📚 Exercise 24E.1 | 练习 24E.1
Exercise 24E.1 in the IB Mathematics course focuses on calculations involving the normal distribution, one of the most important continuous probability models in the syllabus. Whether you are studying Applications and Interpretation or Analysis and Approaches, this exercise builds essential skills in standardising values, using technology to find probabilities, and solving inverse normal problems. The following article unpacks every key concept you need to master this exercise confidently.
IB 数学课程中的练习 24E.1 聚焦于正态分布的相关计算，这是教学大纲中最重要的连续概率模型之一。无论你修读的是「应用与解释」还是「分析与方法」，这一练习都旨在培养标准化数值、运用技术求概率以及解决逆正态问题等核心技能。下文将逐一剖析你需要掌握的所有关键概念，帮助你自信地完成这一练习。
1. The Normal Distribution as a Continuous Model | 作为连续模型的正态分布
A continuous random variable can take any value within an interval. The normal distribution models variables whose data are spread symmetrically around the mean, producing the well‑known bell‑shaped curve. In IB, we write X ~ N(μ, σ²), where μ is the population mean and σ² is the variance.
连续型随机变量可以取某一区间内的任意值。正态分布用于描述那些围绕均值对称分布的数据，它呈现出著名的钟形曲线。在 IB 中我们记作 X ~ N(μ, σ²)，其中 μ 为总体均值，σ² 为方差。
The total area under the curve equals 1, representing the total probability. The curve’s spread is determined by the standard deviation σ: small σ gives a tall, narrow peak; large σ gives a flatter, wider shape.
曲线下的总面积为 1，代表总概率。曲线的离散程度由标准差 σ 决定：σ 较小则曲线高耸狭窄；σ 较大则曲线低平宽阔。
2. Probability Density Function and Its Notation | 概率密度函数及其表示
The probability density function (pdf) of a normal distribution is given by f(x) = (1 / (σ√2π ations you are never required to integrate this function by hand; instead, you will use a calculator’s built‑in normal cumulative distribution function (normalcdf).
正态分布的概率密度函数为 f(x) = (1 / (σ√2π)) × e^[−(x−μ)² / (2σ²)]。在 IB 考试中，你无需手动对该函数积分；而是借助计算器内建的正态累积分布函数（normalcdf）来求得概率。
ative function gives the area to the left of a given value, i.e. the probability P(X ≤ x).
必须分清概率密度函数与累积分布函数的区别。密度函数给出某一点的曲线高度，而累积函数则给出给定值左侧的面积，即概率 P(X ≤ x)。
3. The Standard Normal Distribution Z ~ N(0, 1) | 标准正态分布 Z ~ N(0, 1)
If X ~ l distribution with mean 0 and standard deviation 1. This transformation allows us to compare different normal variables on a common scale and to use standard normal tables where technology is not available.
若 X ~ N(μ, σ²)，则标准化得分 Z = (X − μ) / σ 服从均值为 0、标准差为 1 的标准正态分布。这一变换使我们能够在同一尺度上比较不同的正态变量，并在无法使用技术的情况下借助标准正态分布表求概率。
Although IB students primarily rely on graphical display calculators (GDC), understanding the Z‑score remains essential for interpreting results and for questions that ask you to standardise before proceeding.
尽管 IB 学生主要依靠图形计算器（GDC），但理解 Z 分数对于解读结果以及处理那些需要先标准化再计算的题目依然至关重要。
4. Calculating Z‑Scores Step by Step | 逐步计算 Z 分数
The Z‑ with μ = 170 cm and σ = 8 cm, a height of 182 cm corresponds to Z = (182 − 170) / 8 = 1.5.
Z 分数公式为 Z = (x − μ) / σ。Z 为正值表示 x 在均值之上，负值则表明 x 在均值之下。例如，身高服从正态分布，μ = 170 cm，σ = 8 cm，身高 182 cm 对应的 Z = (182 − 170) / 8 = 1.5。
Always double‑check that you are using the standard deviation σ, not the variance σ². Many mistakes in Exercise 24E.1 arise from confusing these two quantities.
务必检查你使用的是标准差 σ，而非方差 σ²。练习 24E.1 中的许多错误正是源于混淆了这两个量。
5. Finding Probabilities with a GDC | 用图形计算器求概率
Instruments and Casio calculators is normalcdf(lower bound, upper bound, μ, σ). For a left‑il probability P(X ≤ k), set the lower bound to a very small number such as −10⁹⁹ or −10⁹.
