📚 Exercise 18B.1: Normal Distribution Calculations | 练习18B.1：正态分布计算
In IB Mathematics, Exercise 18B.1 often presents a series of problems requiring you to calculate probabilities and percentiles for normally distributed random variables. Mastering these questions is essential for success in the topic of descriptive statistics and probability distributions.
在IB数学中，练习18B.1通常会给出一些需要你计算正态分布随机变量的概率和百分位数的问题。掌握这些题目对于描述统计和概率分布这一主题的成功至关重要。
1. Understanding the Problem Statement | 理解问题陈述
Before diving into calculations, it is crucial to identify the given parameters: the population mean μ and standard deviation σ. Problems may ask for P(X > a), P(X < b), or the value of x given a cumulative probability. Noting whether the question concerns a right-tail, left-tail, or symmetric interval will guide your choice of GDC function.
在开始计算之前，关键在于识别所给参数：总体均值μ和标准差σ。问题可能会要求求P(X > a)、P(X < b)，或在给定累积概率下求x的值。注意问题涉及的是右尾、左尾还是对称区间，这将指导你选择图形计算器的函数。
ks by misreading the inequality. Remember that for a continuous distribution, P(X > a) = 1 – P(X ⩼ a), but P(X ⩾ are identical in normal calculations. Always double-check the direction of the inequality.
许多学生因误读不等号而失分。请记住，对于连续分布，P(X > a) = 1 – P(X ⩼ a)，但在正态计算中P(X ⩾ a)与P(X > a)是相同的。务必反复检查不等号的方向。
2. The Normal Distribution in IB | IB中的正态分布
The normal distribution is a continuous probability distribution modelled by a bell-shaped curve. It is symmetric about the mean μ, and approximately 68% of observations lie within one standard deviation σ of the mean. The total area under the curve equals 1.
正态分布是一种连续型概率分布，其图形呈钟形曲线。它关于均值μ对称，大约68%的观测值落在均值附近一个标准差σ的范围内。曲线下的总面积为1。
In IB examinations, normal distribution problems appear in both Paper 1 (non-calculator) and Paper 2 (calculator). You must be able to use the standard normal table or, more efficiently, your GDC commands to solve them.
在IB考试中，正态分布问题在试卷一（不允许使用计算器）和试卷二（使用计算器）中都会出现。你必须能够使用标准正态表，或者更高效地使用图形计算器命令来求解。
3. Standardization: The Z-Score | 标准化：Z分数
To find probabilities without technology or to interpret results, you convert an x-value to a z-score using the formula below. The resulting z-score follows the standard normal distribution with mean 0 and standard deviation 1, denoted by Z ~ N(0, 1).
要在不使用技术工具的情况下求概率，或为了解释结果，你可以使用下面的公式将x值转换为z分数。所得的z分数服从标准正态分布，均值为0，标准差为1，记作Z ~ N(0, 1)。
z = (x – μ) / σ
For example, if X ~ N(100, 15²) and you need P(X > 115), you compute z = (115 – g a standard normal table or a calculator. The cumulative probability Φ(z) gives P(Z < z).
例如，若X ~ N(100, 15²)，你需要求P(X > 115)，则计算z = (115 – 100) / 15 = 1。然后可用标准正态表或计算器求出P(Z > 1)。累积概率Φ(z)给出P(Z < z)。
4. Using Technology: GDC for Normal Calculations | 使用技术：图形计算器进行正态计算
In IB exams, you are expected to use your GDC for normal distribution tasks. The functions normCdf (for probabilities) and invNorm (for inverse calculations) are your best tools. Set the correct lower and upper bounds; use a very large number like 1E99 for infinity.
在IB考试中，你应该使用图形计算器完成正态分布任务。函数normCdf（用于求概率）和invNorm（用于逆运算）是最好的工具。要设置正确的下限和上限；将无穷大用一个非常大的数如1E99来表示。
For TI-Nspire or TI-84 calculators, normCdf(lower, upper, μ, σ) returns the probability. invNorm(area to the left, μ, σ) returns the x-value that corresponds to the specified cumulative probability. Practice these menus before the exam to avoid syntax errors.
