Question

Most married couples have two or three personality preferences in common. A random sample of 388 married couples found that 120 had three preferences in common. Another random sample of 572 couples showed that 240 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) Find a 95% confidence interval for p₁ – p₂. (Use 3 decimal places.)

lower limit
upper limit

(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 95% 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 can not say that a higher proportion of married couples have three 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 two personality preferences in common.

Answer 1

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.

$\hat{p}_{1}$ is given by,

$\hat{p}_{1} = \frac{120}{388} = 0.3093$

$\hat{p}_{2}$ is given by,

240 0.4196 P2 572

95% confidence interval for p₁ – p₂.

We have,

n1 = 388 n2 = 572

$CI = (\hat{p}_{1}-\hat{p}_{2})\pm z_{\alpha /2}\sqrt{\frac{\hat{p}_{1}-(1-\hat{p}_{1})}{n_{1}}+\frac{\hat{p}_{2}-(1-\hat{p}_{2})}{n_{2}}}$

$CI = (0.3093-0.4196)\pm 1.96* \sqrt{\frac{0.3093*0.6907}{388}+\frac{0.4196*0.5804}{572}}$

$CI = (0.3093-0.4196)\pm 1.96*0.0441$

Above confidence interval states that,

We estimate with 95% confidence that the both population proportion will have preferences between (-0.19674,-0.02386)

b) 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 95% confidence level)?

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 two personality preferences in common.