📚 Exercise 16C.3: Normal Distribution Problems | 练习16C.3：正态分布问题
Exercise 16C.3 focuses on applying the properties of the normal distribution to solve a variety of problems, including finding probabilities, determining unknown means or standard deviations, and working with inverse normal calculations. The normal distribution is one of the most important continuous probability distributions in the IB Mathematics curriculum, used to model many natural phenomena. Mastering these exercises will strengthen your ability to standardise variables, use technology efficiently, and interpret results in real-world contexts.
练习16C.3 着重于应用正态分布的性质解决各类问题，包括求概率、确定未知的均值或标准差，以及进行逆正态分布的计算。正态分布是 IB 数学课程中最重要的连续概率分布之一，用于模拟许多自然现象。掌握这些练习将增强你对变量进行标准化的能力、高效使用技术工具的能力以及在实际背景中解读结果的能力。
1. Understanding the Normal Distribution | 认识正态分布
A normal distribution is a symmetric, bell-shaped curve defined by its mean μ and standard deviation σ. The total area under the curve equals 1, and probabilities correspond to areas under the curve between specified values. The notation X ~ N(μ, σ²) indicates that the random variable X follows a normal distribution with mean μ and variance σ².
正态分布是一条对称的钟形曲线，由其均值 μ 和标准差 σ 定义。曲线下的总面积等于 1，概率对应于指定值之间曲线下的面积。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。
Because the shape depends entirely on μ and σ, we often transform a normal variable into a standard normal variable Z ~ N(0, 1) using the formula:
由于形状完全取决于 μ 和 σ，我们经常使用以下公式将正态变量转换为标准正态变量 Z ~ N(0, 1)：
Z = (X − μ) / σ
This standardisation allows us to use a single reference table or built‑in calculator functions. In IB exams, you are expected to be proficient with both methods.
这一标准化过程使得我们可以使用统一的参考表或计算器内置函数。在 IB 考试中，要求你熟练运用这两种方法。
2. Key Properties of the Normal Curve | 正态曲线的主要性质
The normal curve is symmetric about the mean, so the area to the left of μ is 0.5, and the area to the right is also 0.5. About 68% of the data lies within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations — the empirical rule.
正态曲线关于均值对称，因此均值左侧的面积为 0.5，右侧的面积也为 0.5。大约 68% 的数据落在均值的一个标准差范围内，95% 落在两个标准差范围内，99.7% 落在三个标准差范围内——这是经验法则。
This rule provides quick approximations, but Exercise 16C.3 requires precise probabilities using GDC functions or the standard normal table. Always sketch a diagram to visualise the required area and avoid sign errors.
该法则提供了快速估计，但练习 16C.3 要求使用 GDC 函数或标准正态表得到精确概率。务必画出示意图，将所求面积可视化，以避免符号错误。
3. Using the GDC for Normal Probabilities | 使用 GDC 求正态概率
For a given X ~ N(μ, σ²), your calculator's normal CDF function can find P(a < X < b). The syntax is typically normalcdf(lower bound, upper bound, μ, σ). If there is no lower bound, use a very small number such as −10⁹⁹; for no upper bound, use 10⁹⁹.
对于给定的 X ~ N(μ, σ²)，计算器的正态 CDF 函数可以求出 P(a < X < b)。语法通常为 normalcdf(下限, 上限, μ, σ)。如果没有下限，使用一个非常小的数，如 −10⁹⁹；如果没有上限，则使用 10⁹⁹。
For example, if X ~ N(50, 25) and we want P(X < 45), we compute normalcdf(−10⁹⁹, 45, 50, 5). Remember that σ = √25 = 5, never input the variance. This is one of the most common mistakes in IB problems.
例如，若 X ~ N(50, 25)，我们要求 P(X < 45)，可计算 normalcdf(−10⁹⁹, 45, 50, 5)。请记住 σ = √25 = 5，切勿输入方差。这是 IB 题目中最常见的错误之一。
4. Standardising and Using the Z‑Table | 标准化与使用 Z 值表
normal table to find Φ(z), the cumulative probability P(Z < z). Be careful with negative z‑values; use symmetry Φ(−z) = 1 − Φ(z).
