📚 Exercise 15I: Applications of the Normal Distribution | 练习15I：正态分布应用
In IB Mathematics, Exercise 15I typically challenges students with real‑world problems that involve the normal distribution. This set of exercises extends your ability to model continuous data, calculate probabilities, and find unknown means or standard deviations. The normal distribution is central to statistical inference, and mastering these problems will strengthen both your calculator skills and your conceptual understanding of the bell curve.
在IB数学中，练习15I通常会给学生带来涉及正态分布的实际问题。这组练习旨在扩展你对连续数据进行建模、计算概率以及求解未知均值或标准差的能力。正态分布是统计推断的核心，掌握这些问题将同时提升你的计算器使用技巧和对钟形曲线的概念理解。
1. The Normal Distribution Curve | 正态分布曲线
The normal distribution is a symmetric, bell‑ by two parameters: the mean μ and the standard deviation σ. Every normal random variable X ~ ty that X falls within a given interval is equal to the area over that interval. The curve's peak is at x = μ, and the points of inflection occur at μ ± σ.
正态分布是一条由均值 μ 和标准差 σ 两个参数定义的对称钟形曲线。每个正态随机变量 X ~ N(μ, σ²) 曲线下方的总面积为1，X 落在某个区间内的概率等于该区间上的面积。曲线的峰值位于 x = μ 处，拐点出现在 μ ± σ 处。
In Exercise 15I you will often sketch the curve to visualise the region corresponding to a probability statement such as P(X > k) or P(a < X < b). Understanding the shape helps you check whether your computed probabilities are reasonable, for instance a value three standard deviations above the mean should give a very small tail probability.
在练习15I中，你常常需要画出曲线，将概率语句如 P(X > k) 或 P(a < X < b) 对应的区域形象化。理解图形有助于你检验计算出的概率是否合理，例如比均值高出三个标准差的点，其尾部概率应当非常小。
2. Standardising to the Z‑Score | 标准化与 Z 分数
To work with any normal curve, we convert the random variable X to the standard normal Z ~ h as P(X < 120) given μ = 100 and σ = 15, which becomes P(Z < 1.33) after standardisation.
为了处理任意正态曲线，我们用公式 z = (x − μ) / σ 将随机变量 X 转化为标准正态变量 Z ~ N(0, 1)。标准正态分布的均值为0、标准差为1，其概率值已被广泛编制成表。练习15I经常要求你计算诸如在 μ = 100、σ = 15 时 P(X < 120) 这样的概率，标准化后即变为 P(Z < 1.33)。
The process of standardisation is essential because it allows you to use a single reference distribution. When the exercise provides a data value and asks for its percentile, or when it gives a percentile and asks for the corresponding value, you will move back and forth between the original units and the z‑score.
标准化过程至关重要，因为它让你可以只使用一个参考分布。当练习提供数据值并要求求出百分位数，或给定百分位数而要求对应的数值时，你将需要在原始单位和 z 分数之间来回转换。
3. Using the Inverse Normal Function | 反向正态函数的使用
Many problems in Exercise 15I require the inverse normal calculation: given a left‑tail probability p, find the value k such that P(X < k) = p. Your graphing calculator' lent) expects p, μ, and σ as inputs and returns the boundary k. For example, if the top 10 % of scores are to be awarded a distinction, finding the cut‑off mark demands invNorm(0.9, μ, σ).
练习15I中的许多问题需要反向正态计算：给定左侧尾部概率 p，求满足 P(X < k) = p 的值 k。你的图形计算器的 invNorm 函数（或等效功能）需要输入 p、μ 和 σ，并返回边界值 k。例如，若需对前10%的成绩授予优秀，求截断分数时就需要使用 invNorm(0.9, μ, σ)。
Be careful with right‑ f the question asks for k such that P(X > k) = 0.05, you must first convert this to a left‑ irection of the inequality is a common mistake that Exercise 15I is designed to eliminate.
小心处理右侧尾部概率。如果问题要求解出使 P(X > k) = 0.05 的 k，你必须先将其转化为左侧尾部概率：P(X < k) = 0.95。误读不等式方向是一个常见错误，练习15I正是为了消除这一错误而设计的。
4. Solving for Unknown Mean or Standard Deviation | 求未知均值或标准差
Some of the most demanding questions in Exercise 15I give two probability conditions and ask you to find both μ and σ. For instance, you may be told that 10 % of batteries last less than 400 hours and 25 % last more than 600 hours. By setting up two z‑score equations and solving simultaneously, you can determine the parameters of the population.
