Respuesta :
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. = [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] ~ N(0,1)
where, [tex]\hat p[/tex] = sample proportion of couples who are divorced within five years = [tex]\frac{80}{200}[/tex] = 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 < [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] < 1.645) = 0.90
P( [tex]-1.645 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] < [tex]{\hat p-p}[/tex] < [tex]1.645 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] ) = 0.90
P( [tex]\hat p-1.645 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] < p < [tex]\hat p+1.645 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] ) = 0.90
90% confidence interval for p = [[tex]\hat p-1.645 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex],[tex]\hat p+1.645 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex]]
= [ [tex]0.40-1.645 \times {\sqrt{\frac{0.40(1-0.40)}{200} } }[/tex] , [tex]0.40+1.645 \times {\sqrt{\frac{0.40(1-0.40)}{200} } }[/tex] ]
= [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].
