Question

A random sample of 374 married couples found that 280 had two or more personality preferences in common. In another random sample of 576 married couples, it was found that only 32 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 95% 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 95% 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 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 positive numbers, we can 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.

Answer 1

a)
Here, , n1 = 374 , n2 = 576
p1cap = 0.7487 , p2cap = 0.0556

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.7487 * (1-0.7487)/374 + 0.0556*(1-0.0556)/576)
SE = 0.0244

For 0.95 CI, z-value = 1.96
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.7487 - 0.0556 - 1.96*0.0244, 0.7487 - 0.0556 + 1.96*0.0244)
CI = (0.645 , 0.741)

b)
Cofidence interval contains all the positive values.

the proportion of married couples with two or more personality preferences in common is greater than 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.