练习15I中一些要求最高的问题会给出两个概率条件，让你同时求出 μ 和 σ。例如，你可能会被告知10%的电池寿命低于400小时，25%高于600小时。通过建立两个 z 分数方程并联立求解，就可以确定总体的参数。
μ)/σ and z₂ = (600 − μ)/σ. Then use the standard normal table or calculator to find the z‑ en probabilities. Solving the resulting linear system is a satisfying algebraic challenge.
策略是将每个条件转化为标准化方程：z₁ = (400 − μ)/σ 和 z₂ = (600 − μ)/σ。然后使用标准正态表或计算器找出与给定概率相对应的 z 值。求解所得的线性方程组是一项令人满足的代数挑战。
5. Checking Model Assumptions | 检验模型假设
Real data can only be modelled by a normal distribution if certain conditions hold. The variable should be continuous, the distribution roughly symmetric, and the data should not contain outliers that severely distort the mean and standard deviation. Exercise 15I often begins with a statement such as 'The masses of apples are normally distributed with mean 150 g and standard deviation 20 g', meaning that the model is assumed appropriate.
真实数据只有在满足某些条件时才能用正态分布建模。变量应当是连续的，分布大致对称，并且数据中不应包含严重扭曲均值和标准差的异常值。练习15I常常以“苹果的质量服从均值为150 g、标准差为20 g的正态分布”这样的陈述开头，这意味着假设模型是合适的。
In some IB questions, you may be asked to discuss whether the normal model is suitable. You would look at a box plot or histogram for symmetry, or calculate the percentage of data within one, two, and three standard deviations of the mean and compare with the empirical values 68 %, 95 %, and 99.7 %.
在某些IB问题中，你可能会被要求讨论正态模型是否合适。你可以通过箱线图或直方图观察对称性，或者计算数据落在均值一、二、三个标准差范围内的百分比，并与经验值68%、95%和99.7%进行比较。
6. Mixed Practice with Two Related Normal Variables | 两个相关正态变量的混合练习
woman. This requires constructing a new normal distribution for the difference D = M − W.
练习15I中一个有趣的转折是对两个独立正态随机变量进行比较。例如，男性身高服从 N(176, 6²)，女性身高服从 N(164, 5²)。你可能会被问到随机选取的一名男性比随机选取的一名女性高的概率。这需要为差值 D = M − W 构造一个新的正态分布。
If M ~ N(μₘ, σₘ²) and W ~ N(μ_w, σ_w²) are independent, then D ~ 0). This extension of the normal distribution is powerful and appears frequently in higher‑level IB papers.
如果 M ~ N(μₘ, σₘ²) 且 W ~ N(μ_w, σ_w²) 相互独立，则 D ~ N(μₘ − μ_w, σₘ² + σ_w²)。接着你可以对 D 进行标准化并计算 P(D > 0)。正态分布的这一延伸功能强大，频繁出现在高级别的IB试卷中。
7. Applying the Normal Distribution to Sample Means | 正态分布在样本均值中的应用
ly large sample size n (n ≥ 30 is the usual rule), the sampling distribution of the mean x̄ ven if the population itself is not normal. Its mean is μ and its standard deviation is σ/√n. Exercise 15I includes questions where you calculate the probability that the mean of a sample exceeds a certain value.
中心极限定理告诉我们，当样本容量 n 足够大时（通常 n ≥ 30），即使总体本身不是正态分布，样本均值 x̄ 的抽样分布也近似正态。其均值为 μ，标准差为 σ/√n。练习15I中包含让你计算样本均值超过某个值的概率的问题。
ted, find the probability that their mean time is less than 37 min. You would work with σₓ̄ = 12/√36 = 2 min, then find P(x̄ < 37) using the z‑score (37 − 40)/2 = −1.5.
例如，完成一项任务的时长均值为40分钟、标准差为12分钟。随机选取36名工人，求他们的平均时长小于37分钟的概率。你需要使用 σₓ̄ = 12/√36 = 2 分钟，然后通过 z 分数 (37 − 40)/2 = −1.5 计算 P(x̄ < 37)。
8. Calculator Techniques and Precision | 计算器技巧与精度
IB exams require you to be fluent with your graphing calculator's normal distribution functions: normalcdf (for area between bounds), invNorm (for inverse), and normalpdf (rarely used for probabilities). In Exercise 15I you should practise storing intermediate z‑scores as variables to reduce rounding errors. Giving answers to three significant figures is the standard unless otherwise stated.
