Respuesta :
Answer:
95% confidence interval for the proportion of the adults who were opposed to the death penalty is (0.668, 0.704).
Step-by-step explanation:
We are given that a survey asked whether respondents favored or opposed the death penalty for people convicted of murder. Software shows the results below, where X refers to the number of the respondents who were in favor.
X = 1,790
N = 2,610
[tex]\hat p[/tex] = Sample proportion = X/N = 0.6858
Firstly, the pivotal quantity for 95% 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 = 0.6858
n = sample of respondents = 2,610
p = population proportion
Here for constructing 95% confidence interval we have used One-sample z proportion statistics.
So, 95% confidence interval for the population proportion, p is ;
P(-1.96 < N(0,1) < 1.96) = 0.95 {As the critical value of z at
2.5% level of significance are -1.96 & 1.96}
P(-1.96 < [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] < 1.96) = 0.95
P( [tex]-1.96 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] < [tex]{\hat p-p}[/tex] < [tex]1.96 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] ) = 0.95
P( [tex]\hat p-1.96 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }[/tex] < p < [tex]\hat p+1.96 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] ) = 0.95
95% confidence interval for p = [ [tex]\hat p-1.96 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }[/tex] , [tex]\hat p+1.96 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }[/tex]]
= [ [tex]0.6858-1.96 \times {\sqrt{\frac{0.6858(1-0.6858)}{2610} }[/tex] , [tex]0.6858+1.96 \times {\sqrt{\frac{0.6858(1-0.6858)}{2610} }[/tex] ]
= [0.668 , 0.704]
Therefore, 95% confidence interval for the population proportion of the adults is (0.668, 0.704).
