📚 Exercise 22J – Normal Distribution in Depth | 练习22J：深入正态分布
Exercise 22J is a comprehensive problem set designed for IB Mathematics students to master the normal distribution. Whether you are analysing data in Biology, modelling measurement errors in Physics, or tackling probability questions in the exam, the normal distribution is an indispensable tool. This article walks you through every key concept, technique, and common pitfall that appears in Exercise 22J, with step-by-step explanations and bilingual insights to ensure you can solve each question with confidence.
练习22J是一套为IB数学学生精心设计的综合练习，旨在彻底掌握正态分布。无论你是在生物课中分析数据、在物理课中为测量误差建模，还是在考试里应对概率题，正态分布都是不可或缺的工具。本文带你逐一梳理练习22J涉及的每一个核心概念、解题技巧与常见陷阱，并配以中英双语的逐步讲解，确保你能自信地解出每一道题目。
1. The Normal Curve and Its Properties | 正态曲线及其性质
The normal distribution, often called the bell curve, is a continuous probability distribution symmetric about the mean μ. Its shape is determined entirely by two parameters: the mean μ (location) and the standard deviation σ (spread). The total area under the curve equals 1, representing the entire probability space. Because the curve is symmetric, the mean, median, and mode all coincide at the centre. About 68% of the data lies within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ – the empirical rule that frequently appears in IB exam questions.
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。其形状完全由两个参数决定：均值μ（位置）和标准差σ（离散程度）。曲线下的总面积为1，代表整个概率空间。由于曲线是对称的，均值、中位数和众数都重合在中心。大约68%的数据落在均值±1σ内，95%落在±2σ内，99.7%落在±3σ内——这条经验法则在IB考试中频繁出现。
operties to estimate probabilities without a calculator, or to check the reasonableness of your answers. For instance, if a question states that X ~ ou know that P(40 < X < 60) should be approximately 0.68. Any answer far from this calls for a re-check.
在练习22J中，你将运用这些性质在没有计算器的情况下估算概率，或者检验答案的合理性。例如，若题目给出X ~ N(50, 100)，则σ = 10。你立刻就能知道P(40 < X < 60)应约为0.68。任何与此相差甚远的答案都需要重新检查。
2. Standardising: Z-Scores | 标准化：Z分数
To compare any normal distribution to standard normal tables, we convert a raw value x into a z-score using the formula:
z = (x − μ) / σ
This transformation shifts the distribution to have a mean of 0 and standard deviation of 1, denoted as Z ~ N(0, 1). The z-score tells us how many standard deviations x lies above or below the mean. A positive z indicates a value above the mean, a negative z below. In Exercise 22J, you will calculate z-scores for given μ and σ, and then use the standard normal table to find probabilities.
为了将任意正态分布与标准正态表对照，我们用公式将原始值x转化为z分数：
z = (x − μ) / σ
这一变换将分布平移并缩放至均值为0、标准差为1，记作Z ~ N(0, 1)。z分数告诉我们x在均值之上或之下多少个标准差。z值为正表示高于均值，为负表示低于均值。在练习22J中，你需要根据给定的μ和σ计算z分数，再借助标准正态表求概率。
3. Reading the Standard Normal Table | 查阅标准正态分布表
ful: P(Z ≤ −z) = 1 − Φ(z). In Exercise 22J, you must become fluent in reading the table for both positive and negative z, as well as finding probabilities in regions between two values.
IB公式手册提供了标准正态分布的累积概率表，Φ(z) = P(Z ≤ z)。该表给出了某z值左侧的面积。例如，Φ(1.25) ≈ 0.8944。由于总面积为1，右侧的概率就是1 − Φ(z)。对称性同样实用：P(Z ≤ −z) = 1 − Φ(z)。在练习22J中，你必须熟练查阅正负z值的表格，并能求出两个值之间的区域概率。
Since Φ(−1.5) = 1 − Φ(1.5), you can compute the probability quickly with the table values.
