Respuesta :
Answer:
The confidence interval at the 99% confidence level for the proportion of all adult Americans who believe in astrology is (0.2251, 0.2749). This means that we are 99% sure that the true proportion of all adult Americans who believe in astrology is in this interval.
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]p = 0.25, n = 2003[/tex]
99% confidence level
So [tex]\alpha = 0.01[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.01}{2} = 0.995[/tex], so [tex]Z = 2.575[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.25 - 2.575\sqrt{\frac{0.25*0.75}{2003}} = 0.2251[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.25 + 2.575\sqrt{\frac{0.25*0.75}{2003}} = 0.2749[/tex]
The confidence interval at the 99% confidence level for the proportion of all adult Americans who believe in astrology is (0.2251, 0.2749). This means that we are 99% sure that the true proportion of all adult Americans who believe in astrology is in this interval.
