📚 Exercise 22H.1: Normal Distribution Probabilities | 练习22H.1：正态分布概率
In the IB Mathematics curriculum, Exercise 22H.1 often marks students' first deep dive into the normal distribution — a continuous probability distribution used to model a wide range of real-world phenomena such as heights, test scores, and measurement errors. This exercise set typically asks you to calculate probabilities using the standard normal table, apply the Z‑score formula, and interpret results in context. Understanding the concepts behind these questions is essential for success in both Standard and Higher Level papers.
在IB数学课程中，练习22H.1通常是学生首次深入接触正态分布——一种用于模拟身高、考试成绩和测量误差等多种现实世界现象的连续概率分布。这组练习通常要求你使用标准正态分布表计算概率，应用Z分数公式，并结合实际情境解释结果。理解这些题背后的概念对于标准水平和高级水平考试的成功至关重要。
1. What is Exercise 22H.1 About? | 练习22H.1是关于什么的？
In many IB textbooks, Chapter 22 covers probability distributions, and Section 22H introduces the normal distribution. Exercise 22H.1 contains foundational problems designed to build fluency with the bell curve, standardisation, and table reading.
在许多IB教材中，第22章涵盖概率分布，22H节介绍正态分布。练习22H.1包含基础问题，旨在培养你对钟形曲线、标准化以及查表的熟练程度。
The questions often provide a normal variable X with a given mean μ and standard deviation σ, and ask for probabilities such as P(X < a), P(X > b), or P(c < X < d).
题目通常给出一个服从正态分布的变量X，其均值为μ、标准差为σ，要求计算诸如P(X < a)、P(X > b)或P(c < X < d)等概率。
2. The Normal Distribution Curve | 正态分布曲线
The graph of a normal distribution is a symmetric, bell‑shaped curve centred at the mean μ. The curve extends infinitely in both directions, and the total area under the curve equals 1, representing the total probability.
正态分布的图像是一条关于均值μ对称的钟形曲线。该曲线向两端无限延伸，曲线下的总面积等于1，代表总概率。
Because of symmetry, the mean, median, and mode are all equal. About 68% of observations fall within one standard deviation of the mean, and about 95% within two standard deviations.
由于对称性，均值、中位数和众数都相等。大约68%的观测值落在距离均值一个标准差的范围内，约95%落在两个标准差范围内。
3. Parameters: Mean and Standard Deviation | 参数：均值与标准差
The mean μ determines the central location: shifting μ moves the entire curve left or right without changing its shape. The standard deviation σ controls the spread: a larger σ flattens and widens the curve, while a smaller σ produces a taller, narrower peak.
均值μ决定中心位置：改变μ会使整条曲线左右平移而不改变形状。标准差σ控制离散程度：σ越大，曲线越扁平宽阔；σ越小，曲线越高耸狭窄。
In Exercise 22H.1, you will often be given μ and σ directly, but occasionally you must deduce them from a probability statement.
在练习22H.1中，你通常会直接得到μ和σ，但偶尔也需要根据概率条件反推它们。
4. The Standard Normal Distribution | 标准正态分布
To simplify probability calculations, we convert any normal variable into the standard normal variable Z, which has mean 0 and standard deviation 1. The distribution of Z is called the standard normal distribution and is denoted Z ~ N(0, 1).
为了简化概率计算，我们将任何正态变量转换为标准正态变量Z，其均值为0，标准差为1。Z的分布称为标准正态分布，记作Z ~ N(0, 1)。
The standard normal table gives probabilities for Z, so mastering this conversion is essential for solving all normal distribution exercises.
标准正态分布表给出的是Z的概率，因此掌握这一转换对于解答所有正态分布练习至关重要。
5. The Z‑Score Transformation | Z分数变换
Every normal value x is transformed into a Z‑score using the formula:
每个正态变量的值x都使用以下公式转换为Z分数：
Z = (X − μ) / σ
The numerator shows how far x is from the mean, and dividing by σ expresses that distance in units of standard deviation. A positive Z means x is above the mean; a negative Z means it is below.
分子表示x距离均值有多远，除以σ则用标准差为单位表示这一距离。正Z值表示x高于均值；负Z值表示x低于均值。
6. Using the Standard Normal Table | 使用标准正态分布表
ndard normal table usually gives the cumulative probability Φ(z) = P(Z < z) for positive z‑values. To look up a probability, split the z‑ = 1.25, locate the row 1.2 and the column 0.05; their intersection gives Φ(1.25) = 0.8944.
标准正态分布表通常给出正值z的累积概率Φ(z) = P(Z < z)。查表时将z分数拆分为行（前两位数字）和列（第二位小数）。例如，对于z = 1.25，找到行1.2和列0.05，其交叉点给出Φ(1.25) = 0.8944。
| z | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 |
|---|---|---|---|---|---|---|
| 1.2 | 0.8849 | 0.8869 | 0.8888 | 0.8907 | 0.8925 | 0.8944 |
Always confirm whether your table gives P(Z < z) or the area between 0 and z — klet provides cumulative probabilities from −∞ upwards, so you can directly read P(Z < z).
务必确认你所使用的表格给出的是P(Z < z)还是0到z之间的面积——IB公式手册提供的是从−∞向上的累积概率，因此你可以直接读取P(Z < z)。
7. Calculating P(X < a) — Lower Tail | 计算P(X < a) — 下尾概率
Example: Suppose X ~ N(50, 10²). Find P(X < 65).
Step 1: Standardise: Z = (65 − 50) / 10 = 1.5.
Step 2: Look up Φ(1.5) in the table. Φ(1.5) = 0.9332.
Thus, the probability that X is less than 65 is 0.9332, or 93.32%.
示例：设X ~ N(50, 10²)。求P(X < 65)。
第1步：标准化：Z = (65 − 50) / 10 = 1.5。
第2步：查表得Φ(1.5) = 0.9332。
因此，X小于65的概率为0.9332，即93.32%。
8. Calculating P(X > a) — Upper Tail | 计算P(X > a) — 上尾概率
For an upper tail, use the complement rule: P(X > a) = 1 − P(X < a).
Using the same distribution, P(X > 65) = 1 − P(X < 65) = 1 − 0.9332 = 0.0668.
对于上尾概率，使用补集法则：P(X > a) = 1 − P(X < a)。
沿用同一分布，P(X > 65) = 1 − P(X < 65) = 1 − 0.9332 = 0.0668。
This is very common in Exercise 22H.1 — watch for wording like 'greater than', 'exceeds', or 'at least' (in IB, 'at least' is treated as > for continuous data).
这在练习22H.1中非常常见——注意诸如 "greater than"、"exceeds" 或 "at least" 等措辞（在IB中，对于连续数据，"at least" 视为 > 处理）。
9. Probability Between Two Values | 两值之间的概率
To find P(a < X < b), calculate P(X < b) − P(X < a). For X ~ N(50, 10²), find P(40 < X < 60):
Z₁ = (40 − 50) / 10 = −1; Z₂ = (60 − 50) / 10 = 1.
P(Z < 1) = 0.8413, and P(Z < −1) = 1 − P(Z < 1) = 0.1587.
Therefore, P(40 < X < 60) = 0.8413 − 0.1587 = 0.6826.
要求P(a < X < b)，计算P(X < b) − P(X < a)。对于X ~ N(50, 10²)，求P(40 < X < 60)：
Z₁ = (40 − 50) / 10 = −1；Z
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