Question

A sociologist is studying the effect of having children within the first two years of marriage on the divorce rate. Using hospital birth records, she 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. A 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

Answer 1

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].

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. =
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }` ~ N(0,1)

where,
`\hat p` = sample proportion of couples who are divorced within five years =
`(80)/(200)` = 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

Here for constructing 90% confidence interval we have used One-sample z proportion statistics.

So, 90% confidence interval for the population proportion, p is ;

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 <
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }` < 1.645) = 0.90

P(
`-1.645 * {\sqrt{(\hat p(1-\hat p))/(n) } }` <
`{\hat p-p}` <
`1.645 * {\sqrt{(\hat p(1-\hat p))/(n) } }` ) = 0.90

P(
`\hat p-1.645 * {\sqrt{(\hat p(1-\hat p))/(n) } }` < p <
`\hat p+1.645 * {\sqrt{(\hat p(1-\hat p))/(n) } }` ) = 0.90

90% confidence interval for p = [
`\hat p-1.645 * {\sqrt{(\hat p(1-\hat p))/(n) } }`,
`\hat p+1.645 * {\sqrt{(\hat p(1-\hat p))/(n) } }`]

= [
`0.40-1.645 * {\sqrt{(0.40(1-0.40))/(200) } }` ,
`0.40+1.645 * {\sqrt{(0.40(1-0.40))/(200) } }` ]

= [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].