Exercise 18B: Mastering Normal Distribution for IB Mathematics | IB 数学练习 18B: 掌握正态分布

📚 Exercise 18B: Mastering Normal Distribution for IB Mathematics | IB 数学练习 18B: 掌握正态分布

Exercise 18B in your IB Mathematics course is a crucial set of problems designed to deepen your understanding of the normal distribution. This section typically covers finding probabilities, working with the standard normal variable Z, applying the inverse normal function, and tackling real-world contexts. Whether you are enrolled in Analysis & Approaches or Applications & Interpretation, a solid grasp of these exercises will strengthen your statistics skills for internal assessments and the final exam.

IB 数学课程中的练习 18B 是一组关键题目，旨在加深你对正态分布的理解。这一部分通常涵盖求概率、使用标准正态变量 Z、应用逆正态函数以及处理实际背景问题。无论你选修的是分析与方法还是应用与解释，牢牢掌握这些练习都将为内部评估和最终考试强化统计技能。

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

A normal distribution is a continuous probability distribution modelled by a bell‑shaped curve that is symmetric about the mean μ. The probability density function is given by f(x) = (1/(σ√(2π))) e⁻⁽ˣ⁻μ⁾²/⁽²σ²⁾, where σ is the standard deviation. In IB, we write X ~ N(μ, σ²) to indicate that the random variable X follows a normal distribution with mean μ and variance σ². The total area under the curve equals 1, representing certainty.

正态分布是一种连续型概率分布，其图像是一条关于均值 μ 对称的钟形曲线。概率密度函数为 f(x) = (1/(σ√(2π))) e⁻⁽ˣ⁻μ⁾²/⁽²σ²⁾，其中 σ 是标准差。在 IB 中，我们用 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。曲线下的总面积为 1，代表必然事件。

2. Properties of the Normal Curve | 正态曲线的性质

The normal curve has several key properties you must remember for Exercise 18B. It is symmetric about the mean, so P(X < μ – a) = P(X > μ + a). The mean, median and mode are all equal. As x moves away from the mean, the curve approaches the horizontal axis but never touches it (asymptotic). The points of inflection occur at μ ± σ; this is where the curvature changes direction.

正态曲线具有若干你在练习 18B 中必须牢记的关键性质。它关于均值对称，因此 P(X < μ – a) = P(X > μ + a)。均值、中位数和众数三者相等。当 x 远离均值时，曲线无限趋近于水平轴但并不接触（渐近线）。拐点出现在 μ ± σ 处，此处曲线弯曲方向发生改变。

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

To solve problems without a GDC’s built‑in functions, we often convert an X value to a z‑score using the formula z = (x – μ) / σ. The standard normal variable Z follows Z ~ N(0, 1²). This transformation allows us to use a single table of probabilities. For instance, if X ~ N(70, 5²) and we want P(X < 75), we compute z = (75 – 70)/5 = 1.00 and then read P(Z < 1.00) from the table.

若不用图形计算器的内置函数求解，我们通常使用公式 z = (x – μ) / σ 将 X 值转换为 z 分数。标准正态变量 Z 服从 Z ~ N(0, 1²)。这一变换使我们能使用单一概率表。例如，若 X ~ N(70, 5²) 且需要求 P(X < 75)，则计算 z = (75 – 70)/5 = 1.00，然后从表中读取 P(Z < 1.00)。

4. Using the Standard Normal Table | 使用标准正态分布表

IB data booklets provide a table of cumulative probabilities Φ(z) = P(Z < z) for z ≥ 0. Because of symmetry, you can find probabilities for negative z‑values: Φ(-z) = 1 – Φ(z). For example, Φ(-1.96) = 1 – 0.9750 = 0.0250. When working with P(Z > z), use P(Z > z) = 1 – Φ(z). Always sketch a bell curve to visualise the required area, as this reduces errors in Exercise 18B.