对于TI-Nspire或TI-84计算器，normCdf(下限, 上限, μ, σ) 返回概率值。invNorm(左侧面积, μ, σ) 返回与指定累积概率对应的x值。考前请练习这些菜单操作，以避免语法错误。
5. Worked Example A: Finding P(X > a) | 例题A：求P(X > a)
Given X ~ N(100, 15²), find P(X > 115). Interpret this as the probability that a randomly chosen value exceeds 115.
已知X ~ N(100, 15²)，求P(X > 115)。可以将其解释为随机选取的值超过115的概率。
GDC method: Enter normCdf(115, 1E99, 100, 15). The calculator returns approximately 0.1587. Therefore, there is about a 15.87% chance.
图形计算器方法：输入normCdf(115, 1E99, 100, 15)。计算器返回约0.1587。因此，大约有15.87%的概率。
Standardization check: z = (115 – 100)/15 = 1. P(Z > 1) = 1 – Φ(1) = 1 – 0.8413 = 0.1587. This confirms the same result, showing the logic behind the calculation.
标准化检验：z = (115 – 100)/15 = 1。P(Z > 1) = 1 – Φ(1) = 1 – 0.8413 = 0.1587。这证实了相同的结果，展示了计算背后的逻辑。
6. Worked Example B: Finding P(a < X < b) | 例题B：求P(90 < X < 110)
For the same distribution, find the probability that X lies between 90 and 110. This symmetric interval is commonly tested.
对于相同的分布，求X落在90到110之间的概率。这种对称区间是常考内容。
he output is approximately 0.4950. Alternatively, you can calculate two z-scores: z₁ = (90 – 100)/15 = –0.6667, z₂ = (110 – 100)/15 = 0.6667, and then find P(–0.6667 < Z < 0.6667) = 2 × Φ(0.6667) – 1 ≈ 0.4950.
使用图形计算器，输入normCdf(90, 110, 100, 15)。输出约为0.4950。或者，你也可以计算两个z分数：z₁ = (90 – 100)/15 = –0.6667，z₂ = (110 – 100)/15 = 0.6667，然后求P(–0.6667 < Z < 0.6667) = 2 × Φ(0.6667) – 1 ≈ 0.4950。
Always check that your interval probability is less than 1. An answer near 0.5 is reasonable for a symmetric band around the mean.
切记检查区间概率是否小于1。对于围绕均值的对称带，接近0.5的答案是合理的。
7. Inverse Normal: Finding x Given Probability | 逆正态：已知概率求x
re p is given. This is the percentile problem. Use the invNorm function: invNorm(p, μ, σ) directly gives k.
逆正态问题要求你求出满足P(X < k) = p的k值，其中p是已知的。这就是百分位数问题。使用invNorm函数：invNorm(p, μ, σ) 可直接得出k。
If the question asks for P(X > k) = p, recognise that P(X < k) = 1 – p, so you should enter invNorm(1 – p, μ, σ). Drawing a diagram will help you avoid the common mistake of using p instead of 1 – p.
如果问题是求P(X > k) = p，要认识到P(X < k) = 1 – p，因此你应该输入invNorm(1 – p, μ, σ)。绘制一个示意图将有助于你避免常见错误，即误用p而不是1 – p。
8. Worked Example C: Inverse Normal Calculation | 例题C：逆正态计算
Using X ~ h percentile. On the GDC, use invNorm(0.95, 100, 15). The calculator returns approximately 124.67. So 95% of values are below 124.67.
使用X ~ N(100, 15²)，求满足P(X < k) = 0.95的k值。这意味着k是第95百分位数。在图形计算器上，使用invNorm(0.95, 100, 15)，计算器返回约124.67。因此，95%的值低于124.67。
he problem had asked for P(X > k) = 0.05, the answer would be identical, because the top 5% tail corresponds to the same boundary. Always interpret the wording carefully.
