Year 12 WJEC Statistics: Case Study Practical Exercises | WJEC 12年级统计案例分析实战演练

📚 Year 12 WJEC Statistics: Case Study Practical Exercises | WJEC 12年级统计案例分析实战演练

In the WJEC Year 12 Statistics examination, case study questions demand that you apply a full range of statistical tools to a single realistic scenario. This article walks you through a series of practical exercises, building your confidence in data collection, analysis and interpretation. By working step by step, you will learn how to structure your answers and avoid common pitfalls.

在 WJEC 12年级统计考试中，案例分析题要求你将全套统计工具应用于一个真实情景。本文通过一系列实战演练带你逐步掌握数据收集、分析和解读的方法，帮你建立答题框架，避免常见错误。

1. Defining the Problem – From Question to Variables | 定义问题 – 从问题到变量

Every statistical case study begins with a clear research question. For example, a student might ask, “Do Year 12 students who eat breakfast regularly perform better in mathematics?” This question must be translated into a testable hypothesis by defining the population (all Year 12 students in the school), the response variable (maths test score) and the explanatory variable (categorical: eats breakfast regularly or not).

每一个统计案例研究都始于清晰的研究问题。例如，一个学生可能会问：“经常吃早餐的Year 12学生数学成绩是否更好？”这个问题必须转化为可检验的假设，为此需要定义总体（学校所有Year 12学生）、响应变量（数学测试分数）和解释变量（分类变量：是否经常吃早餐）。

Precise definitions are essential because they remove ambiguity and guide the choice of statistical methods. If the population is poorly defined, the sample may not represent the group you intend to study, and any conclusions will be questionable from the start.

精确的定义至关重要，因为它们能消除歧义并指导统计方法的选择。如果总体定义不清晰，样本可能无法代表你想要研究的群体，从一开始结论就站不住脚。

2. Designing the Sample – Methods and Pitfalls | 设计样本 – 方法与陷阱

Once the research question is framed, you must decide how to collect data. Suppose you plan to survey 100 students about their breakfast habits. The sampling method will determine whether your results can be generalised. Simple random sampling gives every student an equal chance of being chosen but may be impractical in a large school. Stratified sampling divides the population into groups, such as tutor groups or gender, and selects a proportional number from each, which often leads to more precise estimates.

一旦确立了研究问题，你必须决定如何收集数据。假设你计划调查100名学生的早餐习惯，抽样方法将决定结果是否具有推广性。简单随机抽样让每个学生被选中的机会均等，但在大学校中可能不切实际。分层抽样将总体分成若干群体，如导师组或性别，并从每个群体中按比例抽取一定数量，通常能得到更精确的估计。

In contrast, convenience sampling – asking your friends or those who walk past the canteen – introduces serious bias and should be avoided. Always consider non‑response: students who refuse to answer may have different habits from those who participate, and this non‑response bias can distort your findings even if the initial sample was well designed.

相比之下，便利抽样——询问你的朋友或走过食堂的人——会引入严重偏差，应当避免。始终要考虑无应答问题：拒绝回答的学生可能与参与者在习惯上有所不同，即使最初样本设计得再好，这种无应答偏差也能歪曲你的发现。

3. Graphical Representation – Bringing Data to Life | 图形表示 – 让数据生动起来

After collecting the data, visual displays help you spot patterns before any formal analysis. For the breakfast study, you could construct a back‑to‑back stem‑and‑leaf plot to compare the maths scores of breakfast eaters and non‑eaters side by side. This plot preserves the raw data while showing the shape of each distribution.

收集数据后，图形展示可以帮助你在正式分析前发现模式。对于早餐研究，你可以构建背对背茎叶图，将吃早餐与不吃早餐学生的数学成绩并排比较。该图保留了原始数据，同时展示了两组分布的形状。

Box plots are another powerful tool: they display the median, quartiles and any outliers, making it easy to compare the centre and spread of two groups. For continuous data, histograms are common, but you must ensure the class widths are equal or use frequency density on the vertical axis. Every graph must have clearly labelled axes and a descriptive title; otherwise, the examiner cannot be sure what is being shown.

