Year 10 Eduqas Statistics: Case Study Drill | Year 10 Eduqas 统计：案例分析实战演练

📚 Year 10 Eduqas Statistics: Case Study Drill | Year 10 Eduqas 统计：案例分析实战演练

Welcome to this comprehensive case study drill designed for Year 10 students following the Eduqas GCSE Statistics specification. In this article, we will work through a realistic data investigation, from planning and data collection to analysis and evaluation, using a dataset on teenagers’ weekly exercise hours and their self-reported well-being scores. This step-by-step walkthrough will help you master the statistical enquiry cycle and apply your skills to GCSE-style questions.

欢迎参加这个为 Eduqas GCSE 统计学课程设计的综合案例演练。在本文中，我们将通过一个真实的数据调查，从规划和数据收集到分析和评估，使用关于青少年每周锻炼小时数和自报幸福感评分的数据集，逐步完成统计探究。这个循序渐进的讲解将帮助你掌握统计调查周期，并将你的技能应用到 GCSE 风格的考题中。

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1. Defining the Problem and Setting Hypotheses | 定义问题与设定假设

Before any data is collected, a statistician must clearly state the aim of the investigation. For this case study, our research question is: ‘Is there a relationship between the number of hours teenagers spend on physical exercise per week and their self-assessed well-being score on a scale from 1 to 10?’ We suspect that as exercise hours increase, well-being scores also tend to increase. Therefore, our hypothesis is that there is a positive correlation between the two variables.

在收集任何数据之前，统计人员必须清楚地陈述调查目的。在这个案例研究中，我们的研究问题是：“青少年每周用于体育锻炼的小时数与他们在1至10分制上的自评幸福感评分之间是否存在关系？”我们推测，随着锻炼时间的增加，幸福感评分也倾向于提高。因此，我们的假设是这两个变量之间存在正相关关系。

It is important to remember that a hypothesis should be testable. In GCSE Statistics, you will often be asked to suggest a suitable null hypothesis and an alternative hypothesis. The null hypothesis (H₀) would state that there is no correlation between exercise hours and well-being score, while the alternative hypothesis (H₁) would suggest a positive (or negative) correlation. In this case, we are investigating a one-tailed positive correlation.

重要的是要记住，假设应该是可检验的。在 GCSE 统计学中，你经常会被要求提出一个合适的零假设和备择假设。零假设（H₀）会说明锻炼小时数与幸福感评分之间没有相关关系，而备择假设（H₁）则暗示存在正（或负）相关。在本案例中，我们正在研究单尾正相关。

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2. Planning the Survey: Population, Sample and Data Types | 调查规划：总体、样本与数据类型

The target population for this study is all 14- to 16-year-old students in a particular secondary school. Because it is impractical to survey every student, we decide to take a representative sample. We could use stratified sampling by year group and gender to ensure the sample mirrors the school’s composition, or a simple random sample drawn from the school register. For this drill, we assume a simple random sample of 30 students has been selected.

这项研究的目标总体是某所中学所有14至16岁的学生。由于调查每名学生不切实际，我们决定抽取有代表性的样本。我们可以采用按年级和性别分层抽样的方法，确保样本反映学校的构成，或者从学校名册中抽取简单随机样本。在本次演练中，我们假设已选取了30名学生的简单随机样本。

We need to collect two pieces of data from each student: the number of hours of physical exercise per week (to the nearest half hour) and their overall well-being score on a scale of 1 to 10, where 1 means ‘very unhappy’ and 10 means ‘extremely happy’. The exercise variable is continuous (measured on a ratio scale), while the well-being score is discrete but treated as ordinal or scale data for analysis. A well-designed questionnaire would ask for these values in a consistent manner, perhaps with a visual analogue scale for well-being.

我们需要从每名学生那里收集两项数据：每周体育锻炼的小时数（精确到半小时）和他们在1至10分制上的整体幸福感评分，其中1代表“非常不快乐”，10代表“极其快乐”。锻炼变量是连续的（在比率尺度上测量），而幸福感评分是离散的，但分析时可作为定序或尺度数据处理。一份设计良好的问卷应以一致的方式询问这些数值，或许用视觉模拟标尺进行幸福感评分。

Secondary data could also have been used, but for this case study we simulate primary data collection to practise the full cycle. Always remember to consider ethical issues: students’ anonymity is protected, and participation is voluntary.

