Respuesta :
Answer:
The 95% confidence interval would be given (0.622;0.644).
We are confident (95%) that the true proportion of people that said that they change their nail polish once a week is between 0.622 and 0.644
Step-by-step explanation:
Data given and notation
n=7000 represent the random sample taken
X=4431 represent the people that said that they change their nail polish once a week
[tex]\hat p=\frac{4431}{7000}=0.633[/tex] estimated proportion of people that said that they change their nail polish once a week
[tex]\alpha=0.05[/tex] represent the significance level
Confidence =0.95 or 95%
p= population proportion of people that said that they change their nail polish once a week
Solution to the problem
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]0.633 - 1.96 \sqrt{\frac{0.633(1-0.633)}{7000}}=0.622[/tex]
[tex]0.633 + 1.96 \sqrt{\frac{0.633(1-0.633)}{7000}}=0.644[/tex]
And the 95% confidence interval would be given (0.622;0.644).
We are confident (95%) that the true proportion of people that said that they change their nail polish once a week is between 0.622 and 0.644