箱线图是另一项有力的工具：它们展示中位数、四分位数和任何异常值，便于比较两组数据的中心与分散程度。对于连续数据，直方图很常见，但你必须确保组距相等，或在纵轴上使用频率密度。每张图都必须有清晰的坐标轴标签和描述性标题，否则阅卷者无法确定图表展示的内容。

4. Summary Statistics – Central Tendency and Dispersion | 汇总统计量 – 集中趋势与离散程度

Graphs must be supported by numerical summaries. For the breakfast‑eater group, you might calculate a mean maths score of 72 with a standard deviation of 10, while the non‑eater group has a mean of 65 with a standard deviation of 12. Because the mean can be pulled by extreme values, you should also report the median and interquartile range (IQR). The IQR is the difference between the upper and lower quartiles and is resistant to outliers.

图表必须得到数字汇总的支撑。对于吃早餐组，你可能算出平均数学成绩为72，标准差为10；而不吃早餐组平均分为65，标准差为12。由于均值可能被极端值拉偏，你应当同时报告中位数和四分位距（IQR）。四分位距是上四分位数与下四分位数之差，它不受异常值影响。

To compute the sample standard deviation, use the formula:

要计算样本标准差，请使用以下公式：

s = √[ Σ(x_i – x̄)² / (n – 1) ]

These measures provide the foundation for later inference because they quantify both the typical value and the variability within each group.

这些统计量为后续推断奠定了基础，因为它们既量化了每组数据的典型值，又量化了其变异程度。

5. Probability and the Binomial Model | 概率与二项模型

When the explanatory variable is binary, the binomial distribution often becomes the natural model. Suppose that, based on past school data, the proportion of Year 12 students who eat breakfast regularly is 0.5. If you randomly select 8 students, the number who eat breakfast, X, follows a binomial distribution: X ~ B(8, 0.5). You can calculate probabilities using the formula.

当解释变量是二分类变量时，二项分布通常成为自然的模型。假设根据过去的学校数据，Year 12学生中经常吃早餐的比例为0.5。如果你随机选取8名学生，其中吃早餐的人数X服从二项分布：X ~ B(8, 0.5)。你可以用公式计算概率。

For example, the probability that exactly 5 of the 8 students eat breakfast is:

例如，8名学生中恰好有5人吃早餐的概率为：

P(X = 5) = C(8, 5) × 0.5⁵ × 0.5³ = 0.21875

The expected value E(X) = np = 4 and the variance Var(X) = np(1–p) = 2. These properties are used in hypothesis testing when the normal approximation is applied.

期望值E(X) = np = 4，方差Var(X) = np(1–p) = 2。当使用正态近似进行假设检验时，就会用到这些性质。

6. Hypothesis Testing – Making Decisions with Data | 假设检验 – 用数据做决策

A case study typically requires a formal test. Imagine a school claims that exactly 50% of Year 12 students eat breakfast. You survey 100 students and 60 respond that they do. You wish to test, at the 5% significance level, whether the true proportion has increased. Let p be the population proportion. The hypotheses are:

案例分析通常需要进行正式的假设检验。设想一所学校声称恰好50%的Year 12学生吃早餐。你调查了100名学生，60人回答吃早餐。你希望在5%的显著性水平下检验真实比例是否有所上升。设p为总体比例，其假设为：

H₀: p = 0.5 H₁: p > 0.5

The sample proportion p̂ = 60/100 = 0.6. Under H₀, the sampling distribution of p̂ is approximately normal with mean 0.5 and standard deviation √(0.5 × 0.5/100) = 0.05.

样本比例p̂ = 60/100 = 0.6。在H₀下，p̂的抽样分布近似正态，均值为0.5，标准差为√(0.5 × 0.5/100) = 0.05。

The test statistic is:

检验统计量为：

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Year 12 WJEC Statistics: Case Study Practical Exercises | WJEC 12年级统计案例分析实战演练