Question

A random sample of 390 married couples found that 298 had two or more personality preferences in common. In another random sample of 568 married couples, it was found that only 20 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.

Answer 1

To find a 95% confidence interval for p1 - p2, use the formula (p1 - p2) +/- z * sqrt((p1*(1-p1)/n1) + (p2*(1-p2)/n2)), where p1 and p2 are the sample proportions and n1 and n2 are the sample sizes. Find the z-score for the desired confidence level, then calculate the confidence interval.

Step-by-step explanation:

To find a 95% confidence interval for p1 - p2, we can use the formula:

CI = (p1 - p2) +/- z * sqrt((p1*(1-p1)/n1) + (p2*(1-p2)/n2))

Where:

• p1 is the sample proportion of married couples with two or more personality preferences in common
• p2 is the sample proportion of married couples with no personality preferences in common
• n1 is the sample size of the first random sample
• n2 is the sample size of the second random sample
• z is the z-score corresponding to the desired confidence level

In this case, p1 = 298/390, p2 = 20/568, n1 = 390, n2 = 568. To find the z-score, we can use a table or calculator. For a 95% confidence level, the z-score is approximately 1.96. Plugging in these values, we can calculate the confidence interval for p1 - p2.