当不允许使用 GDC 或需要展示解题过程时，需对边界值进行标准化：z = (x − μ)/σ。然后使用标准正态表查找 Φ(z)，即累积概率 P(Z < z)。注意负 z 值；利用对称性 Φ(−z) = 1 − Φ(z)。
Suppose X ~ 333. The probability is Φ(0.8333) − Φ(−0.4167), which simplifies to Φ(0.8333) − [1 − Φ(0.4167)].
假设 X ~ N(100, 144)，求 P(95 < X < 110)。将两端标准化：z₁ = (95−100)/12 = −0.4167，z₂ = (110−100)/12 = 0.8333。概率为 Φ(0.8333) − Φ(−0.4167)，可化简为 Φ(0.8333) − [1 − Φ(0.4167)]。
5. Inverse Normal Calculations | 逆正态分布计算
Often Exercise 16C.3 asks: given a probability, find the corresponding x‑value. This is the inverse normal problem. Use the GDC function invNorm(area, μ, σ). The area is always the cumulative probability to the left of the desired x. If a right‑tail area is given, convert it to left‑tail first.
练习 16C.3 经常会问：给定概率，求对应的 x 值。这就是逆正态分布问题。使用 GDC 函数 invNorm(面积, μ, σ)。这里的面积始终是所求 x 左侧的累积概率。如果给出的是右尾面积，需先转换为左尾面积。
For example, if the top 10% h that P(X > k) = 0.10. First rewrite as P(X < k) = 0.90, then apply invNorm(0.90, 65, 8).
例如，若考试成绩前 10% 将获得 A，且成绩服从正态分布 μ = 65，σ = 8，我们需要找出满足 P(X > k) = 0.10 的分数 k。首先改写为 P(X < k) = 0.90，然后使用 invNorm(0.90, 65, 8)。
6. Finding Unknown Mean or Standard Deviation | 求未知均值或标准差
taneous equations. For instance, if P(X < 20) = 0.1 and P(X > 80) = 0.05, you obtain two z‑scores from the table and solve the system.
有些问题会提供两个对称的概率或概率的组合，使你能够确定 μ 和 σ。利用标准化公式建立方程组。例如，若 P(X < 20) = 0.1 且 P(X > 80) = 0.05，可从表中获得两个 z 分数，然后求解方程组。
Because the standard normal distribution is symmetric, if you are given that a certain percentage lies symmetrically around the mean, the corresponding z‑values are opposites. This property greatly simplifies solving for μ and σ.
由于标准正态分布是对称的，如果已知某一百分比对称地分布在均值周围，那么相应的 z 值互为相反数。这一性质极大地简化了 μ 和 σ 的求解。
7. Typical Exercise 16C.3 Problem: Probabilities from Boundaries | 典型练习 16C.3 题型：由边界求概率
Problem: The lifetimes of a brand of batteries are normally distributed with mean 120 hours and standard deviation 15 hours. Find the probability that a randomly chosen battery lasts between 110 and 140 hours.
题目：某品牌电池的寿命服从均值为 120 小时、标准差为 15 小时的正态分布。求随机选取一节电池的寿命在 110 至 140 小时之间的概率。
Solution: Let X ~ iving Φ(1.3333) − Φ(−0.6667) ≈ 0.9082 − 0.2525 = 0.6557. (Verify your GDC gives the same.)
解答：设 X ~ N(120, 225)。P(110 < X < 140) = normalcdf(110, 140, 120, 15) ≈ 0.495。或者，z₁ = (110−120)/15 = −0.6667，z₂ = (140−120)/15 = 1.3333，得 Φ(1.3333) − Φ(−0.6667) ≈ 0.9082 − 0.2525 = 0.6557。（请验证你的 GDC 结果是否一致。）
8. Typical Problem: Inverse Normal with a Right‑Tail Area | 典型题型：右尾面积的逆正态问题
Problem: The weights of apples in an orchard follow a normal distribution with mean 150 g and standard deviation 25 g. The heaviest 15% of apples are sold as "premium". What is the minimum weight for a premium apple?
题目：某果园的苹果重量服从均值为 150 g、标准差为 25 g 的正态分布。最重的 15% 苹果作为“特优”出售。特优苹果的最小重量是多少？
Solution: We need w such that P(X > w) = 0.15 ⇒ 175.9 g. Therefore, an apple must weigh at least about 176 g to be classified as premium.
