使用表格查找标准正态分布Z的概率
表格在教材和考试中提供，帮助您计算标准正态分布Z的概率。
Tables are provided in this textbook and the exam to help you calculate probabilities for the standard normal distribution, Z.
标准正态变量通常用Z表示，均值为0，标准差为1。常见的写法是：
The standard normal variable is usually denoted by Z and has a mean of 0 and a standard deviation of 1. The common way of writing this is:
标准正态分布记号 / Standard Normal Distribution Notation
μ = 0 和 σ² = 1
μ = 0 and σ² = 1
\[Z \sim N(0, 1^2)\]
～ 表示"分布"，N 表示正态分布
～ means 'is distributed', N for normal
Φ(a) 经常用作 P(Z < a) 的简写记号。
Φ(a) is often used as shorthand for writing P(Z < a).
重要提示 / Important Note
曲线下的总面积 = 1。（概率分布中概率的总和总是等于1）。因此，P(Z < a) 就是曲线在a左侧的面积。
The total area under the curve = 1. (The sum of probabilities in a probability distribution always add up to 1). Thus, P(Z < a) will just be the area under the curve to the left of a.
您可以使用表格或计算器来查找这些概率。正态分布表格在书后可以找到，在考试中的数学公式和统计表格书中也会提供。表格给出不同z值的P(Z < z)。
You can use tables or calculators to find these probabilities. Normal distribution tables are found in the back of the book and will be provided in the Mathematical Formulae and Statistical Tables book in the exam. The tables give P(Z < z) for different values of z.
注意 / Watch Out
对于连续分布，如正态分布，P(Z < z) 和 P(Z ≤ z) 之间没有区别。
For a continuous distribution, such as the normal distribution, there is no difference between P(Z < z) and P(Z ≤ z).
例2：使用正态分布表格 / Example 2: Using Normal Distribution Tables
使用正态分布表格查找：
Use the normal distribution tables to find:
a) P(Z < 1.54) b) P(Z > 2.65)
c) P(Z < -0.75) d) P(-1.20 < Z < 1.40)
解答 / Solution：
a) P(Z < 1.54)
绘制图表并标出所需区域。
Draw a diagram and shade the region required.
查找所需的z值 - 在这种情况下是1.54
Look up the required z value - in this case 1.54
从表格中完整引用该值。
Quote the value in full from the table.
P(Z < 1.54) = 0.9382
b) P(Z > 2.65)
绘制图表并标出所需区域。
Draw the diagram and shade the region required.
表格给出P(Z < 2.65)。因此，您需要计算"1减去这个概率"。
The table gives you P(Z < 2.65). Therefore, you need to calculate '1 minus this probability'.
P(Z > 2.65) = 1 - P(Z < 2.65)
= 1 - 0.996
= 0.004
c) P(Z < -0.75)
绘制图表并标出所需区域。
Draw a diagram and shade the region required.
表格不给出z < 0的值。
The table does not give the values of z < 0.
但是，使用对称性您可以看到概率与P(Z > 0.75)相同。
However, using symmetry you can see the probability is the same as P(Z > 0.75).
P(Z < -0.75) = P(Z > 0.75)
= 1 - 0.7734 （从主表格中找到）
= 1 - 0.7734 (Found from the main table)
= 0.2266
d) P(-1.20 < Z < 1.40)
将问题分解为三个图表。这个图表显示当a < 1.40时的概率。
Split the problem up into three graphs. This graph shows the probability when a < 1.40.
这个图表显示当a < -1.20时的概率。
This graph shows the probability when a < -1.20.
这个图表显示需要计算的面积。在可视化问题后，您可以通过计算P(Z < 1.40) - P(Z < -1.20)来计算概率。
This graph shows the area that needs to be calculated. After visualising the problem you can calculate the probability by calculating P(Z < 1.40) - P(Z < -1.20).
P(-1.20 < Z < 1.40) = P(Z < 1.40) - P(Z < -1.20)
P(Z < 1.40) = 0.9192 （从表格中）
P(Z < 1.40) = 0.9192 (from the tables)
P(Z < -1.20) = 1 - P(Z < 1.20)
= 1 - 0.8849 （从表格中）
= 1 - 0.8849 (from the tables)
= 0.1151
因此：
Therefore:
P(Z < 1.40) - P(Z < -1.20) = 0.9192 - 0.1151 = 0.8041
使用正态分布表格查找概率的步骤 / Steps for Finding Probabilities Using Normal Distribution Tables
表格使用技巧 / Table Usage Tips
常见错误 / Common Errors
检查答案的方法 / Methods for Checking Answers