📚 Exercise 15H: Normal Distribution Mastery | 习题15H：精通正态分布
Welcome to your comprehensive guide to Exercise 15H, focusing on the Normal Distribution—a cornerstone of IB Mathematics. Whether you’re studying Analysis & Approaches or Applications & Interpretation, mastering the bell curve is essential for exam success. This article will revisit key concepts, walk through typical problems you’ll meet in Exercise 15H, and share strategies to tackle them with confidence.
欢迎来到 Exercise 15H 全面指南，聚焦于 IB 数学的核心——正态分布。无论你学习的是分析与方法还是应用与解释，掌握钟形曲线对于考试成功至关重要。本文将重温关键概念，演练你将在 Exercise 15H 中遇到的典型问题，并分享应对这些问题的策略。
1. Introduction to Normal Distribution | 正态分布简介
The normal distribution, often called the bell curve, is a continuous probability distribution symmetrical about the mean μ. Many natural phenomena, such as heights, test scores, and measurement errors, follow this pattern.
正态分布常被称为钟形曲线，是一种关于均值 μ 对称的连续概率分布。许多自然现象，如身高、考试成绩和测量误差，都遵循这一模式。
A normal random variable X ~ N(μ, σ²) has mean μ and variance σ². The curve’s shape is determined by these parameters: μ shifts the center, while σ controls the spread.
正态随机变量 X ~ N(μ, σ²) 具有均值 μ 和方差 σ²。曲线的形状由这些参数决定：μ 决定中心位置，而 σ 控制离散程度。
In IB Math, you will use the normal distribution to model real-world data and calculate probabilities using standardized values.
在 IB 数学中，你将使用正态分布对现实世界数据进行建模，并通过标准化值计算概率。
2. The Probability Density Function | 概率密度函数
The probability density function (PDF) for a normal distribution is given below. While you won’t need to evaluate this function manually in IB, understanding its components helps interpret the curve.
正态分布的概率密度函数 (PDF) 如下所示。在 IB 中你无需手动计算该函数，但理解其组成部分有助于解读曲线。
f(x) = 1/(σ√2π) · e^(-(x-μ)²/(2σ²))
该公式表示为 f(x) = 1/(σ√2π) · e^(-(x-μ)²/(2σ²))。
The total area under the curve equals 1, representing total probability. The curve never touches the horizontal axis, extending infinitely in both directions.
曲线下的总面积为 1，代表总概率为 1。曲线永不触及水平轴，左右两侧无限延伸。
tures: the curve peaks at x = μ, and points of inflection occur at μ ± σ. Approximately 68% of data lies within ±1σ, 95% within ±2σ, and 99.7% within ±3σ.
关键特征：曲线在 x = μ 处达到峰值，拐点出现在 μ ± σ 处。大约 68% 的数据落在 ±1σ 内，95% 在 ±2σ 内，99.7% 在 ±3σ 内。
3. Standard Normal Distribution | 标准正态分布
To compare different normal distributions, we convert to the standard normal distribution Z ~ N(0,1) using the z-score formula.
为了比较不同的正态分布，我们使用 z 分数公式将其转换为标准正态分布 Z ~ N(0,1)。
z = (x - μ)/σ
该公式为 z = (x - μ)/σ。
The z-score measures how many standard deviations an observation is from the mean. A positive z means above the mean, negative below.
z 分数衡量观测值距离均值多少个标准差。z 为正表示高于均值，为负表示低于均值。
In Exercise 15H, many problems start by computing z, then using tables or calculators. Remember to always draw a sketch of the standard normal curve and shade the required area.
在 Exercise 15H 中，许多问题都从计算 z 开始，然后使用表格或计算器求解。务必始终画出标准正态曲线草图并标记所需区域。
4. Using the Z-Table | 使用Z表
that gives Φ(z) = P(Z ≤ z) for positive z. For negative z, use symmetry: Φ(-z) = 1 - Φ(z).
IB 提供累积标准正态分布表，给出正 z 的 Φ(z) = P(Z ≤ z)。对于负 z，利用对称性：Φ(-z) = 1 - Φ(z)。
Example: Find P(Z < 1.25). From the table, Φ(1.25) = 0.8944. So about 89.44% of data lies below 1.25 standard deviations above the mean.
示例：求 P(Z < 1.25)。查表得 Φ(1.25) = 0.8944。因此约 89.44% 的数据落在均值以上 1.25 个标准差以内。
ility from -∞ to z. Most IB tables are cumulative, matching calculator function normalcdf.
始终检查表格是否给出从 -∞ 到 z 的累积概率。IB 多数表格为累积型，与计算器函数 normalcdf 一致。
5. Calculating Probabilities: P(Z < z) | 计算概率：P(Z < z)
To find P(Z < a), simply read Φ(a) from the table (if a > 0) or use symmetry for a < 0. With GDC, use normalcdf(-109, a, 0, 1).
