Question

A random sample of 364 married couples found that 288 had two or more personality preferences in common. In another random sample of 588 married couples, it was found that only 30 had no preferences in common. Let p1 be the population proportion of all married couples who have two or more personality preferences in common. Let p2 be the population proportion of all married couples who have no personality preferences in common. (a) Find a 90% confidence interval for p1 – 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 the 90% 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? We can not make any conclusions using this confidence interval. 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. Because the interval contains only negative numbers, we can say that a higher proportion of married couples have no personality preferences in common.

Answer 1

(a)

n1 = 354

n2 = 588

p1 = 288/354 = 0.813559322

p2 = 30/588 = 0.051020408

% = 90

Pooled Proportion, p = (n1 p1 + n2 p2)/(n1 + n2) = (354 * 0.813559322033898 + 588 * 0.0510204081632653)/(354 + 588) = 0.337579618

q = 1 - p = 1 - 0.337579617834395 = 0.662420382

SE = √(pq * ((1/n1) + (1/n2))) = √(0.337579617834395 * 0.662420382165605 * ((1/354) + (1/588))) = 0.031811937

z- score = 1.644853627

Width of the confidence interval = z * SE = 1.64485362695147 * 0.0318119366832928 = 0.052325979

Lower Limit of the confidence interval = (p1 - p2) - width = 0.762538913870633 - 0.0523259794338647 = 0.710212934

Upper Limit of the confidence interval = (p1 - p2) + width = 0.762538913870633 + 0.0523259794338647 = 0.814864893

The 90% confidence interval is [0.710, 0.815]

(b) We are 90% confident that the true difference in proportion of married couples who have two or more personality preferences in common and married couples who have no personality preferences in common lies in the above interval

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.