📚 Case Study: Investigating Smoking and Lung Capacity | 案例分析：吸烟与肺活量关系探究
This case study examines the relationship between smoking and lung capacity using real‑world inspired data. A health survey recorded forced vital capacity (FVC, in litres) for 20 smokers and 25 non‑smokers. Additionally, for the smoker group, the number of years they have smoked was documented. We aim to (1) test whether the mean FVC differs between smokers and non‑smokers using a two‑sample t‑test, and (2) investigate how smoking duration affects lung capacity via correlation and linear regression. This practical walkthrough integrates key A‑Level Statistics topics: hypothesis testing, data visualisation, assumption checking, correlation, regression and residual analysis.
本案例研究采用模拟真实数据，探讨吸烟与肺活量的关系。一项健康调查记录了20名吸烟者和25名非吸烟者的用力肺活量（FVC，升）。此外，还记录了吸烟者的吸烟年数。我们要完成以下任务：(1) 通过双样本 t 检验，判断吸烟者与非吸烟者的平均 FVC 是否存在显著差异；(2) 利用相关与线性回归，分析吸烟年限对肺活量的影响。本实战演练综合了A‑Level统计课程中的多个关键知识点：假设检验、数据可视化、前提假设核查、相关、回归以及残差分析。
1. Data Description | 数据描述
The raw data are as follows: Non‑smoker FVC (L): 3.9, 4.1, 3.7, 4.2, 3.8, 4.0, 3.6, 4.3, 3.5, 4.4, 3.8, 3.9, 4.0, 3.7, 4.1, 3.6, 4.2, 3.8, 4.3, 3.9, 4.0, 3.7, 4.1, 3.8, 4.2. Smoker FVC (L): 3.2, 2.9, 3.5, 3.0, 2.8, 3.3, 3.1, 2.7, 3.4, 3.0, 2.6, 3.2, 3.1, 2.9, 3.5, 3.3, 2.8, 3.0, 3.4, 2.9. Smoker smoking years: 10, 12, 8, 15, 20, 5, 7, 14, 11, 9, 18, 13, 6, 10, 16, 14, 8, 11, 17, 12.
原始数据如下：非吸烟者 FVC（升）：3.9, 4.1, 3.7, 4.2, 3.8, 4.0, 3.6, 4.3, 3.5, 4.4, 3.8, 3.9, 4.0, 3.7, 4.1, 3.6, 4.2, 3.8, 4.3, 3.9, 4.0, 3.7, 4.1, 3.8, 4.2。吸烟者 FVC（升）：3.2, 2.9, 3.5, 3.0, 2.8, 3.3, 3.1, 2.7, 3.4, 3.0, 2.6, 3.2, 3.1, 2.9, 3.5, 3.3, 2.8, 3.0, 3.4, 2.9。吸烟者吸烟年数：10, 12, 8, 15, 20, 5, 7, 14, 11, 9, 18, 13, 6, 10, 16, 14, 8, 11, 17, 12。
Summary statistics are presented in the table below.
摘要统计量见下表。
| Group | n | Mean | Standard deviation |
|---|---|---|---|
| Non‑smoker | 25 | 3.944 | 0.252 |
| Smoker | 20 | 3.080 | 0.272 |
For the smoker group, the mean number of smoking years is 11.8 with a standard deviation of 4.10.
吸烟组的平均吸烟年数为11.8，标准差为4.10。
2. Visualising the Data | 数据可视化
Before conducting formal tests, it is wise to produce plots. Side‑by‑side boxplots for the two groups reveal that the non‑smoker FVC values are consistently higher, with medians around 3.95 L compared to 3.10 L for smokers. There is slight overlap in the interquartile ranges, but a difference is evident. A scatter plot of FVC against smoking years (for smokers only) suggests a downward trend: as smoking years increase, lung capacity tends to decrease. Visual checks help spot outliers and inform the choice of subsequent analyses.
在进行正式检验之前，明智的做法是绘制图表。两组数据的并列箱线图显示，非吸烟者的 FVC 值普遍更高，中位数约为3.95升，而吸烟者约为3.10升。四分位距有少量重叠，但差异明显。仅针对吸烟者的 FVC 对吸烟年数的散点图显示出下降趋势：随着吸烟年数增加，肺活量趋于下降。可视化检查有助于发现离群值，并为后续分析方法的选择提供依据。
3. Checking Assumptions for the t‑test | 检验 t 检验的假设条件
A two‑sample t‑ version). Independence is given by the study design. Normality can be assessed via Shapiro‑Wilk tests or normal probability plots. For both groups, the p‑values from Shapiro‑ gesting no serious departure from normality. We also test equality of variances using an F‑test: the ratio of sample variances is F = 0.2722 / 0.2522 = 0.0740 / 0.0635 ≈ 1.165, with degrees of freedom (19,24). The two‑tailed p‑ ces. However, because sample sizes are unbalanced and to be rigorous, we will employ Welch's t‑test, which does not assume equal variances.
双样本 t 检验要求观测值独立、各组内近似正态分布，以及（对于合并版本）方差相等。根据研究设计，独立性得以保证。通过 Shapiro‑Wilk 检验或正态概率图可以评估正态性。两组数据的 Shapiro‑Wilk p 值均大于0.10，表明没有严重偏离正态。我们再用 F 检验来检验方差齐性：样本方差之比 F = 0.2722 / 0.2522 = 0.0740 / 0.0635 ≈ 1.165，自由度为(19,24)。双侧 p 值约为0.74，因此我们不拒绝方差相等的原假设。但鉴于样本量不平衡，且为了严谨，我们将采用不假设方差相等的 Welch t 检验。
4. Two‑Sample t‑test (Welch) | 双样本 t 检验（Welch 法）
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