Answer:
90% confidence interval for the proportion of couples who had a child within the first two years of marriage and are divorced within five years is [0.34 , 0.46].
Step-by-step explanation:
We are given that a sociologist selects a random sample of 200 couples who had a child within the first two years of marriage.
Following up on these couples, she finds that 80 couples are divorced within five years.
Firstly, the pivotal quantity for 90% confidence interval for the population proportion is given by;
P.Q. = ~ N(0,1)
where, = sample proportion of couples who are divorced within five years = = 0.40
n = sample of couples who had a child within the first two years of marriage = 200
p = population proportion of couples who had a child within the first two years of marriage and are divorced within five years
<em>Here for constructing 90% confidence interval we have used One-sample z proportion statistics.</em>
<u>So, 90% confidence interval for the population proportion, p is ;</u>
P(-1.645 < N(0,1) < 1.645) = 0.90 {As the critical value of z at 5% level
of significance are -1.645 & 1.645}
P(-1.645 < < 1.645) = 0.90
P( < < ) = 0.90
P( < p < ) = 0.90
<u>90% confidence interval for p</u> = [,]
= [ , ]
= [0.34 , 0.46]
Therefore, 90% confidence interval for the proportion of couples who had a child within the first two years of marriage and are divorced within five years is [0.34 , 0.46].