A random sample of386married couples found that280had two or more personality preferences in common. In another random sample of554married couples, it was found that only32had no preferences in common. Letp₁be the population proportion of all married couples who have two or more personality preferences in common. Letp₂be the population proportion of all married couples who have no personality preferences in common.

(a) Find a90%confidence interval forp₁-p₂. (Use 3 decimal places.)

lower limit

upper limit

(b) Explain the meaning of the confidence interval in part (a) 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 (at the90%confidence level) about the proportion of married couples with two or more personality preferences in common compared with the proportion of married couples sharing no personality preferences in common?

Because the interval contains only positive numbers, we can say that a higher proportion of married couples have two or more personality preferences in common.

Because the interval contains both positive and negative numbers, we can not say that a higher proportion of married couples have two or more personality preferences in common.

We can not make any conclusions using this confidence interval.

Because the interval contains only negative numbers, we can say that a higher proportion of married couples have no personality preferences in common.