Exercise 22E.2: Normal Distribution | 练习题 22E.2：正态分布

📚 Exercise 22E.2: Normal Distribution | 练习题 22E.2：正态分布

Exercise 22E.2 in the IB Mathematics Analysis & Approaches course reinforces essential skills for working with the normal distribution. This article breaks down key concepts, calculator techniques, and typical problems you will encounter. Whether you are preparing for Paper 1 or Paper 2, mastering these exercises will sharpen your ability to model continuous random variables, compute probabilities, and interpret real-world data under the normal curve.

IB 数学分析与方法课程中的练习题 22E.2 旨在强化处理正态分布的基本技能。本文拆解了关键概念、计算器技巧以及你会遇到的典型题目。无论你是在为试卷一还是试卷二备考，掌握这些练习都将提升你对连续随机变量建模、计算概率以及在实际数据中解读正态曲线的能力。

1. What is a Normal Distribution? | 什么是正态分布？

A normal distribution is a symmetric, bell-shaped continuous probability distribution defined by two parameters: the mean μ and the standard deviation σ. Its curve is unimodal, with the peak at the mean, and extends infinitely in both directions. The total area under the curve equals 1, and the probability of the random variable X lying in an interval corresponds to the area under the curve over that interval.

正态分布是一种对称的钟形连续概率分布，由两个参数定义：均值 μ 和标准差 σ。其曲线是单峰的，峰值出现在均值处，并向两侧无限延伸。曲线下的总面积为 1，随机变量 X 落在某个区间内的概率等于该区间上曲线下的面积。

2. The Standard Normal Distribution and Z-Scores | 标准正态分布与 Z 分数

The standard normal distribution has mean 0 and standard deviation 1, denoted Z ~ N(0, 1). To convert a value x from any normal variable X ~ N(μ, σ²) to a Z-score, use the formula:

z = (x − μ) / σ

This transformation standardises the variable, making it possible to use standard normal tables or calculator functions to find probabilities. Always check whether you are working with the raw score or the Z-score when interpreting results.

标准正态分布的均值为 0，标准差为 1，记作 Z ~ N(0, 1)。要将任意正态变量 X ~ N(μ, σ²) 的值 x 转换为 Z 分数，可使用公式：

z = (x − μ) / σ

这一变换将变量标准化，使得我们可以利用标准正态表或计算器函数来求概率。在解读结果时，务必确认你使用的是原始分数还是 Z 分数。

3. Calculator Functions for Normal Probabilities | 用于正态概率的计算器函数

For IB Mathematics, you are expected to use a GDC efficiently. Common functions include normalcdf (or normCdf) and invNorm. normalcdf(lower, upper, μ, σ) returns the probability P(lower < X < upper). invNorm(area, μ, σ) gives the value k such that P(X < k) = area. Many GDC models also allow tail settings, so it is essential to draw a quick sketch and label the region of interest before using the calculator.

在 IB 数学中，你需要熟练使用图形计算器。常用函数包括 normalcdf（或 normCdf）和 invNorm。normalcdf(下限, 上限, μ, σ) 返回概率 P(下限 < X < 上限)。invNorm(面积, μ, σ) 给出满足 P(X < k) = 面积 的 k 值。许多计算器型号还允许设置尾部方向，因此在使用计算器之前，画一个简图并标出目标区域至关重要。

4. Inverse Normal Calculations | 逆正态计算

Inverse normal problems ask you to find a boundary value given a cumulative probability. For example, if X ~ N(100, 15²) and you need the value k such that P(X < k) = 0.8, apply invNorm(0.8, 100, 15). Pay attention to wording: sometimes you are given an upper‑tail probability, e.g. P(X > k) = 0.05, which means P(X < k) = 0.95 before using invNorm.

逆正态问题要求你根据给定的累积概率求边界值。例如，若 X ~ N(100, 15²) 且需要满足 P(X < k) = 0.8 的 k 值，可应用 invNorm(0.8, 100, 15)。注意措辞：有时会给出上尾概率，如 P(X > k) = 0.05，这意味着在使用 invNorm 前要先转化为 P(X < k) = 0.95。

5. Typical Problem 1: Direct Probability | 典型问题 1：直接概率

A common Exercise 22E.2 question reads: “If X ~ N(52, 36), find P(X < 58).” Here μ = 52 and σ = √36 = 6. Using normalcdf with lower bound −∞ (or −10⁹⁹), upper bound 58, mean 52, standard deviation 6 yields the probability. Remember that σ² is the variance, so take the square root to get σ.

练习题 22E.2 中一个常见的问题是：“若 X ~ N(52, 36)，求 P(X < 58)。”此处 μ = 52，σ = √36 = 6。使用 normalcdf，下限为 −∞（或 −10⁹⁹），上限为 58，均值为 52，标准差为 6，即可得到概率。记住 σ² 是方差，因此需要取平方根得到 σ。

6. Typical Problem 2: Finding k for a Given Probability | 典型问题 2：给定概率求 k

You might see: “X ~ N(75, 8²). Find k such that P(X > k) = 0.15.” Since the calculator’s invNorm works with left‑tail probabilities, rewrite as P(X < k) = 1 − 0.15 = 0.85. Then invNorm(0.85, 75, 8) gives the result. Always sketch the curve and shade the correct tail to avoid errors.