IB 公式手册提供了一张关于 z ≥ 0 的累积概率表 Φ(z) = P(Z < z)。利用对称性，你可以求出负 z 值的概率：Φ(-z) = 1 – Φ(z)。例如，Φ(-1.96) = 1 – 0.9750 = 0.0250。当计算 P(Z > z) 时，使用 P(Z > z) = 1 – Φ(z)。务必画出钟形曲线将所需面积可视化，这样可以减少练习 18B 中的错误。

5. Calculating Probabilities for a Given Normal Distribution | 计算给定正态分布的概率

In Exercise 18B, you will often be asked to find P(a < X < b) for X ~ N(μ, σ²). First standardise both boundaries: z₁ = (a – μ)/σ and z₂ = (b – μ)/σ. Then P(a < X < b) = Φ(z₂) – Φ(z₁). If the distribution is on a GDC, you can also use normCdf(a, b, μ, σ) directly. Always check whether the boundary values are inclusive or exclusive – for continuous distributions it makes no difference, but the wording may guide region definitions.

在练习 18B 中，你经常需要针对 X ~ N(μ, σ²) 计算 P(a < X < b)。先将两个边界标准化：z₁ = (a – μ)/σ，z₂ = (b – μ)/σ。那么 P(a < X < b) = Φ(z₂) – Φ(z₁)。如果使用图形计算器，你也可以直接使用 normCdf(a, b, μ, σ)。务必检查边界值是包含还是不包含——对于连续分布而言没有区别，但题目措辞可为区域定义提供指引。

6. Inverse Normal: Finding Quantiles | 逆正态：求分位数

If a question gives the probability and asks for the corresponding value x, you need the inverse normal function. For Z ~ N(0, 1²), find the z‑score such that Φ(z) = p using the table or invNorm(p) on a GDC. Then transform back: x = μ + zσ. In context, an IB problem might say ‘Find the height exceeded by 10% of students’ – this means P(X > h) = 0.10, so P(X < h) = 0.90 and you find the 90th percentile.

若题目给出概率并要求对应的值 x，则需要使用逆正态函数。对于 Z ~ N(0, 1²)，可通过查表或计算器 invNorm(p) 找到满足 Φ(z) = p 的 z 分数。然后反变换：x = μ + zσ。在具体情景中，IB 题目可能会说“求出被 10% 学生超过的身高”——这意味着 P(X > h) = 0.10，因此 P(X < h) = 0.90，需求第 90 百分位数。

7. The Empirical Rule (68–95–99.7) | 经验法则 (68-95-99.7)

The empirical rule provides quick probability estimates that are especially useful for checking answers. Approximately 68% of observations fall within μ ± 1σ, about 95% within μ ± 2σ, and nearly 99.7% within μ ± 3σ. You might see questions in Exercise 18B that test your ability to apply this rule without a calculator, such as ‘In a normal distribution with mean 50 and standard deviation 5, what percentage of data lies between 45 and 55?’ The answer is approximately 68%.

经验法则能提供快速概率估计，特别适合用来检验答案。大约 68% 的观测值落在 μ ± 1σ 内，约 95% 落在 μ ± 2σ 内，近乎 99.7% 落在 μ ± 3σ 内。练习 18B 中可能会见到无需计算器就能应用该法则的题目，比如“在均值为 50、标准差为 5 的正态分布中，数据落在 45 到 55 之间的百分比是多少？”答案约为 68%。

8. Applications: Real-Life Problems | 应用：实际问题

Exercise 18B usually contains contextual problems, for instance the weights of cereal boxes, examination scores, or the lengths of bolts. You must extract μ and σ from the wording, define the random variable clearly, and decide which probability is required. Always pay attention to wording like ‘more than’, ‘at most’, ‘between’, or ‘exactly’ – though ‘exactly’ for a continuous distribution gives zero probability, sometimes the question implies a rounding interval.

练习 18B 通常包含实际情境题，例如麦片盒的重量、考试成绩或螺栓的长度。你必须从措辞中提取 μ 和 σ，明确定义随机变量，并判断需要哪种概率。要时刻关注诸如“大于”、“至多”、“介于”或“恰好”等用词——尽管连续分布下“恰好”的概率为零，但有时题目隐含一个舍入区间。

9. Combining Normal Distributions: Sums and Differences | 正态分布的线性组合

If X ~ N(μ₁, σ₁²) and Y ~ N(μ₂, σ₂²) are independent, then a linear combination aX + bY also follows a normal distribution with mean aμ₁ + bμ₂ and variance a²σ₁² + b²σ₂². This property is frequently examined in later parts of Exercise 18B. For example, the total mass of 4 apples, each with mean 150 g and standard deviation 12 g, is normally distributed with μ = 4×150 = 600 g and σ = √(4×12²) = 24 g.

