Respuesta :
Answer:
a) [tex] \hat p = \frac{X}{n}= \frac{31}{72}= 0.431[/tex]
b) [tex]0.431 - 1.96 \sqrt{\frac{0.431(1-0.431)}{72}}=0.317[/tex]
[tex]0.431 + 1.96 \sqrt{\frac{0.431(1-0.431)}{72}}=0.545[/tex]
And the 95% confidence interval would be given (0.317;0.545).
Step-by-step explanation:
Part a
The best estimator for the population proportion is the sample proportion given by:
[tex] \hat p = \frac{X}{n}= \frac{31}{72}= 0.431[/tex]
Where X represent the adults in the sample that support the death penalty and n the sample size selected
Part b
The confidence interval would be given by this formula
[tex]\hat p \pm z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
For the 95% confidence interval the value of [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2=0.025[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.
[tex]z_{\alpha/2}=1.96[/tex]
And replacing into the confidence interval formula we got:
[tex]0.431 - 1.96 \sqrt{\frac{0.431(1-0.431)}{72}}=0.317[/tex]
[tex]0.431 + 1.96 \sqrt{\frac{0.431(1-0.431)}{72}}=0.545[/tex]
And the 95% confidence interval would be given (0.317;0.545).
