📚 IGCSE CAIE Statistics: Unit Test Mock Paper Walkthrough | IGCSE CAIE 统计：单元测试模拟卷解析
This walkthrough works through a CAIE IGCSE Statistics unit test mock paper. The questions cover data types, sampling, frequency tables, histograms, averages, spread, probability, Venn diagrams, scatter graphs and moving averages. For every question, you will see examiner-style working and the final answer you should write down.
本文解析一份 CAIE IGCSE 统计单元测试模拟卷。题目涵盖数据类型、抽样、频数表、直方图、平均数、离散程度、概率、维恩图、散点图和移动平均数。每道题都给出符合考官要求的解题步骤与最终答案。
1. Types of Data | 数据类型
Mock question 1: A survey records the following information about students: number of siblings, travel time to school in minutes, favourite subject and height in cm. Classify each variable as qualitative, quantitative discrete or quantitative continuous.
模拟题 1：某调查记录了学生的以下信息：兄弟姐妹数量、上学所需时间（分钟）、最喜欢的科目和身高（cm）。请将每个变量分类为定性变量、定量离散变量或定量连续变量。
Worked solution:
解题步骤：
- Number of siblings: quantitative discrete, because it can only take whole-number values such as 0, 1, 2.
- Travel time to school: quantitative continuous, because time can take any value in an interval.
- Favourite subject: qualitative, because it is a non-numerical category.
- Height: quantitative continuous, because height can be measured to any level of accuracy.
兄弟姐妹数量：定量离散变量，因为只能取 0、1、2 等整数值。上学所需时间：定量连续变量，因为时间可以在一定区间内取任意值。最喜欢的科目：定性变量，因为它是非数值类别。身高：定量连续变量，因为身高可以测量到任意精度。
2. Sampling Methods | 抽样方法
Mock question 2: A school has 1100 students. The table shows the number of students in each year group. The head teacher wants to take a stratified sample of 110 students for a survey. Calculate the number of students to select from each year group, and explain why stratified sampling is suitable here.
模拟题 2：某学校有 1100 名学生。下表显示各年级学生人数。校长想抽取 110 名学生进行分层抽样调查。请计算每个年级应抽取的学生人数，并说明为什么此处适合采用分层抽样。
Worked solution:
解题步骤：
| Year group | Students | Sample size |
|---|---|---|
| Year 7 | 240 | 24 |
| Year 8 | 260 | 26 |
| Year 9 | 220 | 22 |
| Year 10 | 200 | 20 |
| Year 11 | 180 | 18 |
| Total | 1100 | 110 |
The sampling fraction is 110 ÷ 1100 = 0.1 = 10%. Multiply each year group size by 0.1, so Year 7 gives 240 × 0.1 = 24, Year 8 gives 26, Year 9 gives 22, Year 10 gives 20 and Year 11 gives 18.
抽样比例为 110 ÷ 1100 = 0.1 = 10%。每个年级人数乘以 0.1，因此 Year 7 抽 240 × 0.1 = 24 人，Year 8 抽 26 人，Year 9 抽 22 人，Year 10 抽 20 人，Year 11 抽 18 人。
Stratified sampling is suitable because it guarantees that each year group is represented in the sample in proportion to its size, making the sample more representative than a simple random sample.
此处适合分层抽样，因为它能保证每个年级在样本中的比例与总体一致，使样本比简单随机抽样更具代表性。
3. Frequency Tables and Mode | 频数表与众数
Mock question 3: Twenty students scored the following marks out of 5 in a short test: 3, 4, 2, 5, 3, 2, 4, 3, 5, 3, 1, 2, 4, 3, 5, 3, 2, 4, 3, 2. Construct a frequency table and write down the mode.
模拟题 3：20 名学生在一次小测验中的分数（满分 5 分）如下：3, 4, 2, 5, 3, 2, 4, 3, 5, 3, 1, 2, 4, 3, 5, 3, 2, 4, 3, 2。请构建频数表并写出众数。
Worked solution:
解题步骤：
| Score | Frequency |
|---|---|
| 1 | 1 |
| 2 | 5 |
| 3 | 7 |
| 4 | 4 |
| 5 | 3 |
The mode is 3, because it has the highest frequency of 7. Always remember to give the data value, not the frequency, when stating the mode.
众数是 3，因为它的频数最高，为 7 次。注意，写众数时要写数据值，而不是频数。
4. Histograms and Frequency Density | 直方图与频数密度
Mock question 4: The ages of people in a cinema are grouped as shown in the table. Calculate the frequency density for each class and explain how you would use these values to draw a histogram.
模拟题 4：某电影院观众的年龄分组如下表所示。请计算每个区间的频数密度，并说明如何用这些数值绘制直方图。
Worked solution:
解题步骤：
The formula for frequency density is:
频数密度公式为：
Frequency density = frequency ÷ class width
| Age, a (years) | Frequency | Class width | Frequency density |
|---|---|---|---|
| 0 ≤ a < 10 | 12 | 10 | 12 ÷ 10 = 1.2 |
| 10 ≤ a < 20 | 18 | 10 | 18 ÷ 10 = 1.8 |
| 20 ≤ a < 35 | 15 | 15 | 15 ÷ 15 = 1.0 |
| 35 ≤ a < 60 | 10 | 25 | 10 ÷ 25 = 0.4 |
On a histogram, frequency density is plotted on the vertical axis and age is plotted on the horizontal axis. The area of each bar is proportional to the frequency of that class, so unequal class widths are shown correctly.
在直方图中，纵轴表示频数密度，横轴表示年龄。每个直方的面积与该区间的频数成正比，因此即使组距不相等，也能正确表示频数分布。
5. Estimating the Mean from Grouped Data | 分组数据平均数的估计
Mock question 5: Use the grouped age data from Question 4 to estimate the mean age of the people in the cinema.
模拟题 5：使用第 4 题中的年龄分组数据，估计电影院观众的平均年龄。
Worked solution:
解题步骤：
First find the midpoint of each class, then multiply by the frequency.
先找出每个区间的中点，再乘以频数。
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