The heights of a large group of men are normally distributed with a mean of 178cm and a standard deviation of 4cm. A man is selected at random from this group.
a) 求他身高超过185cm的概率。(2分)
a) Find the probability that he is taller than 185cm. (2 marks)
b) 求随机选择的三名男性身高都小于180cm的概率。(3分)
b) Find the probability that three men, selected at random, are all less than 180cm tall. (3 marks)
c) 门框制造商希望确保少于0.005的男性需要弯腰通过门框。基于这个群体，求门框的最小高度（精确到厘米）。(2分)
c) A manufacturer of door frames wants to ensure that fewer than 0.005 men have to bend down to pass through the frame. On the basis of this group, find the minimum height of a door frame to the nearest centimetre. (2 marks)
The random variable X ~ N(24, σ²). Given that P(X > 30) = 0.05, find:
a) σ的值 (2分)
a) the value of σ (2 marks)
b) P(X < 20) (1分)
b) P(X < 20) (1 mark)
c) 使得P(X > d) = 0.01的d值 (2分)
c) the value of d so that P(X > d) = 0.01 (2 marks)
解 / Solution:
解题步骤 / Solution Steps
a) P(X > 30) = P(Z > (30-24)/σ) = 0.05
P(Z > 6/σ) = 0.05
在百分比点表中查找p=0.05对应的z值：z = 1.645
6/σ = 1.645
σ = 6/1.645 = 3.65
b) P(X < 20) = P(Z < (20-24)/3.65) = P(Z < -1.10)
= 1 - P(Z < 1.10) = 1 - 0.8643 = 0.1357
c) P(X > d) = P(Z > (d-24)/3.65) = 0.01
(d-24)/3.65 = 2.326
d = 24 + 3.65 × 2.326 = 32.5
答案：a) σ = 3.65; b) 0.136; c) d = 32.5
Question 4: 分位数问题
随机变量X ~ N(μ, σ²)。X的下四分位数是20，上四分位数是40。
The random variable X ~ N(μ, σ²). The lower quartile of X is 20 and the upper quartile is 40.
a) 求μ和σ。(3分)
a) Find μ and σ. (3 marks)
b) 求10%到90%的百分位距。(3分)
b) Find the 10% to 90% interpercentile range. (3 marks)
解 / Solution:
解题步骤 / Solution Steps
a) 下四分位数：P(X < 20) = 0.25
上四分位数：P(X < 40) = 0.75
建立方程组：
(20-μ)/σ = -0.6745
(40-μ)/σ = 0.6745
解方程组：μ = 30, σ = 14.8
b) 10%百分位数：P(X < x₁) = 0.10
90%百分位数：P(X < x₂) = 0.90
x₁ = 30 - 1.282 × 14.8 = 11.0
x₂ = 30 + 1.282 × 14.8 = 49.0
百分位距 = 49.0 - 11.0 = 38.0
答案：a) μ = 30, σ = 14.8; b) 38.0
Question 5: 综合应用题
心理学家给学生两个不同的测试。第一个测试均值为80，标准差为10，学生得分为85。
A psychologist gives a student two different tests. The first test has a mean of 80 and a standard deviation of 10, and the student scores 85.
a) 求在第一个测试中得分85或更高的概率。(2分)
a) Find the probability of scoring 85 or more on the first test. (2 marks)
b) 第二个测试均值为100，标准差为15。学生得分为105。求在第二个测试中得分105或更高的概率。(2分)
b) The second test has a mean of 100 and a standard deviation of 15. The student scores 105. Find the probability of a score of 105 or more on the second test. (2 marks)
c) 说明并给出理由，学生的两个测试分数中哪个更好。(2分)
c) State, giving a reason, which of the student's two test scores was better. (2 marks)