在前面的章节中，我们学习了如何根据给定的z值查找概率。现在我们要学习相反的过程：根据给定的概率值查找对应的z值。这在统计学中被称为"反向查找"或"分位数查找"。
In previous sections, we learned how to find probabilities given z-values. Now we will learn the reverse process: finding the corresponding z-value given a probability value. This is called "inverse lookup" or "quantile lookup" in statistics.
给定概率p，查找z值使得P(Z > z) = p
Given probability p, find z-value such that P(Z > z) = p
或者 / Or: P(Z < z) = p
百分比点表（Percentage Points Table）是专门用于反向查找的工具。它给出了各种概率值p = P(Z > z)对应的z值。这个表应该优先使用，因为它专门为此目的设计。
The Percentage Points Table is a specialized tool for inverse lookup. It provides z-values corresponding to various probability values p = P(Z > z). This table should be used preferentially as it is specifically designed for this purpose.
百分比点表示例 / Example of Percentage Points Table
记住对称性：如果P(Z > 1.286) = 0.1，那么P(Z < -1.286) = 0.1也成立。
Remember symmetry: if P(Z > 1.286) = 0.1, then P(Z < -1.286) = 0.1 also holds.
在查找z值时，我们需要判断z值是正数还是负数。以下是判断规则：
When looking up z-values, we need to determine whether the z-value is positive or negative. Here are the rules:
• 如果P(Z < a) > 0.5，那么a > 0
• If P(Z < a) > 0.5, then a > 0
• 如果P(Z < a) < 0.5，那么a < 0
• If P(Z < a) < 0.5, then a < 0
• 如果P(Z > a) < 0.5，那么a > 0
• If P(Z > a) < 0.5, then a > 0
• 如果P(Z > a) > 0.5，那么a < 0
• If P(Z > a) > 0.5, then a < 0
求常数a的值，使得P(Z < a) = 0.7517
Find the value of constant a such that P(Z < a) = 0.7517
解 / Solution:
1. 绘制图表帮助可视化曲线
1. Draw a diagram to help visualize the curve
2. 在主表中查找P(Z < z) = 0.7517对应的值
2. Look in the main table to find the value that gives P(Z < z) = 0.7517
3. 这将给出a的值
3. This will give the value of a
答案：a = 0.68
求常数a的值，使得P(Z > a) = 0.100
Find the value of constant a such that P(Z > a) = 0.100
解 / Solution:
1. 绘制图表
1. Draw a diagram
2. 注意P(Z > a) < 0.5，所以a将是正数
2. Note that P(Z > a) < 0.5, so a will be positive
3. 检查百分比点表，看p = 0.100是否列出
3. Check the percentage points table to see if p = 0.100 is listed
答案：a = 1.2816
求常数a的值，使得P(Z > a) = 0.0322
Find the value of constant a such that P(Z > a) = 0.0322
解 / Solution:
1. 绘制图表
1. Draw a diagram
2. p = 0.0322没有在百分比点表中列出
2. p = 0.0322 is not listed in the percentage points table
3. 需要计算1 - 0.0322 = 0.9678，然后使用主表
3. Need to calculate 1 - 0.0322 = 0.9678, then use the main table
4. 在主表中查找z使得P(Z < z) = 0.9678
4. In the main table, find z so that P(Z < z) = 0.9678
答案：a = 1.85
求常数a的值，使得P(Z < a) = 0.1075
Find the value of constant a such that P(Z < a) = 0.1075
解 / Solution:
1. 绘制图表
1. Draw a diagram
2. 注意如果P(Z < a) < 0.5，那么a将是负数
2. Note that if P(Z < a) < 0.5, then a will be negative
3. 使用对称性帮助查找a
3. Use symmetry to help find a
4. 这将是1 - 0.1075 = 0.8925
4. This will be 1 - 0.1075 = 0.8925
5. 由于p = 0.1075不在主表中，使用主表查找z值
5. Since p = 0.1075 is not in the main table, use the main table to find the z-value
6. P(Z < z) = 0.8925，所以z = 1.24
6. P(Z < z) = 0.8925, so z = 1.24
7. 因此a = -z = -1.24
7. Therefore a = -z = -1.24
答案：a = -1.24
使用表查找P(Z < a) = 0.75
Use the tables to find P(Z < a) = 0.75
解 / Solution:
1. 绘制图表帮助可视化曲线
1. Draw a diagram to help visualize the curve
2. 在主表中查找0.75时，你会发现当a = 0.67时概率是0.7486，当a = 0.68时概率是0.7517
2. When looking at the main table for 0.75, you will find when a = 0.67 the probability is 0.7486 and when a = 0.68 the probability is 0.7517
3. 使用a = 0.67，因为它是最接近0.75的值
3. Use a = 0.67 as it is the value closest to 0.75
答案：a = 0.67
注意考试中类似这样的问题。当概率值不在表中时，选择最接近的值。
Watch out for questions like these in the exams. When the probability value is not in the table, choose the closest value.
反向查找z值的标准步骤：
Standard steps for inverse z-value lookup:
-
绘制图表 / Draw a diagram: 帮助可视化问题
Helps visualize the problem -
判断z值符号 / Determine z-value sign: 使用上述规则判断z是正数还是负数
Use the above rules to determine if z is positive or negative -
选择查找方法 / Choose lookup method:
- 优先使用百分比点表（如果概率值在表中）
- Preferentially use percentage points table (if probability value is in the table)
- 否则使用主表进行反向查找
- Otherwise use main table for inverse lookup
-
应用对称性 / Apply symmetry: 当需要时使用正态分布的对称性质
Use symmetry properties of normal distribution when needed -
验证答案 / Verify answer: 检查答案是否合理
Check if the answer is reasonable
-
符号错误 / Sign errors: 忘记判断z值的正负号
Forgetting to determine the sign of the z-value -
表的选择 / Table selection: 没有优先使用百分比点表
Not prioritizing the use of percentage points table -
对称性应用 / Symmetry application: 在需要时没有正确应用对称性
Not correctly applying symmetry when needed -
近似值选择 / Approximation selection: 当概率值不在表中时，没有选择最接近的值
Not choosing the closest value when probability is not in the table
始终先检查百分比点表，因为它专门为此目的设计。只有在概率值不在百分比点表中时，才使用主表进行反向查找。
Always check the percentage points table first, as it is specifically designed for this purpose. Only use the main table for inverse lookup when the probability value is not in the percentage points table.