Respuesta :
Answer:
(0.3798, 0.4602)
Step-by-step explanation:
Let p be the true proportion of US adults who believe raising the minimum wage will help the economy. A point estimate for p is [tex]\hat{p}=0.42[/tex] and a good aproximation (because we have a large sample) for the standard deviation of [tex]\hat{p}[/tex] is [tex]\sqrt{\hat{p}(1-\hat{p})/n}=\sqrt{0.42(0.58)/1000}=0.0156[/tex]. Therefore, a 99% confidence interval for p is given by [tex]\hat{p}\pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}[/tex] where [tex]\alpha=0.01[/tex] and [tex]z_{\alpha/2}[/tex] is the [tex]\alpha/2[/tex]th quantile of the standard normal distribution, so we have, [tex]0.42\pm z_{0.005}0.0156[/tex], i.e., [tex]0.42\pm(-2.5758)(0.0156)[/tex] or equivalently (0.3798, 0.4602).
The 99% confidence interval for the true proportion of US adults who believe this is (0.38, 0.46).
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]\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm M[/tex]
The margin of error is:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the z-score that has a p-value of [tex]\frac{1+\alpha}{2}[/tex].
In this problem:
- Survey of 1000 adults, thus [tex]n = 1000[/tex]
- Proportion of 42%, thus [tex]\pi = 0.42[/tex]
- 99% confidence level, thus z has a p-value of [tex]\frac{1 + 0.99}{2} = 0.995[/tex], thus z = 2.575.
Then:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
[tex]M = 2.575\sqrt{\frac{0.42(0.58)}{1000}}[/tex]
[tex]M = 0.04[/tex]
The interval is:
[tex]M - \pi = 0.42 - 0.04 = 0.38[/tex]
[tex]M + \pi = 0.42 + 0.04 = 0.46[/tex]
It is (0.38, 0.46).
A similar problem is given at https://brainly.com/question/22618167
