Most married couples have two or three personality preferences in common. A rand

Question

Most married couples have two or three personality preferences in common. A random sample of 376 married couples found that 130 had three preferences in common. Another random sample of 554 couples showed that 220 had two personality preferences in common Let p_1 be the population proportion of all married couples who have three personality preferences in common. Let p_2 be the population proportion of all married couples who have two personality preferences in common. (a) Find a 99% confidence interval for p_1 - p_2. (Use 3 decimal places.) (b) Examine the confidence interval in part (a) 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 99% confidence level)? Because the interval contains only positive numbers, we can say that a higher proportion of married couples have three personality preferences in common. Because the interval contains both positive and negative numbers, we cannot say that a higher proportion of married couples have three personality preferences in common. We cannot 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 two personality preferences in common.

Explanation / Answer

a)
Confidence Interval for Diffrence of Proportion
CI = (p1 - p2) ± Z a/2 Sqrt(p1(1-p1)/n1 + p2(1-p2)/n2 )------------------(1)
Proportion 1
Probability Succuses( X1 )=130
No.Of Observed (n1)=376
P1= X1/n1=0.3457

P^1=1-0.3457=0.6543

Proportion 2
Probability Succuses(X2)=220
No.Of Observed (n2)=554
P2= X2/n2=0.3971   

p^2=1-0.3971=0.6029

Z alpha/2 = 2.58 from confidence interval z table

substtuting these values in equation
C.I = (0.3457-0.3971) ±Z alpha/2 * Sqrt( (0.3457*0.643/376) + (0.3971*0.6029/554) )
=(0.3457-0.3971) ± 2.58* Sqrt(0.001)
=-0.0514-0.0819,-0.0514+0.0819
=[-0.1333,0.0305]

b) option b
confidence interval contain all positive, all negative
Because the interval contains both positive and negative numbers, we can not say
that a higher proportion of married couples have three personality preferences in common

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