若 X ~ N(μ₁, σ₁²) 与 Y ~ N(μ₂, σ₂²) 独立，那么线性组合 aX + bY 也服从正态分布，均值为 aμ₁ + bμ₂，方差为 a²σ₁² + b²σ₂²。练习 18B 的后半部分经常考查这一性质。例如，每个苹果均重 150 g、标准差 12 g，4 个苹果的总质量服从正态分布，μ = 4×150 = 600 g，σ = √(4×12²) = 24 g。

10. Normal Approximation to the Binomial Distribution | 正态近似二项分布

When n is large and p is close to 0.5, a binomial distribution B(n, p) can be approximated by a normal distribution N(np, np(1-p)). For IB purposes, the conditions are np > 5 and n(1-p) > 5. Questions in Exercise 18B might ask you to use a continuity correction: e.g., to approximate P(X ≥ 30), use P(X > 29.5) under the normal curve. This adjustment improves accuracy because a discrete distribution is being modelled by a continuous one.

当 n 很大且 p 接近 0.5 时，二项分布 B(n, p) 可用正态分布 N(np, np(1-p)) 来近似。IB 要求的条件是 np > 5 且 n(1-p) > 5。练习 18B 中可能要求使用连续性校正：例如，要近似计算 P(X ≥ 30)，在正态曲线下应使用 P(X > 29.5)。这一调整可提高准确性，因为是在用连续分布对离散分布建模。

11. Common Mistakes and Tips for Exercise 18B | 练习 18B 常见错误与技巧

Students frequently forget to square the standard deviation when writing variance, or they confuse population standard deviation with sample values in a hypothesis test context. Another pitfall is reading the z‑table incorrectly – always check whether the table gives the area to the left or the right of z. Also, when using an inverse normal function, ensure your GDC is set to ‘tail left’ unless the problem states otherwise. Drawing a labelled diagram is the single best defence against these mistakes.

学生常常在写出方差时忘记将标准差平方，或者在假设检验情境中混淆总体标准差和样本值。另一个陷阱是读错 z 表——务必核实表提供的是 z 左侧还是右侧的面积。此外，使用逆正态函数时，确保图形计算器设置为“左侧尾部”，除非题目另有说明。绘制带标注的示意图是防范这些错误的最佳办法。

12. Practice Questions (similar to Exercise 18B) | 练习题（类似练习 18B）

Try these exercises to sharpen your skills:

• The masses of adult cats are normally distributed with mean 4.2 kg and standard deviation 0.35 kg. Find the proportion of cats weighing less than 3.8 kg.
• A machine fills bottles such that volumes are N(500 ml, 8² ml²). What volume is exceeded by 2.5% of the bottles?
• Scores on an IB exam are normally distributed with mean 68 and standard deviation 9. Find the probability that a randomly chosen student scores between 60 and 80.
• Given that P(Z > k) = 0.0735, find k.
• X ~ N(10, 2²) and Y ~ N(15, 3²) are independent. Find P(X + Y > 28).

Use both GDC methods and standard normal tables where possible, and compare your answers.

尝试以下练习来打磨技能：

• 成年猫的体重服从均值为 4.2 kg、标准差为 0.35 kg 的正态分布。求体重低于 3.8 kg 的猫的比例。
• 一台灌装机灌装瓶子，容量服从 N(500 ml, 8² ml²)。求被 2.5% 的瓶子超过的容量。
• IB 某次考试成绩服从均值为 68、标准差为 9 的正态分布。求随机抽取一名学生其成绩介于 60 到 80 的概率。
• 已知 P(Z > k) = 0.0735，求 k。
• X ~ N(10, 2²) 与 Y ~ N(15, 3²) 独立。求 P(X + Y > 28)。

尽可能同时使用计算器方法和标准正态表，并比较答案。

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