采用95%置信水平。
3. Data Collection and Preparation | 数据收集与整理
The data were extracted automatically from the point-of-sale system linked to the loyalty programme. Before analysis, we checked for missing values: 3% of transactions lacked an age entry; these were imputed using the median age stratified by store_location. We also identified an outlier in spend_per_visit (a transaction of GBP 987, likely a large catering order) and decided to winsorise it at the 99th percentile to reduce undue influence.
数据通过连接至会员系统的销售点终端自动提取。分析前，我们检查了缺失值：3%的交易缺少年龄信息，这些缺失值按门店位置分层的年龄中位数进行了插补。我们还发现了一个异常值——一笔987英镑的交易（很可能是一笔大额餐饮订单），决定将其在99百分位处缩尾处理，以降低不当影响。
4. Exploratory Data Analysis | 探索性数据分析
We began by examining the distributions. The histogram of spend_per_visit for non-app users was slightly right-skewed with a median of GBP 18.40 and IQR 12.30, while app users showed a median of GBP 23.60 and IQR 14.10. A side-by-side boxplot suggested a visible shift in location and spread. Summary statistics by group are presented in Table 1.
我们首先检查分布情况。非应用用户消费金额的直方图略呈右偏，中位数为18.40英镑，四分位距12.30；应用用户的中位数为23.60英镑，四分位距14.10。并排箱线图显示位置和离散度存在明显差异。分组汇总统计量见表1。
Table 1: Summary statistics
| Group | n | Mean | SD | Min | Q1 | Median | Q3 | Max |
|---|---|---|---|---|---|---|---|---|
| App user | 210 | 25.80 | 11.45 | 4.20 | 17.80 | 23.60 | 31.90 | 68.50 |
| Non-user | 290 | 20.10 | 9.87 | 2.90 | 12.70 | 18.40 | 25.00 | 57.20 |
Before formal testing, we checked normality: Q-Q plots indicated moderate departures in the tails, but with n > 200 in each group the Central Limit Theorem assures robustness of the two-sample t-test.
在正式检验之前，我们检查了正态性：Q-Q图显示尾部存在中度偏离，但由于每组样本量均大于200，中心极限定理保证了双样本t检验的稳健性。
5. Probability Models and Distribution Fitting | 概率模型与分布拟合
To verify that the sampling distribution of the mean difference is approximately normal, we considered the theoretical underpinning. Given the moderately skewed raw data, a log-transformation was applied: ln_spend = ln(spend_per_visit). The transformed variable for both groups passed the Anderson-Darling normality test (p = 0.32 for app users, p = 0.18 for non-users). Thus, we could base inference on the logged data and back-transform results for interpretation.
为验证均值差的抽样分布近似正态，我们考虑了理论基础。鉴于原始数据中度偏斜，进行了对数变换：ln_spend = ln(spend_per_visit)。变换后的变量在两组中均通过Anderson-Darling正态性检验（应用用户p = 0.32；非应用用户p = 0.18）。因此，我们可以基于对数变换数据进行推断，并在解释时反向变换结果。
6. Sampling and Estimation | 抽样与估计
A two-sample t-test on the ln_spend variable gave the following 95% confidence interval for the difference in means (app – non-app): (0.147, 0.306). Back-transforming via exponentiation, we obtained a 95% CI for the ratio of geometric means: exp(0.147) = 1.158 to exp(0.306) = 1.358. This suggests that the typical spend of an app user is between 15.8% and 35.8% higher than that of a non-app user, holding other factors constant.
对ln_spend变量进行双样本t检验，得到均值差（应用用户减非应用用户）的95%置信区间为（0.147, 0.306）。通过指数反向变换，我们得到几何均值比的95%置信区间：exp(0.147) = 1.158 至 exp(0.306) = 1.358。这表明在控制其他因素的情况下，应用用户的标准消费金额比非应用用户高15.8%到35.8%。
7. Hypothesis Testing | 假设检验
We formally tested H₀: μ_app ≤ μ_non-app against H₁: μ_app > μ_non-app using one-tailed independent-samples t-test on the log scale. The test statistic was t = 5.24 with df = 498, yielding p < 0.001. Thus, we reject H₀ and conclude that app users have a significantly higher mean spend. We also performed a non-parametric Mann-Whitney U test as a sensitivity check; the result remained significant (U = 38,240, p < 0.001).
