7.3 Using Tables to Find the Value of z Given a Probability

7.3 反向查找z值 - 练习题

练习说明 / Exercise Instructions

以下练习题将帮助你掌握反向查找z值的技能。建议按顺序完成，每道题都要仔细分析解题步骤。

The following exercises will help you master the skill of inverse z-value lookup. It is recommended to complete them in order, carefully analyzing the solution steps for each problem.

基础练习 / Basic Exercises

练习1 / Exercise 1

求常数a的值，使得P(Z < a) = 0.3336

Find the value of constant a such that P(Z < a) = 0.3336

解 / Solution:

1. 绘制图表帮助理解

1. Draw a diagram to help understand

2. 由于P(Z < a) < 0.5，所以a < 0

2. Since P(Z < a) < 0.5, so a < 0

3. 使用对称性：P(Z < a) = 0.3336，所以P(Z > -a) = 0.3336

3. Use symmetry: P(Z < a) = 0.3336, so P(Z > -a) = 0.3336

4. 因此P(Z < -a) = 1 - 0.3336 = 0.6664

4. Therefore P(Z < -a) = 1 - 0.3336 = 0.6664

5. 在主表中查找0.6664对应的z值

5. Look up the z-value corresponding to 0.6664 in the main table

答案：a = -0.43

练习2 / Exercise 2

求常数a的值，使得P(Z > a) = 0.6879

Find the value of constant a such that P(Z > a) = 0.6879

解 / Solution:

1. 绘制图表

1. Draw a diagram

2. 由于P(Z > a) > 0.5，所以a < 0

2. Since P(Z > a) > 0.5, so a < 0

3. 计算P(Z < a) = 1 - 0.6879 = 0.3121

3. Calculate P(Z < a) = 1 - 0.6879 = 0.3121

4. 使用对称性：P(Z < a) = 0.3121，所以P(Z > -a) = 0.3121

4. Use symmetry: P(Z < a) = 0.3121, so P(Z > -a) = 0.3121

5. 在百分比点表中查找p = 0.3121对应的z值

5. Look up the z-value corresponding to p = 0.3121 in the percentage points table

答案：a = -0.49

练习3 / Exercise 3

求常数a的值，使得P(Z > a) = 0.1112

Find the value of constant a such that P(Z > a) = 0.1112

解 / Solution:

1. 绘制图表

1. Draw a diagram

2. 由于P(Z > a) < 0.5，所以a > 0

2. Since P(Z > a) < 0.5, so a > 0

3. 优先在百分比点表中查找p = 0.1112

3. First check the percentage points table for p = 0.1112

4. 如果不在百分比点表中，计算P(Z < a) = 1 - 0.1112 = 0.8888

4. If not in percentage points table, calculate P(Z < a) = 1 - 0.1112 = 0.8888

5. 在主表中查找0.8888对应的z值

5. Look up the z-value corresponding to 0.8888 in the main table

答案：a = 1.22

练习4 / Exercise 4

求常数a的值，使得P(-a < Z < a) = 0.5820

Find the value of constant a such that P(-a < Z < a) = 0.5820

解 / Solution:

1. 绘制图表

1. Draw a diagram

2. 由于P(-a < Z < a) = 0.5820，所以P(Z < a) = 0.5 + 0.5820/2 = 0.791

2. Since P(-a < Z < a) = 0.5820, so P(Z < a) = 0.5 + 0.5820/2 = 0.791

3. 在主表中查找0.791对应的z值

3. Look up the z-value corresponding to 0.791 in the main table

4. 由于0.791不在表中，选择最接近的值0.7881（对应z = 0.80）

4. Since 0.791 is not in the table, choose the closest value 0.7881 (corresponding to z = 0.80)

答案：a = 0.80

进阶练习 / Advanced Exercises

练习5 / Exercise 5

对于标准正态分布Z ~ N(0, 1²)，求满足以下条件的a值：

For the standard normal distribution Z ~ N(0, 1²), find values of a such that:

a) P(Z < a) = 0.9082

b) P(Z > a) = 0.0314

c) P(Z > a) = 0.1500

解 / Solution:

