Respuesta :
Answer:
98% confidence interval for the proportion of all American females who support the death penalty for convicted murders is [0.59 , 0.64].
Step-by-step explanation:
We are given that the results of the most recent survey; in a sample of 3100 females, 62% said that they were in favor of the death penalty for convicted murders.
Firstly, the pivotal quantity for 98% 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 females who were in favor of the death penalty for convicted murders = 62%
n = sample of females = 3100
p = population proportion of all American females who support the death penalty
Here for constructing 98% confidence interval we have used One-sample z proportion statistics.
So, 98% confidence interval for the population proportion, p is ;
P(-2.33 < N(0,1) < 2.33) = 0.98 {As the critical value of z at 1% level
of significance are -2.33 & 2.33}
P(-2.33 < [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] < 2.33) = 0.98
P( [tex]-2.33 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] < [tex]{\hat p-p}[/tex] < [tex]2.33 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] ) = 0.98
P( [tex]\hat p-2.33 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] < p < [tex]\hat p+2.33 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] ) = 0.98
98% confidence interval for p = [ [tex]\hat p-2.33 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] , [tex]\hat p+2.33 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] ]
= [ [tex]0.62-2.33 \times {\sqrt{\frac{0.62(1-0.62)}{3100} } }[/tex] , [tex]0.62+2.33 \times {\sqrt{\frac{0.62(1-0.62)}{3100} } }[/tex] ]
= [0.59 , 0.64]
Therefore, 98% confidence interval for the proportion of all American females who support the death penalty for convicted murders is [0.59 , 0.64].