我们在对数尺度上用单尾独立样本t检验正式检验 H₀: μ_app ≤ μ_non-app 与 H₁: μ_app > μ_non-app。检验统计量 t = 5.24，自由度 498，p < 0.001。因此，我们拒绝 H₀，得出结论：应用用户的平均消费金额显著更高。作为敏感性检查，我们还进行了非参数Mann-Whitney U检验，结果仍显著（U = 38,240, p < 0.001）。
8. Regression Analysis and Prediction | 回归分析与预测
To isolate the app effect from potential confounders, we fitted a multiple linear regression model:
ln_spend = β₀ + β₁·app_user + β₂·store_location + β₃·age + β₄·basket_size + ε
where app_user is coded 1 for app user, 0 otherwise, and store_location is coded 1 for urban, 0 for suburban. The estimated coefficients with standard errors are summarised below:
为了从潜在混杂因素中分离出应用效应，我们拟合了多元线性回归模型：
ln_spend = β₀ + β₁·app_user + β₂·store_location + β₃·age + β₄·basket_size + ε
其中 app_user 对应用户编码为1，否则为0；store_location 市区编码为1，郊区为0。系数估计值及标准误总结如下：
| Predictor | Coefficient | SE | t-value | p-value |
|---|---|---|---|---|
| Intercept | 1.829 | 0.095 | 19.25 | <0.001 |
| app_user | 0.238 | 0.041 | 5.80 | <0.001 |
| store_location | 0.172 | 0.038 | 4.53 | <0.001 |
| age | 0.003 | 0.001 | 3.00 | 0.003 |
| basket_size | 0.145 | 0.009 | 16.11 | <0.001 |
The adjusted R² was 0.41, indicating a moderate fit. All predictors were significant. The coefficient for app_user remained positive and significant even after controlling for location, age, and basket size. Exponentiating the coefficient gives a multiplicative effect of e^(0.238) ≈ 1.269, so app users spend approximately 26.9% more on average, ceteris paribus. We also checked residual diagnostics: the residuals were roughly symmetric with no strong patterns, confirming model adequacy.
调整R²为0.41，表明拟合度中等。所有预测变量均显著。即使控制了门店位置、年龄和购物篮大小，app_user的系数仍为正且显著。将系数取指数得到乘数效应 e^(0.238) ≈ 1.269，因此平均而言，在其他条件不变的情况下，应用用户消费额高出约26.9%。我们还检查了残差诊断：残差大致对称且无明显模式，验证了模型适用性。
9. Interpretation and Limitations | 结果解读与局限性
The analysis provides strong evidence that the loyalty app is associated with higher spending. However, observational data cannot prove causation. Users may have been inherently more engaged shoppers. Moreover, the sample only covers one month, so seasonal effects or app novelty may inflate the effect. The urban/suburban classification is coarse; finer geospatial or store-level variables could improve the model. Finally, winsorising outliers and imputing missing ages introduce some uncertainty, although sensitivity analyses showed robust conclusions.
分析提供了有力证据表明会员应用与更高消费相关。然而，观察性数据无法证明因果关系。用户可能本身就是更投入的购物者。此外，样本仅覆盖一个月，季节效应或应用的新鲜感可能夸大了效应。市区/郊区的分类较为粗糙；更精细的地理空间或门店层面变量可以改善模型。最后，对异常值的缩尾处理和对缺失年龄的插补引入了一些不确定性，尽管敏感性分析显示结论稳健。
10. Conclusions and Recommendations | 结论与建议
We conclude with high confidence that app users exhibit higher mean spending, and this association withstands adjustment for key demographic and transactional variables. FreshMart should continue investing in the app and consider strategies to increase adoption, especially in suburban stores where baseline spending is lower. Future studies should adopt a randomised rollout or cohort design to strengthen causal claims and incorporate longer time frames to assess sustained impact.
我们有高度信心得出结论：应用用户的平均消费更高，并且这种关联在调整了关键人口统计和交易变量后仍然存在。FreshMart应继续投资该应用，并考虑采取策略提高使用率，尤其是在基准消费较低的郊区门店。未来研究应采用随机推广或队列设计以强化因果推断，并纳入更长时间范围以评估持续影响。
Published by TutorHao | Statistics Revision Series | aleveler.com
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