📚 Exercise 22H.2: Normal Distribution Applications | 练习22H.2：正态分布应用
Exercise 22H.2 from the IB Mathematics curriculum challenges students to apply the normal distribution in a variety of probability contexts. This article serves as a comprehensive revision guide, detailing the underlying theory, worked examples, and common examiner tips so you can master normal distribution problems with confidence.
IB 数学课程中的练习 22H.2 要求学生在多种概率情境中应用正态分布。本文提供全面的复习指导，详解基础理论、典型例题以及常见的考官建议，助你自信地掌握正态分布问题。
1. Understanding the Normal Distribution | 理解正态分布
A normal distribution is a continuous probability distribution whose probability density function is a symmetric, bell‑shaped curve. It is fully described by two parameters: the population mean μ and the population standard deviation σ. The notation X ~ N(μ, σ²) indicates that the random variable X follows a normal distribution with mean μ and variance σ².
正态分布是一种连续概率分布，其概率密度函数是对称的钟形曲线。它由两个参数完全描述：总体均值 μ 和总体标准差 σ。符号 X ~ N(μ, σ²) 表示随机变量 X 服从均值为 μ、方差为 σ² 的正态分布。
Properties of the normal curve include symmetry about the mean, an area of exactly 1 under the curve, and the fact that approximately 68% of the data lie within one standard deviation of the mean, 95% within two, and 99.7% within three.
正态曲线的性质包括关于均值对称、曲线下面积为 1，以及大约 68% 的数据落在均值的一个标准差内，95% 落在两个标准差内，99.7% 落在三个标准差内。
2. The Standard Normal Distribution and Z‑Scores | 标准正态分布与 Z 值
To compare values from different normal distributions or to use statistical tables, we transform any normal variable X into the standard normal variable Z ~ N(0, 1). The z‑score measures how many standard deviations an observation x is above or below the mean.
为了比较来自不同正态分布的数据或使用统计表，我们将任意正态变量 X 转换为标准正态变量 Z ~ N(0, 1)。z 值衡量观测值 x 距均值有多少个标准差。
z = (x – μ) / σ
For example, if X ~ N(50, 10²) and x = 65, then z = (65 – 50) / 10 = 1.5, indicating that 65 is 1.5 standard deviations above the mean.
例如，若 X ~ N(50, 10²) 且 x = 65，则 z = (65 – 50) / 10 = 1.5，表明 65 在均值上方 1.5 个标准差处。
3. Using Technology or Tables to Find Probabilities | 用技术或表格求概率
In IB examinations, you are expected to use your graphing calculator (GDC) to compute normal probabilities directly without converting to z‑scores unless specifically asked. The relevant functions are typically normalcdf for cumulative probabilities and invNorm for inverse calculations.
在 IB 考试中，除非特别要求，否则你应该使用图形计算器 (GDC) 直接计算正态概率，而无需转换为 z 值。相关函数通常为 normalcdf（累积概率）和 invNorm（逆运算）。
When using a printed standard normal table, you first compute the z‑score and then look up the cumulative area to the left of that z‑value. Remember that most tables give P(Z < z) for positive z.
使用纸质标准正态表时，你需先计算 z 值，然后查找该 z 值左侧的累积面积。请记住，大多数表格给出的是正 z 值的 P(Z < z)。
| GDC command | 用来计算 |
| normalcdf(lower, upper, μ, σ) | P(lower < X < upper) |
| invNorm(area, μ, σ) | 给定左侧面积求 x 值 |
4. Example 1 – Probability Below a Given Value | 例 1：低于某值的概率
Problem: The masses of apples harvested from an orchard are normally distributed with mean 150 g and standard deviation 20 g. Find the probability that a randomly selected apple has a mass less than 175 g.
问题：果园收获的苹果质量服从正态分布，均值为 150 g，标准差为 20 g。求随机选取一个苹果质量小于 175 g 的概率。
Let X ~ N(150, 20²). We require P(X < 175). Using the GDC: normalcdf(–∞, 175, 150, 20) gives approximately 0.8944. Alternatively, compute z = (175 – 150)/20 = 1.25, and from the table, P(Z < 1.25) = 0.8944.