求 P(Z < a)，若 a > 0 可直接从表中读取 Φ(a)；若 a < 0 则利用对称性。使用图形计算器时，可用 normalcdf(-109, a, 0, 1)。
Example: X ~ 15²). Find P(X < 115). First z = (115-100)/15 = 1. Then P(Z < 1) = 0.8413. So 84.13% chance.
示例：X ~ N(100, 15²)。求 P(X < 115)。首先 z = (115-100)/15 = 1。则 P(Z < 1) = 0.8413，即概率为 84.13%。
In Exercise 15H, you’ll meet similar questions: given a normal variable and a threshold, find the probability of being below it.
Exercise 15H 中你会遇到类似问题：给定正态变量和一个阈值，求低于该阈值的概率。
6. Calculating Probabilities: P(Z > z) and P(a < Z < b) | 计算概率：P(Z > z) 和 P(a < Z < b)
1 - Φ(a). For P(a < Z < b), subtract: Φ(b) - Φ(a). Always sketch to avoid mistakes.
对于 P(Z > a)，使用互补概率：P(Z > a) = 1 - Φ(a)。对于 P(a < Z < b)，则相减：Φ(b) - Φ(a)。始终画草图以避免错误。
Example: X ~ 0)/10 = 1.0. Then P = Φ(1.0) - Φ(-0.5) = Φ(1.0) - (1 - Φ(0.5)) = 0.8413 - 0.3085 = 0.5328.
示例：X ~ N(50, 10²)。求 P(45 < X < 60)。z₁ = -0.5, z₂ = 1.0。则概率为 Φ(1.0) - Φ(-0.5) = 0.8413 - 0.3085 = 0.5328。
Many Exercise 15H items require careful z-score calculation before using the table. Double-check your arithmetic.
Exercise 15H 中的许多题目都需要先仔细计算 z 分数再查表。请仔细检查算术。
7. Inverse Normal Distribution | 逆正态分布
ue. Use the table to find the z such that Φ(z) = given probability, then solve x = μ + zσ.
逆正态问题给出概率，要求对应的 x 值。利用表格找出满足 Φ(z) = 给定概率的 z，然后求解 x = μ + zσ。
If the table provides cumulative left-tail probabilities, for a right-tail probability p, find z for 1 - p. Your GDC has an invNorm function for direct calculation.
如果表格提供左尾累积概率，对于右尾概率 p，需要查找 1 - p 对应的 z。图形计算器有 invNorm 函数可直接计算。
Example: X ~ ind k such that P(X < k) = 0.95. From table, z ≈ 1.645. Then k = 70 + 1.645×5 = 78.225.
示例：X ~ N(70, 5²)。求 k 使得 P(X < k) = 0.95。查表得 z ≈ 1.645，则 k = 70 + 1.645×5 = 78.225。
Exercise 15H includes inverse normal problems where you interpret percentiles or quality control cut-offs.
Exercise 15H 包含逆正态问题，涉及百分位数或质量控制分界点。
8. Real-Life Applications | 实际应用
The normal distribution models exam scores, packaging weights, and biological measurements. Exercise 15H may ask: "If test scores are normally distributed with mean 65 and standard deviation 8, what proportion of students score above 80?"
正态分布可以对考试成绩、包装重量和生物测量值建模。Exercise 15H 可能会问："如果考试成绩服从均值为 65、标准差为 8 的正态分布，那么分数超过 80 的学生比例是多少？"
Another typical question: quality control in manufacturing, where lengths must be within specification limits. Use normal probabilities to find the proportion of defective items.
另一个典型问题是制造业的质量控制，其中长度必须在规格限内。使用正态概率求不合格品的比例。
In IB exam papers, you are expected to identify the distribution, standardize, use technology accurately, and interpret the result in context.
在 IB 考试中，你需要识别分布类型、标准化、准确使用技术并在情境中解释结果。
9. Common Mistakes | 常见错误
Mistake 1: Forgetting to square the standard deviation when writing variance. N(μ, σ²) takes variance, not σ.
错误 1：写方差时忘记对标准差平方。N(μ, σ²) 使用的是方差，而非 σ。
le incorrectly for negative z-values. Always use symmetry: P(Z < -a) = 1 - P(Z < a).
错误 2：处理负 z 值时误用 z 表。始终利用对称性：P(Z < -a) = 1 - P(Z < a)。
Mistake 3: Confusing left-tail and right-tail probabilities. Draw a diagram and shade the required area.
错误 3：混淆左尾和右尾概率。画出示意图并标记所需区域。
Mistake 4: Not standardizing correctly. Ensure you subtract μ
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