Respuesta :
Answer:
a) [tex]\hat p=\frac{351}{606}=0.579[/tex] estimated proportion of posters that were identified as Santa Fe black-on-white
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]0.579 - 1.96 \sqrt{\frac{0.579(1-0.579)}{606}}=0.540[/tex]
[tex]0.579 + 1.96 \sqrt{\frac{0.579(1-0.579)}{606}}=0.618[/tex]
And the 95% confidence interval would be given (0.540;0.618).
Step-by-step explanation:
Data given and notation
n=606 represent the random sample taken
X=351 represent the posters that were identified as Santa Fe black-on-white
Part a
[tex]\hat p=\frac{351}{606}=0.579[/tex] estimated proportion of posters that were identified as Santa Fe black-on-white
[tex]\alpha=0.05[/tex] represent the significance level (no given, but is assumed)
z would represent the statistic (variable of interest)
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]0.579 - 1.96 \sqrt{\frac{0.579(1-0.579)}{606}}=0.540[/tex]
[tex]0.579 + 1.96 \sqrt{\frac{0.579(1-0.579)}{606}}=0.618[/tex]
And the 95% confidence interval would be given (0.540;0.618).