也可以使用二手数据，但在本案例研究中我们模拟一手数据收集，以便练习整个统计周期。要始终记得考虑伦理问题：保护学生的匿名性，且参与是自愿的。

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3. Collecting and Recording the Data | 收集与记录数据

After administering the questionnaire, we obtain numerical data from 30 students. The dataset is recorded in a table with three columns: Student ID, Exercise (hours/week), and Well-being Score (1-10). For convenience, we only display the first 10 rows below, but the entire dataset will be used for calculations.

在发放问卷后，我们从30名学生那里获得了数值数据。数据集被记录在一个三列表中：学生编号、锻炼（小时/周）和幸福感评分（1-10）。为方便起见，我们在下面仅显示前10行，但整个数据集将用于计算。

Student ID	Exercise (hours)	Well-being Score
1	0.5	2.0
2	1.0	2.5
3	1.5	3.0
4	2.0	3.8
5	2.5	4.2
6	3.0	4.5
7	3.5	5.1
8	4.0	5.5
9	4.5	6.0
10	5.0	6.4

The full dataset of 30 observations is used for all subsequent calculations. The raw data should be checked for obvious errors or outliers – none were found in this exercise.

全部30个观测值的完整数据集将用于后续所有的计算。应检查原始数据是否存在明显错误或异常值——本次练习中未发现任何异常。

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4. Univariate Analysis: Exercise Hours | 单变量分析：锻炼小时数

We begin by summarising the exercise hours (x) on their own. The mean number of hours per week is calculated as:

我们首先单独对锻炼小时数（x）进行汇总。每周平均小时数计算如下：

x̄ = Σxᵢ / n = 147.5 / 30 ≈ 4.92 hours

x̄ = Σxᵢ / n = 147.5 / 30 ≈ 4.92 小时

The median (Q₂) is the middle value when the data are sorted. After ordering the 30 observations, the median lies between the 15th and 16th values, giving 5.0 hours. This suggests a roughly symmetric distribution, as mean ≈ median.

中位数（Q₂）是数据排序后的中间值。将30个观测值排序后，中位数位于第15和第16个值之间，得出5.0小时。这表明分布大致对称，因为均值≈中位数。

The five-number summary helps build a box plot:

五数概括有助于构建箱线图：

Minimum = 0.0 hrs, Q₁ = 2.5 hrs, Median = 5.0 hrs, Q₃ = 7.5 hrs, Maximum = 10.0 hrs. Interquartile range (IQR) = Q₃ − Q₁ = 5.0 hrs.

最小值 = 0.0小时，第一四分位数 Q₁ = 2.5小时，中位数 = 5.0小时，第三四分位数 Q₃ = 7.5小时，最大值 = 10.0小时。四分位距（IQR）= Q₃ − Q₁ = 5.0小时。

The standard deviation (s) measures the spread about the mean. For the sample data, s is found using:

标准差（s）衡量数据围绕均值的离散程度。对于样本数据，s 通过以下公式求得：

s = √[ Σ(xᵢ − x̄)² / (n − 1) ]

s = √[ Σ(xᵢ − x̄)² / (n − 1) ]

Using the full dataset, Σ(xᵢ − x̄)² = 219.64, so s = √(219.64 / 29) ≈ √7.57 ≈ 2.75 hours. Since the standard deviation is moderately large relative to the mean, there is considerable variation in exercise habits among students.

使用完整数据集，Σ(xᵢ − x̄)² = 219.64，因此 s = √(219.64 / 29) ≈ √7.57 ≈ 2.75小时。由于标准差相对于均值较大，说明学生们的锻炼习惯存在相当大的差异。

A box plot would show the box from 2.5 to 7.5 hours with the median line at 5.0, and whiskers extending to 0 and 10. There are no extreme outliers according to the 1.5 × IQR rule.

箱线图将显示箱子从2.5到7.5小时，中位线在5.0处，须线延伸至0和10。根据 1.5 × IQR 规则，没有极端异常值。

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5. Univariate Analysis: Well-being Score | 单变量分析：幸福感评分

Now we turn to the well-being score (y). The summary statistics are:

现在我们来看幸福感评分（y）。汇总统计量如下：

ȳ = Σyᵢ / n = 193.8 / 30 ≈ 6.46

ȳ = Σyᵢ / n = 193.8 / 30 ≈ 6.46

Minimum = 1.5, Q₁ = 4.2, Median = 6.8, Q₃ = 8.5, Maximum = 10.0. The IQR = 4.3. The mean is slightly lower than the median, suggesting a slight negative

Published by TutorHao | Year 10 统计 Revision Series | aleveler.com

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