令 X ~ N(150, 20²)。需要求 P(X < 175)。使用 GDC：normalcdf(–∞, 175, 150, 20) 得到约 0.8944。或计算 z = (175 – 150)/20 = 1.25，查表得 P(Z < 1.25) = 0.8944。
Interpretation: there is about an 89.4% chance that an apple weighs less than 175 g.
解释：苹果重量低于 175 g 的概率约为 89.4%。
5. Example 2 – Probability Above a Given Value | 例 2：高于某值的概率
Problem: Using the same apple distribution, determine the proportion of apples that weigh more than 130 g.
问题：沿用相同的苹果分布，求质量超过 130 g 的苹果所占比例。
∞, 150, 20), yielding about 0.8413. The symmetry of the normal curve means P(X > 130) = 1 – P(X < 130).
我们需要 P(X > 130)。在 GDC 上使用 normalcdf(130, ∞, 150, 20)，得到约 0.8413。正态曲线的对称性意味着 P(X > 130) = 1 – P(X < 130)。
The result, 0.8413, tells us that roughly 84.1% of the apples weigh over 130 g.
结果 0.8413 告诉我们，大约 84.1% 的苹果重量超过 130 g。
6. Example 3 – Probability Between Two Values | 例 3：介于两个值之间的概率
Problem: For X ~ N(150, 20²), calculate the probability that an apple weighs between 140 g and 160 g.
问题：对于 X ~ N(150, 20²)，计算一个苹果重量在 140 g 到 160 g 之间的概率。
0) ≈ 0.3829. This interval lies symmetrically around the mean and represents roughly a one‑standard‑deviation band; indeed 68% would be from 130 to 170, so a narrower 140–160 band naturally yields a smaller probability.
P(140 < X < 160) = normalcdf(140, 160, 150, 20) ≈ 0.3829。此区间关于均值对称，大致相当于半个标准差的范围；确实，68% 落在 130–170 之间，因此较窄的 140–160 区间自然给出更小的概率。
Use your GDC to check that P(130 < X < 170) ≈ 0.6827, confirming the empirical rule.
请用你的 GDC 验证 P(130 < X < 170) ≈ 0.6827，以确认经验法则。
7. Example 4 – Inverse Normal: Finding a Cut‑Off Value | 例 4：逆正态：求临界值
Problem: The top 10% of apples are classified as “premium”. Find the minimum mass required for an apple to be premium, given X ~ N(150, 20²).
问题：质量排名前 10% 的苹果被列为“特级”。已知 X ~ N(150, 20²)，求苹果达到特级所需的最小质量。
0) on the GDC gives k ≈ 175.63 g. Thus an apple must weigh at least 175.6 g to be premium.
我们需要 k 使得 P(X > k) = 0.10，也即 P(X < k) = 0.90。在 GDC 上使用 invNorm(0.90, 150, 20) 得到 k ≈ 175.63 g。因此，苹果必须至少重 175.6 g 才能成为特级。
Always check the direction of the inequality: invNorm gives the value with a specified area to the left.
务必检查不等式的方向：invNorm 给出指定左侧面积对应的值。
8. The Empirical Rule and Quick Estimates | 经验法则与快速估算
Even without a calculator, you can make informed estimates using the 68–95–99.7 rule. For any normal distribution, approximately 95% In the apple example, μ ± 2σ gives 150 ± 40, i.e. 110 to 190 g. This means it is rare (2.5% in each tail) to find an apple lighter than 110 g or heavier than 190 g.
即使没有计算器，你也可以借助 68-95-99.7 法则进行有依据的估算。对于任何正态分布，大约 95% 的观测值落在 μ ± 2σ 内。在苹果的例子中，μ ± 2σ 得到 150 ± 40，即 110 至 190 g。这意味着低于 110 g 或高于 190 g 的苹果非常少见（每尾各 2.5%）。
These benchmarks help you quickly assess whether your computed probabilities are reasonable. For Exercise 22H.2, always perform a sanity check with the empirical rule after finishing a calculation.