解答：我们需要 w 使得 P(X > w) = 0.15 ⇒ P(X < w) = 0.85。invNorm(0.85, 150, 25) 得出 w ≈ 175.9 g。因此，苹果至少需要重约 176 g 才能被归类为特优。
9. Combined Probability Scenarios | 组合概率情景
find p = P(single item satisfies condition), then compute P(at least one) = 1 − (1 − p)ⁿ.
练习 16C.3 可能会出现这样的问题：选取一个包含 n 个物件的样本，计算至少有一个符合标准的概率。这会将正态分布与二项分布结合起来：首先求出 p = P(单个物件满足条件)，然后计算 P(至少一个) = 1 − (1 − p)ⁿ。
over 140 hours is 1 − (1 − 0.0918)³ ≈ 0.2509. Always define the binomial variable clearly.
例如，若一节电池寿命超过 140 小时的概率为 0.0918，而你购买了三节电池，则至少有一节寿命超过 140 小时的概率为 1 − (1 − 0.0918)³ ≈ 0.2509。务必清晰地定义二项分布变量。
10. Checking Conditions for Normality | 检验正态性的条件
In IB problems, you must confirm that the normal model is appropriate. Usually the context states that the data are normally distributed, but sometimes you are given a mean and standard deviation from a large sample and are told that the underlying distribution is normal. Be aware that real‑world data may not be exactly normal, but the Central Limit Theorem assures normality for sample means if the sample size is large enough.
在 IB 问题中，你必须确认正态模型是合适的。通常题目会明确说明数据服从正态分布，但有时会给出大样本的均值和标准差，并告知总体分布是正态的。需要注意的是，现实世界的数据可能并不完全服从正态分布，但中心极限定理保证，只要样本量足够大，样本均值的分布近似正态。
When solving Exercise 16C.3, assume the normal distribution holds unless instructed otherwise. Drawing a normal probability plot or checking skewness and outliers is beyond the scope of this exercise set.
在解答练习 16C.3 时，除非另有说明，均假定正态分布成立。绘制正态概率图或检验偏度和异常值超出了本练习集的范围。
11. Common Pitfalls and How to Avoid Them | 常见陷阱及避免方法
Many students mistakenly input the variance σ² instead of the standard deviation σ into their GDC. Always double‑check: the normalcdf and invNorm functions require σ. Also, when using the table, rounding z‑scores too early can lead to inaccurate probabilities. Keep at least three decimal places for z.
许多学生错误地将方差 σ² 而不是标准差 σ 输入 GDC。务必反复检查：normalcdf 和 invNorm 函数需要的是 σ。此外，使用表格时过早对 z 分数进行四舍五入会导致概率不准确。z 值至少保留三位小数。
continuous, so P(Z = a) = 0, and inequalities do not matter for continuous distributions.
另一个陷阱是当概率以顶部或底部的百分比形式给出时，混淆了 P(X > a) 与 P(X < a)。画出示意图并标记面积，可避免误解。最后，记住 Z 始终是连续的，因此 P(Z = a) = 0，对于连续分布而言，不等式符号无影响。
12. Summary and Further Practice | 总结与拓展练习
Exercise 16C.3 consolidates essential skills in handling normal distributions: standardisation, using the GDC, inverse normal queries, and solving for unknown parameters. Proficiency in these techniques will prepare you for exam questions that integrate normal distributions into broader contexts, such as hypothesis testing or regression.
练习 16C.3 巩固了处理正态分布的基本技能：标准化、使用 GDC、逆正态查询以及求解未知参数。熟练运用这些技巧将有助于你应对将正态分布融入更广泛情境的考试题目，例如假设检验或回归分析。
Continue practicing with mixed problems that require combining normal with other distributions. Always reflect on whether the answer is reasonable in the given real‑world context — for instance, a probability cannot be negative or exceed 1, and a lifetime cannot be negative.
继续练习需要将正态分布与其他分布结合的混合题目。始终反思在给定的实际情境下答案是否合理——例如，概率不能为负或超过 1，寿命不能为负。
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