Respuesta :
Answer:
The point estimate of this proportion is [tex]\pi = 0.5857[/tex]
The 95% confidence interval to estimate the proportion of Americans who feel that the environment is a major issue with them is (0.5584, 0.6130).
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].
For this problem, we have that:
[tex]n = 1255, \pi = \frac{735}{1255} = 0.5857[/tex]
The point estimate of this proportion is [tex]\pi = 0.5857[/tex]
95% confidence level
So [tex]\alpha = 0.05[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.5857 - 1.96\sqrt{\frac{0.5857*0.4143}{1255}} = 0.5584[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.5857 + 1.96\sqrt{\frac{0.5857*0.4143}{1255}} = 0.6130[/tex]
The 95% confidence interval to estimate the proportion of Americans who feel that the environment is a major issue with them is (0.5584, 0.6130).
Answer:
The best point of estimate is given by:
[tex]\hat p =\frac{X}{n}= \frac{735}{1255}=0.586[/tex]
[tex]0.586 - 1.96\sqrt{\frac{0.586(1-0.586)}{1255}}=0.559[/tex]
[tex]0.586 + 1.96\sqrt{\frac{0.586(1-0.586)}{1255}}=0.613[/tex]
And we have 95% of confidence that the true proportion of Americans who feel that the environment is a major issue with them is between 0.559 and 0.613
Step-by-step explanation:
The interest is the real proportion of Americans who feel that the environment is a major issue with them. The best point of estimate is given by:
[tex]\hat p =\frac{X}{n}= \frac{735}{1255}=0.586[/tex]
Confidence interval
The confidence interval for the true proportion of interest is given by this:
[tex]\hat p \pm z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]
We want an interval at 95% of confidence, so then the significance level is [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2 =0.025[/tex]. And the critical value for this case are:
[tex]z_{\alpha/2}=-1.96, z_{1-\alpha/2}=1.96[/tex]
Replacing into the confidence interval formula we got:
[tex]0.586 - 1.96\sqrt{\frac{0.586(1-0.586)}{1255}}=0.559[/tex]
[tex]0.586 + 1.96\sqrt{\frac{0.586(1-0.586)}{1255}}=0.613[/tex]
And we have 95% of confidence that the true proportion of Americans who feel that the environment is a major issue with them is between 0.559 and 0.613