查阅表格时，一定要快速画出钟形曲线并涂上所求区域。这一可视化步骤可以避免许多符号错误。例如，P(−1.5 < Z < 0.5) = Φ(0.5) − Φ(−1.5)。因为Φ(−1.5) = 1 − Φ(1.5)，你就能迅速用表格值算出概率。
4. Calculating Probabilities for Any Normal Distribution | 计算任意正态分布的概率
Once you are comfortable with Z, you can handle any X ~ is: standardise the boundary value(s), consult the table, and combine probabilities. For X ~ /15 = 1.00. Then P(X > 115) = 1 − Φ(1.00) ≈ 0.1587. Exercise 22J contains many variations: 'greater than', 'less than', 'between', and 'outside' ranges. Always convert to z-scores before using the table.
一旦你熟悉了Z，就可以处理任意X ~ N(μ, σ²)了。一般方法是：将边界值标准化，查表，再组合概率。设X ~ N(100, 15²)，求P(X > 115)，计算z = (115 − 100)/15 = 1.00。然后P(X > 115) = 1 − Φ(1.00) ≈ 0.1587。练习22J包含多种变体：'大于'、'小于'、'介于'和'区间外'的概率。务必先转化为z分数再查表。
If the variance is given instead of the standard deviation, take the square root first. If the question involves a sample mean, recall that the standard deviation of the sample mean is σ/√n. These IB-level nuances are woven into Exercise 22J to test your full understanding.
如果给出的是方差而非标准差，记得先开平方。如果问题涉及样本均值，请记住样本均值的标准差是σ/√n。这些IB级别的细微之处被编织进练习22J中，以检验你的全面理解。
5. Inverse Normal: Finding the Cut-off Value | 逆正态：求临界值
robability in the body of the normal table to find the z-score. For example, if the top 10% z ≈ 1.2816. Then x = μ + zσ. Exercise 22J has several inverse questions, often phrased as 'find the guaranteed time' or 'find the minimum mark for an A grade'.
逆正态问题把过程倒过来：给定概率，找到对应的x值。首先在正态分布表内查找概率以获取z分数。例如，若要找出分布中前10%对应的临界值，我们需要k使得P(X > k) = 0.10，因此P(Z ≤ z) = 0.90，查表得z ≈ 1.2816。最后由x = μ + zσ求出x。练习22J中有多道逆问题，通常表述为'求保证时间'或'求获得A等级的最低分数'。
When using technology, many IB questions expect you to use the inverse normal function on your GDC (graphic display calculator). However, the problem set Exercise 22J often mixes table-based and calculator-based methods to ensure you understand the underlying logic, not just button-pressing.
在使用技术工具时，许多IB题目期望你能用图形计算器（GDC）的逆正态函数。然而，练习22J常常混合表格法与计算器法，以确保你理解底层逻辑，而不是只会按按钮。
6. Determining Unknown Mean or Standard Deviation | 确定未知均值或标准差
μ and σ. Here you must set up a system of two equations using the standardised form. Let the corresponding z-scores from the inverse table be z₁ and z₂. Then:
(30 − μ) / σ = z₁,  (75 − μ) / σ = z₂
Solve these simultaneously. Typically subtraction eliminates μ first. This algebraic skill is essential for IB papers, and Exercise 22J provides sufficient practice to build fluency.
练习22J中最具挑战性的一些题目会给出两个概率条件，比如P(X < 30) = 0.15和P(X > 75) = 0.08，要求你求出μ和σ。此时你需要用标准化形式建立两个方程。设通过逆查表得到的z分数分别为z₁和z₂，则有：
(30 − μ) / σ = z₁,  (75 − μ) / σ = z₂
联立求解。通常先用减法消去μ。这一代数技能对IB考试至关重要，练习22J提供了充足的训练来帮助你熟练掌握。
7. Normal Approximation to Binomial Distribution | 二项分布的正态近似
hould be greater than 5. Exercise 22J includes problems where you must decide whether the approximation is appropriate, then calculate probabilities using the normal model. This links two major topics and highlights the power of the normal distribution as a limiting case.
当试验次数n很大且p不太接近0或1时，二项分布B(n, p)可用正态分布近似，其中μ = np，σ² = np(1−p)。经验法则要求np和nq（q = 1−p）均大于5。练习22J中会呈现一些题目，要求你先判断近似是否合适，再用正态模型计算概率。这连接了两大主题，并凸显了正态分布作为极限情况的力量。
8. Applying Continuity Correction | 应用连续性校正
a + 0.5). Exercise 22J carefully drills this adjustment, without which marks are often lost. Always check whether the question explicitly requires the correction or allows direct GDC binomial calculations for comparison.