你可能会看到这样的题目：“X ~ N(75, 8²)。求 k 使得 P(X > k) = 0.15。”由于计算器的 invNorm 处理的是左尾概率，需改写为 P(X < k) = 1 − 0.15 = 0.85。然后用 invNorm(0.85, 75, 8) 得出结果。一定要画出曲线并给正确的尾部涂上阴影，以避免错误。

7. Typical Problem 3: Real‑life Application | 典型问题 3：实际应用

Exercise 22E.2 frequently includes contextual problems. For instance: “The weights of apples from an orchard are normally distributed with mean 150 g and standard deviation 25 g. Find the probability that a randomly chosen apple weighs between 130 g and 180 g.” Set X ~ N(150, 25²), then compute normalcdf(130, 180, 150, 25). Interpreting the result in the context of the problem is just as important as the calculation.

练习题 22E.2 经常包含实际情境题。例如：“某果园苹果的重量服从均值为 150 g、标准差为 25 g 的正态分布。求随机选取的一个苹果重量介于 130 g 与 180 g 之间的概率。”设 X ~ N(150, 25²)，然后计算 normalcdf(130, 180, 150, 25)。在题目情境中解读结果与计算本身同样重要。

8. Sum of Independent Normal Variables | 独立正态变量之和

If X₁ ~ N(μ₁, σ₁²) and X₂ ~ N(μ₂, σ₂²) are independent, then their sum S = X₁ + X₂ is also normally distributed with mean μ₁ + μ₂ and variance σ₁² + σ₂². A typical Exercise 22E.2 extension: “The masses of two components are independent normal variables. Find the probability that the total mass exceeds a given value.” Use the combined parameters and the same normalcdf logic.

若 X₁ ~ N(μ₁, σ₁²) 与 X₂ ~ N(μ₂, σ₂²) 相互独立，则它们的和 S = X₁ + X₂ 也服从正态分布，均值为 μ₁ + μ₂，方差为 σ₁² + σ₂²。练习题 22E.2 中的一个典型扩展是：“两个组件的质量是独立的正态变量。求总质量超过某一给定值的概率。”使用合并后的参数并沿用相同的 normalcdf 逻辑即可求解。

9. Normal Approximation to the Binomial | 二项分布的正态近似

For large n, a binomial distribution B(n, p) can be approximated by N(np, np(1 − p)), provided np and n(1 − p) are both greater than 5. You may be asked to apply a continuity correction, e.g. P(X ≤ a) becomes P(Y < a + 0.5) where Y ~ N(np, np(1 − p)). This topic appears in some Exercise 22E.2 sets under the HL curriculum.

当 n 很大时，二项分布 B(n, p) 可用正态分布 N(np, np(1 − p)) 进行近似，前提是 np 和 n(1 − p) 均大于 5。你可能需要应用连续性校正，例如将 P(X ≤ a) 转化为 P(Y < a + 0.5)，其中 Y ~ N(np, np(1 − p))。该主题出现在 HL 课程的部分 22E.2 练习题中。

10. Sketching and Labeling the Bell Curve | 绘制并标注钟形曲线

Even when using a GDC, sketching the normal curve with the mean, standard deviation boundaries, and shaded probability region is a good habit. It helps you verify whether you have selected the correct tail and whether your answer is reasonable. The IB often awards method marks for a clear, labelled sketch.

即使在使用图形计算器时，绘制正态曲线并标注均值、标准差边界以及阴影概率区域也是一个好习惯。这有助于检验你是否选择了正确的尾部，以及答案是否合理。IB 通常会对清晰且带有标注的草图给予方法分。

11. Common Pitfalls | 常见易错点

• Confusing variance and standard deviation. N(μ, σ²) uses variance as the second parameter; always extract σ correctly.
• Forgetting to convert the upper‑tail probability before using invNorm.
• Misapplying continuity correction in normal approximation, leading to inaccurate boundaries.
• Rounding too early during multi‑step calculations, which can accumulate errors.

• 混淆方差与标准差。 N(μ, σ²) 的第二个参数是方差；务必正确提取 σ。
• 在使用 invNorm 之前忘记转换上尾概率。
• 在正态近似中错误使用连续性校正，导致边界不准确。
• 在多步计算中过早四舍五入，可能累积误差。

12. Practice and Exam Tips | 练习与考试技巧

Work through every part of Exercise 22E.2 by writing down the distribution notation, the calculator input, and a concluding sentence in context. Time yourself as you would in an exam, and always double‑check whether the question asks for a probability or a specific value. The normal distribution is one of the most heavily assessed topics, so fluency with both the concepts and the calculator operations is essential.

完成练习题 22E.2 的每一部分时，写下分布记号、计算器输入以及在题目语境中的结论句。像在考试中一样给自己计时，并反复检查题目要求的是概率还是具体数值。正态分布是考查最频繁的主题之一，因此对概念和计算器操作的熟练掌握至关重要。

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