Beanstalks. In some cities tall people who want to meet and socialize with other tall people can join Beanstalk Clubs. To qualify, a man must be over 6'2" tall, and a woman over 5'10". According to the National Health Survey, heights of adults may have a Normal model with mean heights of 69.1 for men and 64.0 for women. The respective standard deviations are 2.8 and 2.5.

a) You're probably not surprised to learn that men are generally taller than women, but what does the greater standard deviation for men's heights indicate?

b) Are men or women more likely to qualify for Beanstalk membership?

c) Beanstalk members believe that height is an important factor when people select their spouses. To investigate, we select at random a married man and, independently, a married woman. Define two random variables, and use them to express how many inches taller the man is than the woman.

d) What's the mean of this difference?

e) What's the standard deviation of this difference?

f) What's the probability that the man is taller than the woman (that the difference in heights is greater than 0)?

g) Suppose a survey of married couples reveals that 92% of the husbands were taller than their wives. Based on your answer to part f, do you believe that people's choice of spouses is independent of height? Explain.