由于二项分布是离散的，而正态分布是连续的，必须运用连续性校正。若要近似二项分布的P(X ≤ a)，应使用正态分布求P(X < a + 0.5)。类似地，P(X ≥ a)转化为P(X > a − 0.5)；对于严格的P(X = a)，需计算P(a − 0.5 < X < a + 0.5)。练习22J专门针对这一调整进行训练，不做校正往往会丢分。务必看清题目是明确要求校正，还是允许直接用GDC二项分布计算作为对照。
9. Real-World Contexts in IB Exam | IB考试中的实际背景
IB questions frequently embed normal distribution into realistic scenarios: battery lifetimes, examination marks, heights of plants, machinery tolerances, or filling processes. Exercise 22J mirrors this style. For instance, a problem might state: 'The masses of apples from an orchard are normally distributed with mean 180 g and standard deviation 25 g. Grade A apples weigh more than 210 g. Find the proportion of Grade A apples.' Such contexts require careful reading to extract μ, σ, and the probability statement. Translating words into mathematical notation is half the battle.
IB题目经常将正态分布嵌入真实场景：电池寿命、考试成绩、植物高度、机器公差或灌装过程。练习22J正反映了这种风格。例如，一道题可能这样描述：'某果园苹果的质量服从正态分布，均值为180 g，标准差为25 g。A级苹果重量超过210 g。求A级苹果的比例。' 这类情景需要仔细阅读以提取μ、σ及概率表述。将文字转化为数学符号是成功的一半。
10. Step-by-Step Approach to Exercise 22J | 练习22J的逐步解题法
A systematic approach will greatly reduce errors. First, write down X ~ N(μ, σ²) or N(μ, σ²) as given. Second, identify what is required – a probability, a value, or a parameter. Third, standardise if necessary, and draw a labelled sketch. Fourth, use the normal table or GDC to obtain probabilities or z-scores. Fifth, for inverse problems, rearrange the formula correctly. Finally, write a concluding statement in context, rounding to the specified precision (usually 3 or 4 significant figures). Exercise 22J is designed so that each step reinforces discipline.
采用系统方法会大大减少错误。首先，写下题设的X ~ N(μ, σ²)。其次，明确要求的是概率、某个值还是参数。第三，必要时进行标准化，并画出带标注的草图。第四，使用正态表或GDC求出概率或z分数。第五，对于逆问题，正确地重新排列公式。最后，根据背景写出结论句，并按指定精度（通常3或4位有效数字）四舍五入。练习22J的设计正是为了巩固每一步的规范性。
11. Common Pitfalls and How to Avoid Them | 常见陷阱及规避方法
ways check: if variance is 36, then σ = 6. Another frequent error is forgetting to convert 'greater than' to 'less than' when reading the table – l can lead to wrong z-sign. In the normal approximation, omitting continuity correction is a classic mistake. Exercise 22J contains pitfalls at every turn, making it an excellent diagnostic tool.
许多学生因混淆方差σ²与标准差σ而失分。务必检查：若方差为36，则σ = 6。另一个常见错误是查表时忘记将'大于'转化为'小于'——请记住P(Z > z) = 1 − Φ(z)。此外，做逆正态时，未检查区域是左尾还是右尾就直接读表，会导致z符号错误。在正态近似中，遗漏连续性校正也是一个经典失误。练习22J处处暗藏陷阱，这使它成为极佳的诊断工具。
12. Summary and Key Takeaways | 总结与核心要点
Exercise 22J encapsulates the essential skills for mastering normal distribution in IB Mathematics: understanding the bell curve, standardising, using the normal table and GDC, solving inverse problems, finding unknown parameters, and applying the normal approximation with continuity correction. Practice repeatedly, always sketch the curve, and interpret your answer in context. Once you complete this exercise with full understanding, you will be well-prepared for any exam question involving the normal distribution.
练习22J浓缩了在IB数学中掌握正态分布所需的关键技能：理解钟形曲线、标准化、使用正态表和GDC、求解逆问题、寻找未知参数，以及用连续性校正进行正态近似。反复练习，始终画出曲线草图，并结合背景解读答案。当你带着充分理解完成这套练习时，你已为任何涉及正态分布的考试题目做好了十足准备。
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