使用表格查找标准正态分布Z的概率 - 练习题
使用正态分布表格查找以下概率：
Use the normal distribution tables to find the following:
a) P(Z > 1.27) b) P(Z > -1.66) c) P(Z < -2.28)
d) P(0 < Z < 1.31) e) P(1.30 < Z < 1.89) f) P(-2.8 < Z < -1.6)
对于标准正态分布 Z ~ N(0, 1²)，求：
For the standard normal distribution Z ~ N(0, 1²), find:
a) P(Z < 2.12) b) P(Z < 1.36) c) P(Z > 0.84)
d) P(Z < -0.38) e) P(-2.30 < Z < 0) f) P(Z < -1.63)
g) P(-2.16 < Z < -0.85) h) P(-1.57 < Z < 1.57)
某工厂生产的零件长度X服从正态分布N(10, 0.5²)。
The length X of parts produced by a factory follows a normal distribution N(10, 0.5²).
a) 求P(X < 10.5)
a) Find P(X < 10.5)
b) 求P(X > 9.8)
b) Find P(X > 9.8)
c) 求P(9.5 < X < 10.5)
c) Find P(9.5 < X < 10.5)
学生考试成绩X服从正态分布N(75, 15²)。
Student exam scores X follow a normal distribution N(75, 15²).
a) 求P(X > 90)
a) Find P(X > 90)
b) 求P(X < 60)
b) Find P(X < 60)
c) 求P(70 < X < 85)
c) Find P(70 < X < 85)
d) 如果90%的学生成绩高于某个分数，求这个分数。
d) If 90% of students score above a certain mark, find this mark.
某地区成年男性身高X服从正态分布N(175, 8²)。
The height X of adult males in a region follows a normal distribution N(175, 8²).
a) 求P(X > 185)
a) Find P(X > 185)
b) 求P(X < 165)
b) Find P(X < 165)
c) 求P(170 < X < 180)
c) Find P(170 < X < 180)
d) 如果5%的男性身高超过某个值，求这个值。
d) If 5% of males are taller than a certain value, find this value.
a) P(Z > 1.27)
P(Z > 1.27) = 1 - P(Z < 1.27) = 1 - 0.8980 = 0.1020
b) P(Z > -1.66)
P(Z > -1.66) = P(Z < 1.66) = 0.9515
c) P(Z < -2.28)
P(Z < -2.28) = P(Z > 2.28) = 1 - P(Z < 2.28) = 1 - 0.9887 = 0.0113
d) P(0 < Z < 1.31)
P(0 < Z < 1.31) = P(Z < 1.31) - P(Z < 0) = 0.9049 - 0.5 = 0.4049
e) P(1.30 < Z < 1.89)
P(1.30 < Z < 1.89) = P(Z < 1.89) - P(Z < 1.30) = 0.9706 - 0.9032 = 0.0674
f) P(-2.8 < Z < -1.6)
P(-2.8 < Z < -1.6) = P(Z < -1.6) - P(Z < -2.8)
= P(Z > 1.6) - P(Z > 2.8) = (1 - 0.9452) - (1 - 0.9974) = 0.0548 - 0.0026 = 0.0522
a) P(Z < 2.12) = 0.9830
b) P(Z < 1.36) = 0.9131
c) P(Z > 0.84) = 1 - P(Z < 0.84) = 1 - 0.7995 = 0.2005
d) P(Z < -0.38) = P(Z > 0.38) = 1 - P(Z < 0.38) = 1 - 0.6480 = 0.3520
e) P(-2.30 < Z < 0) = P(Z < 0) - P(Z < -2.30) = 0.5 - P(Z > 2.30) = 0.5 - (1 - 0.9893) = 0.4893
f) P(Z < -1.63) = P(Z > 1.63) = 1 - P(Z < 1.63) = 1 - 0.9484 = 0.0516
g) P(-2.16 < Z < -0.85) = P(Z < -0.85) - P(Z < -2.16) = P(Z > 0.85) - P(Z > 2.16)
= (1 - 0.8023) - (1 - 0.9846) = 0.1977 - 0.0154 = 0.1823
h) P(-1.57 < Z < 1.57) = P(Z < 1.57) - P(Z < -1.57) = P(Z < 1.57) - P(Z > 1.57)
= 0.9418 - (1 - 0.9418) = 0.9418 - 0.0582 = 0.8836
首先标准化：Z = (X - μ)/σ = (X - 10)/0.5
First standardize: Z = (X - μ)/σ = (X - 10)/0.5
a) P(X < 10.5)
Z = (10.5 - 10)/0.5 = 1
P(X < 10.5) = P(Z < 1) = 0.8413
b) P(X > 9.8)
Z = (9.8 - 10)/0.5 = -0.4
P(X > 9.8) = P(Z > -0.4) = P(Z < 0.4) = 0.6554
c) P(9.5 < X < 10.5)
Z₁ = (9.5 - 10)/0.5 = -1, Z₂ = (10.5 - 10)/0.5 = 1
P(9.5 < X < 10.5) = P(-1 < Z < 1) = P(Z < 1) - P(Z < -1)
= 0.8413 - P(Z > 1) = 0.8413 - (1 - 0.8413) = 0.6826
标准化：Z = (X - 75)/15
Standardize: Z = (X - 75)/15
a) P(X > 90)
Z = (90 - 75)/15 = 1
P(X > 90) = P(Z > 1) = 1 - P(Z < 1) = 1 - 0.8413 = 0.1587
b) P(X < 60)
Z = (60 - 75)/15 = -1
P(X < 60) = P(Z < -1) = P(Z > 1) = 0.1587
c) P(70 < X < 85)
Z₁ = (70 - 75)/15 = -1/3 ≈ -0.33, Z₂ = (85 - 75)/15 = 2/3 ≈ 0.67
P(70 < X < 85) = P(-0.33 < Z < 0.67) = P(Z < 0.67) - P(Z < -0.33)
= 0.7486 - P(Z > 0.33) = 0.7486 - (1 - 0.6293) = 0.3779
d) 90%的学生成绩高于某个分数
P(Z > z) = 0.1，所以P(Z < z) = 0.9
从表格查找：z ≈ 1.28
X = 75 + 1.28 × 15 = 94.2
所以90%的学生成绩高于94.2分
标准化：Z = (X - 175)/8
Standardize: Z = (X - 175)/8
a) P(X > 185)
Z = (185 - 175)/8 = 1.25
P(X > 185) = P(Z > 1.25) = 1 - P(Z < 1.25) = 1 - 0.8944 = 0.1056
b) P(X < 165)
Z = (165 - 175)/8 = -1.25
P(X < 165) = P(Z < -1.25) = P(Z > 1.25) = 0.1056
c) P(170 < X < 180)
Z₁ = (170 - 175)/8 = -0.625, Z₂ = (180 - 175)/8 = 0.625
P(170 < X < 180) = P(-0.625 < Z < 0.625) = P(Z < 0.625) - P(Z < -0.625)
= 0.7340 - P(Z > 0.625) = 0.7340 - (1 - 0.7340) = 0.4680
d) 5%的男性身高超过某个值
P(Z > z) = 0.05，所以P(Z < z) = 0.95
从表格查找：z ≈ 1.645
X = 175 + 1.645 × 8 = 188.16
所以5%的男性身高超过188.16cm