这些基准帮助你快速评估计算出的概率是否合理。对于练习 22H.2，在完成计算后，始终用经验法则进行合理性检查。
9. Common Pitfalls When Solving Exercise 22H.2 | 解答练习 22H.2 的常见误区
Mixing up σ and σ²: The normal distribution is written N(μ, σ²). When entering parameters into your GDC, use the standard deviation σ, not the variance. Double‑check that you have entered 20, not 400, for our apple example.
混淆 σ 和 σ²：正态分布写作 N(μ, σ²)。在 GDC 中输入参数时，使用标准差 σ 而非方差。再检查一下，在苹果示例中你输入的是 20，而不是 400。
Tail confusion: Questions often ask for “greater than” or “at least” und, set the upper bound to a very large number (e.g. 1E99) to represent ∞. Similarly, for “less than” use lower bound –1E99.
尾部混淆：问题常要求“大于”或“至少”。如果使用 normalcdf 时设有下界，把上界设为一个非常大的数（如 1E99）以表示 ∞。同样地，“小于”时下界用 –1E99。
Not verifying conditions: The normal model is only appropriate when the underlying data are continuous and roughly symmetric with no extreme outliers. Exercise 22H.2 mostly provides clearly normal contexts, but always read the question stem carefully.
未验证条件：只有当基础数据是连续的、大致对称且没有极端异常值时，正态模型才适用。练习 22H.2 大多数情况下提供明确的正态背景，但请始终仔细阅读题干。
10. Practice Problems and Exam Technique | 练习题与应试技巧
To consolidate your skills, attempt these variations similar to Exercise 22H.2. Write out the random variable, the GDC command, and a concluding sentence in context.
为巩固你的技能，请尝试以下与练习 22H.2 类似的变体。写出随机变量、GDC 命令以及结合背景的结论句。
- English: The lengths of a batch of pencils are normally distributed with mean 18.0 cm and standard deviation 0.4 cm. Find the probability that a pencil is longer than 18.5 cm.
- 中文：一批铅笔的长度服从正态分布，均值为 18.0 cm，标准差为 0.4 cm。求一支铅笔长度超过 18.5 cm 的概率。
- English: The IQ scores of a population are N(100, 15²). What IQ score separates the lowest 5% from the rest?
- 中文：某人群的智商分数服从 N(100, 15²)。哪个 IQ 分数能将最低的 5% 与其余人区分开？
In the exam, show your working: state the distribution, write the probability statement, record the GDC command and the result, and finally answer in words. This earns full method marks even if a keystroke error occurs.
在考试中，请展示步骤：写明分布、写出概率表达式、记录 GDC 命令及结果，最后用文字作答。即使按键失误，这样也能获得完整的方法分。
11. Linking Normal Distribution to Other Topics | 正态分布与其他专题的联系
The normal distribution often appears in IB questions alongside binomial distribution via normal approximations, or in hypothesis testing when the test statistic is assumed normal. Understanding how to standardise, find critical values, and interpret probabilities is essential for these higher‑order tasks.
正态分布经常与二项分布（通过正态近似）或假设检验（当检验统计量假定为正态时）一起出现在 IB 题目中。理解如何进行标准化、如何求临界值以及如何解释概率，对于完成这些高阶任务至关重要。
You may also see normal distribution used in sampling distributions of the sample mean, where the variance becomes σ²/n. Keep this relationship in mind when revising topics like the central limit theorem.
你还可能在样本均值的抽样分布中见到正态分布，此时方差变为 σ²/n。在复习中心极限定理等专题时，请记住这一关系。
12. Summary and Final Thoughts | 总结与最后思考
Mastering Exercise 22H.2 requires a solid grasp of the normal curve, accurate use of your GDC’s normalcdf and invNorm functions, and the discipline to interpret results in context. Whether you are calculating proportions or determining thresholds, always sketch the curve, shade the relevant area, and double‑check your parameters.
掌握练习 22H.2 需要牢固把握正态曲线、准确使用 GDC 的 normalcdf 和 invNorm 函数，以及结合背景解读结果的自律。无论你是计算比例还是确定阈值，都请画出曲线草图、涂出相关区域并仔细核对参数。
With regular practice, you will find that normal distribution problems become a reliable source of marks on your IB Mathematics assessment.
通过定期练习，你会发现正态分布问题将成为你在 IB 数学评估中可靠的得分点。
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