📚 Investigation 1: Sampling Simulation | 探究一：抽样模拟
Sampling simulation is a cornerstone of statistical exploration in IB Mathematics, particularly for internal assessment (IA) projects. By using random number generators to repeatedly draw samples from a population, students can empirically verify the Central Limit Theorem, explore the behaviour of sample means, and gain deep insights into inferential statistics. This investigation guides you through designing and conducting a sampling simulation, analysing the resulting sampling distribution, and reflecting on the implications for real-world data analysis.
抽样模拟是 IB 数学中统计探索的基石，尤其常用于内部评估（IA）课题。通过使用随机数生成器反复从总体中抽取样本，学生能够通过实证验证中心极限定理，探索样本均值的行为，并深入理解推断统计。本探究将引导你设计并实施一项抽样模拟，分析所得的抽样分布，并反思其对真实世界数据分析的启示。
1. Introduction to Sampling Simulation | 抽样模拟简介
In the IB Mathematics Analysis & Approaches (AA) and Applications & Interpretation (AI) courses, the concept of a sampling distribution is fundamental. A simulation allows us to mimic the process of taking many random samples from a population, compute a statistic (such as the mean) for each, and examine the distribution of that statistic without exhausting real-world data. This investigation often serves as an ideal IA topic because it combines theoretical knowledge with practical technology use.
在 IB 数学（分析与方法和应用与解释）课程中，抽样分布的概念是基础。模拟使我们能够模仿从总体中反复抽取随机样本的过程，计算每个样本的统计量（如均值），并检查该统计量的分布，而无需穷尽真实数据。这一探究常成为理想的 IA 选题，因为它将理论知识与技术实践相结合。
2. Defining the Population and Sampling Parameters | 定义总体与抽样参数
his population, the theoretical mean μ = 0.5 and the theoretical standard deviation σ = 1/√12 30, and 50, and the number of repeated samples B = 1000 (or larger for smoother results).
首先我们需要确定一个总体分布。初次模拟的常见选择是区间 [0, 1] 上的均匀分布。该总体的理论均值 μ = 0.5，理论标准差 σ = 1/√12 ≈ 0.288675。明确你的抽样策略：有放回的简单随机抽样，样本量 n = 5、10、30 和 50，重复抽样次数 B = 1000（或更大以使结果更平滑）。
3. Generating Random Samples with Technology | 使用技术生成随机样本
You can perform the simulation using a graphic display calculator (GDC) such as the TI-84 Plus CE, NumWorks, or software like Excel, GeoGebra, or Python. The key is to generate n random numbers, compute their mean, and store the result. Repeating this B times yields a list of sample means.
你可以使用图形计算器（GDC）如 TI-84 Plus CE、NumWorks，或 Excel、GeoGebra、Python 等软件进行模拟。关键在于生成 n 个随机数，计算其均值，并存储结果。重复 B 次即得到一个样本均值列表。
- Python (very effective for large B): Use
numpy.random.uniform(0,1,n).mean()inside a loop. - TI-84 (using seq and rand): For n=5, B=1000, you can store in L1 with a clever sequence or write a short program.
- Excel: Use
RAND()to generate values,AVERAGE()to calculate the mean, and drag down for 1000 rows.
- Python（对大量重复非常有效）： 在循环中使用
numpy.random.uniform(0,1,n).mean()。 - TI-84（使用 seq 与 rand）： 对于 n=5，B=1000，可以通过巧妙的序列或编写简短程序存入 L1。
- Excel： 使用
RAND()生成数值，AVERAGE()计算均值，并向下拖动 1000 行。
4. Computing and Storing Sample Means | 计算并存储样本均值
For each of the B trials, record the sample mean. After the simulation, you will have B values that represent the sampling distribution of the mean for that particular n. Organise your data in a list or column. It is crucial to keep separate lists for each sample size to compare their distributions later.
对 B 次试验中的每一次，记录其样本均值。模拟结束后，你将得到代表该特定样本量下均值抽样分布的 B 个数值。将数据组织到列表或列中。务必为每个样本量保留单独的列表，以便稍后比较它们的分布。
5. Visualising the Sampling Distribution with Histograms | 用直方图可视化抽样分布
istogram should mirror the uniform population shape. As n increases, the histogram becomes bell-shaped and symmetric, centering around 0.5. Use appropriate bin widths and label axes clearly: x-axis 'Sample Mean', y-axis 'Frequency'. Overlay the normal curve if your tool supports it.
为每个 n 的样本均值创建频率直方图。当 n=1 时，直方图应呈现均匀总体的形状。随着 n 增大，直方图变为钟形且对称，中心在 0.5 附近。使用合适的组距并清晰标注坐标轴：x 轴“样本均值”，y 轴“频率”。如果你的工具支持，可叠加正态曲线。
6. The Central Limit Theorem in Practice | 实践中的中心极限定理
The Central Limit Theorem (CLT) states that for sufficiently large n, the distribution of the sample mean approximates a normal distribution with mean μ and standard deviation σ/√n, regardless of the population's shape (provided the population has finite variance). Your simulation should confirm this.
中心极限定理（CLT）指出，当 n 足够大时，无论总体形状如何（只要总体方差有限），样本均值的分布近似于均值为 μ、标准差为 σ/√n 的正态分布。你的模拟应能证实这一点。
Let us compare the theoretical standard error with the simulated standard deviation of the means:
让我们比较理论标准误与模拟均值的标准差：
| Sample Size n | Theoretical SE (σ/√n) | Simulated SD of Means |
| 5 | 0.1291 | ~0.1285 |
| 10 | 0.0913 | ~0.0908 |
| 30 | 0.0527 | ~0.0529 |
| 50 | 0.0408 | ~0.0410 |
x̄ ~ N(0.5, 0.2887² / n) approximately
Close agreement between the theoretical and simulated values validates the CLT. Small discrepancies arise from the finite number of simulations.
理论值与模拟值的紧密一致性验证了中心极限定理。微小的差异源于有限的模拟次数。
7. The Effect of Sample Size on Variability | 样本量对变异性的影响
As n grows, the standard error σ/√n This illustrates why larger samples provide more precise estimates of the population mean.
随着 n 增大，标准误 σ/√n 减小，意味着样本均值更紧密地聚集在 μ 周围。这在逐渐变窄的直方图中显而易见。n=50 时的分布比 n=5 时集中得多，这阐明了为什么更大样本能提供更精确的总体均值估计。
8. Exploring Skewed Populations | 探索偏态总体
ain noticeably skewed. However, as n increases to 30 or above, the distribution of the sample mean becomes approximately normal, illustrating the CLT's independence from population shape.
一项有力的探究应将模拟扩展至非均匀的偏态总体。例如，考虑参数 λ=1 的指数分布，总体均值 μ=1，标准差 σ=1
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图