a) P(Z < a) = 0.9082

由于P(Z < a) > 0.5，所以a > 0

Since P(Z < a) > 0.5, so a > 0

在主表中查找0.9082对应的z值

Look up the z-value corresponding to 0.9082 in the main table

答案：a = 1.33

b) P(Z > a) = 0.0314

由于P(Z > a) < 0.5，所以a > 0

Since P(Z > a) < 0.5, so a > 0

优先在百分比点表中查找p = 0.0314

First check the percentage points table for p = 0.0314

答案：a = 1.86

c) P(Z > a) = 0.1500

由于P(Z > a) < 0.5，所以a > 0

Since P(Z > a) < 0.5, so a > 0

在百分比点表中查找p = 0.1500

Look up p = 0.1500 in the percentage points table

答案：a = 1.04

练习6 / Exercise 6

求常数a的值，使得P(0 < Z < a) = 0.3554

Find the value of constant a such that P(0 < Z < a) = 0.3554

解 / Solution:

1. 绘制图表

1. Draw a diagram

2. 由于P(0 < Z < a) = 0.3554，所以P(Z < a) = 0.5 + 0.3554 = 0.8554

2. Since P(0 < Z < a) = 0.3554, so P(Z < a) = 0.5 + 0.3554 = 0.8554

3. 在主表中查找0.8554对应的z值

3. Look up the z-value corresponding to 0.8554 in the main table

4. 由于0.8554不在表中，选择最接近的值0.8554（对应z = 1.06）

4. Since 0.8554 is not in the table, choose the closest value 0.8554 (corresponding to z = 1.06)

答案：a = 1.06

练习7 / Exercise 7

求常数a的值，使得P(-a < Z < a) = 0.80

Find the value of constant a such that P(-a < Z < a) = 0.80

解 / Solution:

1. 绘制图表

1. Draw a diagram

2. 由于P(-a < Z < a) = 0.80，所以P(Z < a) = 0.5 + 0.80/2 = 0.90

2. Since P(-a < Z < a) = 0.80, so P(Z < a) = 0.5 + 0.80/2 = 0.90

3. 在主表中查找0.90对应的z值

3. Look up the z-value corresponding to 0.90 in the main table

4. 由于0.90不在表中，选择最接近的值0.8997（对应z = 1.28）

4. Since 0.90 is not in the table, choose the closest value 0.8997 (corresponding to z = 1.28)

答案：a = 1.28

综合练习 / Comprehensive Exercises

练习8 / Exercise 8

一个质量控制工程师需要确定产品的合格标准。如果产品的某个指标服从标准正态分布，且要求95%的产品合格，求合格标准的上限。

A quality control engineer needs to determine product qualification standards. If a certain indicator of the product follows a standard normal distribution and 95% of products need to be qualified, find the upper limit of the qualification standard.

解 / Solution:

1. 设合格标准的上限为a，则P(Z < a) = 0.95

1. Let the upper limit of qualification standard be a, then P(Z < a) = 0.95

2. 由于P(Z < a) > 0.5，所以a > 0

2. Since P(Z < a) > 0.5, so a > 0

3. 在主表中查找0.95对应的z值

3. Look up the z-value corresponding to 0.95 in the main table

4. 由于0.95不在表中，选择最接近的值0.9495（对应z = 1.64）

4. Since 0.95 is not in the table, choose the closest value 0.9495 (corresponding to z = 1.64)

答案：合格标准的上限为1.64

Answer: The upper limit of qualification standard is 1.64

练习提示 / Exercise Tips

始终先绘制图表帮助理解问题
Always draw a diagram first to help understand the problem
注意判断z值的正负号
Pay attention to determining the sign of z-values
优先使用百分比点表
Prioritize the use of percentage points table
当概率值不在表中时，选择最接近的值
When probability value is not in the table, choose the closest value
验证答案的合理性
Verify the reasonableness of the answer

进一步练习建议 / Further Practice Recommendations

完成以上练习后，建议：

After completing the above exercises, it is recommended to:

尝试自己设计类似的问题
Try designing similar problems yourself
练习不同类型的概率表达式
Practice different types of probability expressions
熟练掌握表格的使用
Master the use of tables
注意考试中的时间管理
Pay attention to time management in exams

7.3 Using Tables to Find the Value of z Given a Probability - 练习题