如果问题要求P(X > k) = 0.05，答案将是相同的，因为顶端5%的尾部对应相同的界限。始终仔细解释问题的措辞。
9. Common Mistakes and Misconceptions | 常见错误与误解
One frequent error is confusing the standard deviation with the variance. In IB problems, the variance σ² might be given; you must take the square root to obtain σ before entering values into your GDC. Using variance instead of standard deviation will produce wildly incorrect results.
一个常见错误是混淆标准差和方差。在IB题目中，方差σ²可能已被给出；在将数值输入图形计算器之前，你必须先开平方求出σ。使用方差代替标准差将产生严重错误的结果。
Another mistake is failing to recognise that normal probabilities are continuous. Avoid adding or subtracting 0.5 (continuity correction), as that is used for binomial approximations, not for pure normal calculations.
另一个错误是未能认识到正态概率是连续的。避免加减0.5（连续性校正），因为那用于二项近似，而不是纯粹的正态计算。
Finally, be careful with standardisation when using tables. Always draw a sketch and shade the required area to avoid subtracting in the wrong order.
最后，使用表格时要注意标准化。始终绘制草图并涂出所需区域，以避免以错误的顺序相减。
10. Checking Your Answers | 检查答案
After obtaining a probability, check if it lies between 0 and 1. For a right-tail probability like P(X > mean), the answer should be 0.5. If you get a value above 0.5 for a value above the mean, verify that you used the correct tail.
在得到概率后，检查它是否在0到1之间。对于像P(X > 均值)这样的右尾概率，答案应为0.5。如果你对高于均值的值得到大于0.5的概率，请核实你是否使用了正确的尾部。
ble for a high percentile; a value of 30 would suggest an error in input or understanding.
对于逆正态结果，确保x值是合理的。如果μ = 100、σ = 15，那么130这个值对于高分位数是合理的；而30则表明在输入或理解上存在错误。
11. Summary of Steps for Exercise 18B.1 | 练习18B.1步骤总结
First, identify the mean μ and standard deviation σ. Second, determine the type of problem: right-tail, left-tail, interval, or inverse. Third, choose the correct GDC function: normCdf for probability, invNorm for x-values. Fourth, execute the calculation and round appropriately (usually three significant figures). Fifth, check your answer against a quick mental estimate.
首先，识别均值μ和标准差σ。其次，判断问题类型：右尾、左尾、区间或逆运算。第三，选择正确的图形计算器函数：probability用normCdf，x值用invNorm。第四，完成计算并适当四舍五入（通常取三位有效数字）。第五，对照快速心算估计值检查你的答案。
If you are required to show working for a non-calculator paper, include the standardisation steps and the use of the standard normal table. Always sketch the normal curve and shade the relevant region.
如果要求你在非计算器试卷上展示解题过程，请包含标准化步骤和标准正态表的使用。务必绘制正态曲线并涂出相关区域。
12. Practice Tips for Exercise 18B.1 | 练习18B.1的练习建议
Work through past paper problems that involve mixed normal distribution questions. Time yourself to ensure you can set up and solve each problem within the typical 2–3 minutes allocated per mark. Familiarity with the GDC menus will save valuable time during exams.
练习历年真题中涉及混合正态分布问题的部分。给自己计时，确保能在每分2–3分钟的典型时间内建立并解决每个问题。熟悉图形计算器菜单将为你节省宝贵的考试时间。
Memorise the critical values: approximately 68% within ±1σ, 95% within ±2σ, and 99.7% within ±3σ. These rules of thumb will help you sanity-check your results quickly.
记住关键数值：大约68%的值落在±1σ范围内，95%落在±2σ范围内，99.7%落在±3σ范围内。这些经验法则能帮助你快速验证结果的合理性。
Finally, create your own practice set similar to Exercise 18B.1: choose a mean and standard deviation, and formulate questions covering all the categories discussed above. Self-testing is one of the most effective revision strategies.
最后，仿照练习18B.1创建你自己的练习题集：选择一个均值和一个标准差，并提出涵盖上述所有类别的问题。自测是最有效的复习策略之一。
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