IB考试要求你熟练使用图形计算器的正态分布功能：normalcdf（用于求边界间的面积）、invNorm（反向求解）以及 normalpdf（很少用于概率）。在练习15I中，你应该练习将中间 z 分数存储为变量，以减少舍入误差。除非另有说明，答案通常保留三位有效数字。
Be mindful that the calculator expects standard deviation, not variance, when you enter σ. A common slip is to input σ² instead of σ. Also, set your calculator to the correct tail direction: some software uses right‑tail by default, so double‑check the syntax.
注意当输入 σ 时，计算器要求的是标准差而不是方差。一个常见的失误是输入了 σ² 而非 σ。此外，要将计算器设置在正确的尾部方向：有些软件默认使用右侧尾部，因此请仔细检查语法。
9. Answering 'Show That' and Justification Questions | 回答“证明”及论证类问题
Exercise 15I contains structured questions that ask you to 'show that' a particular z‑score is 1.25 or that the cutoff mark is 72. Here you must document your steps clearly: state the probability statement, write the standardisation equation, and substitute the values. Simply writing the final answer without working will not earn full marks.
练习15I包含要求你“证明”某个 z 分数为1.25或截断分数为72的结构化问题。在这里，你必须清晰记录解题步骤：陈述概率语句，写出标准化方程，并代入数值。只写出最终答案而不展示过程将无法获得全部分数。
You should also be prepared to justify why a distribution might be modelled as normal. For instance, you might refer to the symmetry of the data, the absence of extreme outliers, or the theoretical context (measurement errors often follow a normal distribution).
你还应准备好论证为什么某个分布可以被建模为正态分布。例如，你可以提及数据的对称性、不存在极端异常值，或者理论背景（测量误差通常服从正态分布）。
10. Real‑Life Contexts and Interpretation | 实际情境与解释
elate your final probability back to the context. Instead of just writing P = 0.067, state 'There is a 6.7 % chance that a randomly selected apple weighs more than 180 g.' This demonstrates deep understanding.
练习15I中的问题深深植根于各种情境：考试分数、产品重量、反应时间或元件寿命。一定要将最终概率与情境联系起来。不要只写 P = 0.067，而应陈述“随机选取一个苹果重量超过180克的可能性为6.7%”。这展示出深刻的理解。
Interpretation also extends to decisions: if a probability is very low (e.g. less than 0.05), you might conclude that an event is unusual. In quality control contexts, such a finding might suggest that a machine needs recalibration. Connecting numbers to real‑world actions is a key IB skill.
解释还包括决策：如果概率非常低（例如小于0.05），你可能得出该事件不寻常的结论。在质量控制情境中，这样的发现可能表明机器需要重新校准。将数字与实际行动联系起来是一项关键的IB技能。
11. Common Pitfalls to Avoid | 需要避免的常见陷阱
- Confusing the standard deviation of the population σ with the standard deviation of the sample mean σ/√n.
- Using the z‑table value for the wrong tail without subtracting from 1.
- Forgetting to square the standard deviation when writing X ~ N(μ, σ²).
- Applying the normal model to discrete data without a continuity correction (though IB often omits this).
- 混淆总体标准差 σ 与样本均值的标准差 σ/√n。
- 使用 z 表时误取尾部值而未从1中减去。
- 在写 X ~ N(μ, σ²) 时忘记给标准差加上平方。
- 在未进行连续性校正的情况下对离散数据应用正态模型（尽管IB常忽略这一点）。
By recognising these pitfalls, you can approach Exercise 15I with confidence and turn potential mistakes into learning opportunities.
通过识别这些陷阱，你可以自信地应对练习15I，并将潜在的错误转变为学习的机会。
12. Summary and Exam Readiness | 总结与备考
Exercise 15I is a comprehensive resource that consolidates the core skills of the normal distribution topic. From basic standardisation to inverse solving and parameter estimation, every type of question reinforces your statistical thinking. The key to success is systematic practice: write down each step, use your calculator wisely, and always interpret your answer in context.
练习15I是一份综合性的资源，巩固了正态分布主题的核心技能。从基本的标准化到反向求解和参数估计，每一类问题都在强化你的统计思维。成功的关键在于系统的练习：写下每一个步骤，明智地使用计算器，并且始终将答案放在具体情境中解释。
As you complete these problems, you will be ready not only for the normal distribution questions in the IB exam but also for the broader expectation of reasoning under uncertainty that defines the IB mathematical experience.
当你完成这些问题时，你不仅为IB考试中的正态分布题目做好了准备，也为IB数学体验所要求的在不确定性下进行推理这一更广泛的目标做好了准备。
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