2. Most married couples have two or three personality preferences in common. A random sample of

375 married couples and found that 132 had three preferences in common. Another random sample

of 571 couples showed that 217 had two personality preferences in common. Let p1 be the

population proportion of all married couples who have three personality preferences in common.

Let p2 be the population proportion of all married couples who have two personality preferences in

common.

a) Can a normal distribution be used to approximate the p1 p2 distribution? Explain.

b) Find a 90% confidence interval for p1 p2. Show all work that leads to your answer

c) Examine the confidence interval in part (b) and explain what it means in the context of this

problem. Does the confidence interval contain all positive, all negative, or both positive and

negative numbers? What does this tell you about the proportion of married couples with three

personality preferences in common compared with the proportion of couples with two

preferences in common (at the 90% confidence level)?

d) Suppose a marriage counselor believes that the proportion of married couples with three

personality preferences in common is significantly different from the known value of 35%. A

survey randomly given to her current and former clients found 173 of 432 couples who shared

three personality preferences. Does this indicate that the proportion is significantly different

(either higher or lower)? Use a 5% level of significance to test her claim and show all work.