Question

In 2013, the Public Religion Research Institute conducted a survey of 1,033 adults, 18 years of age or older, in the continental United States. One of the questions on their survey was as follows:
On any given Sunday, are you more likely to only be in church, more likely to only be watching football, doing both, or doing neither?

(a) To only be in church [269 respondents selected this answer]

(b) To only be watching football [175 respondents selected this answer]

(c) Doing both [217 respondents selected this answer]

(d) Doing neither [372 respondents selected this answer]
Create a 95% confidence interval to estimate the actual percentage of adults in the U.S., aged 18 and older, that are "likely" to ONLY be in church on any given Sunday.

Answer 1

Using the z-distribution, it is found that the 95% confidence interval to estimate the actual percentage of adults in the U.S., aged 18 and older, that are "likely" to ONLY be in church on any given Sunday is (23.36%, 28.72%).

In a sample with a number n of people surveyed with a probability of a success of $\pi$, and a confidence level of $\alpha$, we have the following confidence interval of proportions.

$\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}$

In which z is the z-score that has a p-value of $\frac{1+\alpha}{2}$.

269 out of 1033 respondents said they were likely to only be in church, hence:

$n = 1033, \pi = \frac{269}{1033} = 0.2604$

95% confidence level, hence$\alpha = 0.95$, z is the value of Z that has a p-value of $\frac{1+0.95}{2} = 0.975$, so $z = 1.96$.  

The lower limit of this interval is:

$\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.2604 - 1.96\sqrt{\frac{0.2604(0.7396)}{1033}} = 0.2336$

The upper limit of this interval is:

$\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.2604 + 1.96\sqrt{\frac{0.2604(0.7396)}{1033}} = 0.2872$

As a percentage:

0.2336 x 100% = 23.36%

0.2872 x 100% = 28.72%

The 95% confidence interval to estimate the actual percentage of adults in the U.S., aged 18 and older, that are "likely" to ONLY be in church on any given Sunday is (23.36%, 28.72%).

A similar problem is given at https://brainly.com/question/25743435