现代 IB 考试要求你熟练运用 normalcdf 功能。在大多数德州仪器和卡西欧计算器上，其语法为 normalcdf(下界, 上界, μ, σ)。若要计算左尾概率 P(X ≤ k)，可将下界设为一个非常小的数，如 −10⁹⁹ 或 −10⁹。
For a right‑ er bound to a large number, e.g. 10⁹⁹. The calculator then returns the area under the normal curve between the chosen bounds, giving the required probability.
若需求右尾概率 P(X ≥ k)，则将上界设为一个很大的数，例如 10⁹⁹。计算器便返回所选界限之间正态曲线下的面积，即所求概率。
- On TI‑84: Press [2nd] [VARS], choose 2:normalcdf(, and enter lower, upper, μ, σ).
- 在 TI‑84 上：按 [2nd] [VARS]，选择 2:normalcdf(，输入下界、上界、μ、σ。
- On Casio fx‑CG50: Go to STAT mode, press F5(DIST), then F1(NORM) and F2(Ncd), and input the appropriate boundaries.
- 在 Casio fx‑CG50 上：进入 STAT 模式，按 F5(DIST)，再按 F1(NORM) 和 F2(Ncd)，输入相应边界。
6. Probability for Standard Intervals | 标准区间的概率
σ) ≈ 0.954, and P(μ − 3σ ≤ X ≤ μ + 3σ) ≈ 0.997. These are often called the empirical or 68–95–99.7 rule, and they provide a quick check for calculator results.
正态计算中有三个对称区间尤为重要：P(μ − σ ≤ X ≤ μ + σ) ≈ 0.683，P(μ − 2σ ≤ X ≤ μ + 2σ) ≈ 0.954，P(μ − 3σ ≤ X ≤ μ + 3σ) ≈ 0.997。它们常被称为经验法则或 68–95–99.7 规则，可用于快速检验计算器结果。
ility that a randomly chosen value lies within a given range, such as P(45 ≤ X ≤ 55) for X ~ N(50, 25). Recognise that the standard deviation here is √25 = 5. Enter normalcdf(45, 55, 50, 5) to obtain approximately 0.6827.
在练习 24E.1 中，你可能会被要求求某随机值落在给定范围内的概率，例如 X ~ N(50, 25) 时 P(45 ≤ X ≤ 55)。注意此时标准差为 √25 = 5。输入 normalcdf(45, 55, 50, 5) 便可得出约 0.6827。
| Interval / 区间 | Probability / 概率 | Z‑bounds / Z 界限 |
|---|---|---|
| μ ± 1σ | ≈ 0.683 | −1 ≤ Z ≤ 1 |
| μ ± 2σ | ≈ 0.954 | −2 ≤ Z ≤ 2 |
| μ ± 3σ | ≈ 0.997 | −3 ≤ Z ≤ 3 |
7. Inverse Normal Distribution Problems | 逆正态分布问题
An inverse normal problem gives you a probability (or a percentile) and asks you to find the corresponding x‑value. In IB, this is handled by the invNorm (or InvNormal) function, with syntax invNorm(area to the left, μ, σ).
逆正态问题会给出一个概率（或百分位数），要求你找出对应的 x 值。在 IB 中，这通过 invNorm（或 InvNormal）功能完成，语法为 invNorm(左侧面积, μ, σ)。
For example, if X ~ N(70, 9) and you need the value below which 90% The result, approximately 73.84, is the 90th percentile. Always ensure you input the left‑tail area — if the question gives the upper 10%, use 1 − 0.10 = 0.90.
例如，X ~ N(70, 9)，需要求低于它 90% 数据所在的值，计算 invNorm(0.9, 70, 3) 即可。结果约为 73.84，即第 90 百分位数。请务必确保输入的是左尾面积——若题目给出的是上侧 10%，应使用 1 − 0.10 = 0.90。
invNorm(0.90, μ, σ) → x such that P(X ≤ x) = 0.90
invNorm(0.90, μ, σ) → 满足 P(X ≤ x) = 0.90 的 x
8. Context‑Based Application Questions | 基于情境的应用题
IB exam questions embed normal distributions in real‑world contexts: IQ scores, battery lifetimes, lengths of fish, examination marks, or filling volumes of bottles. You must extract μ and σ from the problem statement, identify whether the question asks for a probability or a percentile, and clearly state your answer with units.
IB 考试题目会将正态分布嵌入实际情境：智商分数、电池寿命、鱼的长度、考试成绩或瓶子灌装体积等。你必须从题干中提取 μ 和 σ，判断题目要求的是概率还是百分位数，并用单位清晰作答。
A typical Exercise 24E.1 problem might read: 'The mass of apples harvested from an orchard follows a normal distribution with mean 150 g and standard deviation 20 g. Find the proportion of apples with mass between 130 g and 170 g.' Here you simply compute normalcdf(130, 170, 150, 20).
练习 24E.1 中的一道典型题目可能是：「某果园收获的苹果质量服从均值为 150 g、标准差为 20 g 的正态分布。求质量介于 130 g 和 170 g 之间的苹果比例。」此时直接计算 normalcdf(130, 170, 150, 20) 即可。
9. Interpreting Calculator Outputs and Drawing Diagrams | 解读计算器输出与绘制示意图
A sketch of the normal curve, with the mean and the region of interest shaded, can prevent many misunderstanding errors. Label the x‑axis clearly, mark μ, and indicate boundaries like x₁, x₂. Even in a calculator‑focused exercise, this visual check helps confirm you have chosen the correct bounds.
绘制正态曲线的草图并标出均值和关注的阴影区域，能够避免许多理解性错误。清晰标注 x 轴，标示 μ，并注明 x₁、x₂ 等边界。即便在侧重计算器的练习中，这种视觉检验也有助于确认你选择了正确的界限。
When using a GDC, always check whether you are working with the correct distribution type and that the standard deviation (not variance) is entered. Students often mis‑enter σ where σ² is given, leading to dramatically wrong answers.
使用图形计算器时，务必检查是否选中了正确的分布类型，并确认输入的是标准差（而非方差）。学生常常在给出方差 σ² 时误输入 σ，导致答案严重错误。
10. Common Pitfalls in Exercise 24E.1 | 练习 24E.1 常见陷阱
- Confusing population standard deviation and sample standard deviation — IB questions specify the population parameters.
- 混淆总体标准差与样本标准差——IB 题目会明确指出总体参数。
- Using the wrong tail area in invNorm: if the question says 'top 5%', use 0.95 as the input.
- 在 invNorm 中使用了错误的尾部面积：若题目要求「上侧 5%」，应输入 0.95。
- total area under the curve is 1; symmetry can often simplify P(X ≥ μ + k) to P(X ≤ μ − k).
- 忘记曲线下总面积为 1；对称性常可将 P(X ≥ μ + k) 简化为 P(X ≤ μ − k)。
- Rounding intermediate Z‑values prematurely — store them in the calculator memory instead.
- 过早对中间 Z 值四舍五入——应将其存储于计算器内存中。
- Entering 'normalpdf' instead of 'normalcdf' — the pdf gives height, not cumulative probability.
- 误输入 'normalpdf' 而非 'normalcdf'——pdf 给出的是高度，而非累积概率。
11. Connecting Exercise 24E.1 to the IB Internal Assessment | 将练习 24E.1 与 IB 内部评估相连接
Many students choose to analyse a normally distributed variable for their Internal Assessment (IA). Mastery of the skills in Exercise 24E.1 — calculating probabilities, constructing interval estimates, and interpreting percentiles — provides the exact toolkit needed to rigorously test hypotheses about a population mean using normal models.
不少学生选择在内部评估（IA）中分析一个服从正态分布的变量。掌握练习 24E.1 中的技能——计算概率、构建区间估计以及解读百分位数——刚好提供了利用正态模型严格检验关于总体均值的假设所需的全套工具。
When writing your IA, you will frequently use inverse normal functions to find critical values and normalcdf to determine p‑values. Practising Exercise 24E.1 thoroughly therefore pays dividends beyond the written exam.
在撰写 IA 时，你会频繁地运用逆正态函数求临界值，并用 normalcdf 确定 p 值。因此，彻底完成练习 24E.1 将给书面考试之外的环节带来巨大回报。
12. Summary and Self‑Check Questions | 总结与自测问题
Exercise 24E.1 consolidates the fundamental normal‑ mal problems. To gauge your readiness, ask yourself: Can I explain the meaning of N(μ, σ²)? Can I use my GDC to find P(a ≤ X ≤ b) and the value of x for a given left‑tail area? Can I interpret my answer in the context of a practical problem?
练习 24E.1 巩固了 IB 大纲所要求的基本正态分布技能：运用技术求概率、标准化数值以及解决逆正态问题。为了检验自己的掌握程度，请自问：我能解释 N(μ, σ²) 的涵义吗？我能使用图形计算器求 P(a ≤ X ≤ b) 以及给定左尾面积所对应的 x 值吗？我能结合实际情境解读我的答案吗？
If you can confidently answer yes to these, you are well prepared not only for Exercise 24E.1 but also for the normal distribution components in your IB end‑of‑topic test and final examinations.
如果你能自信地回答「能」，那么你不仅为练习 24E.1 做好了准备，也为 IB 章节测验和最终考试中的正态分布部分奠定了扎实的基础。
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