Respuesta :
Answer:
There is 95% confidence that the population proportion of people from Vermont who exercised for at least 30 minutes a day 3 days a week is between 55.9% and 74.7%.
Step-by-step explanation:
We have to answer the population proportion for Vermont.
We can only do it by a confidence interval, as we only have information from a sample.
This sample, of size n=100, has a proportion p=0.653.
The degrees of freedom are:
[tex]df=n-1=100-1=99[/tex]
We will calculate a 95% confidence interval, which for df=99 has a critical value of t of t=1.984.
The margin of error can be calculated as:
[tex]E=t*\sigma_p=t\sqrt{\dfrac{p(1-p)}{n}}=1.984\sqrt{\dfrac{0.653*0.347}{100}}\\\\\\E=1.984*\sqrt{0.00226}=1.984*0.0476=0.094[/tex]
Then, the upper and lower bounds of the confidence interval are:
[tex]LL=p-E=0.653-0.094=0.559\\\\UL=p+E=0.653+0.094=0.747[/tex]
Then, we can say that there is 95% confidence that the population proportion of people from Vermont who exercised for at least 30 minutes a day 3 days a week is between 55.9% and 74.7%.
Using a confidence interval, we can be 95% confident that the value of the population proportion of people from Vermont who exercised for at least 30 minutes a day 3 days a week is (0.56, 0.746).
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 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].
Proportion of 65.3% from a sample of 100, hence two parameters are [tex]\pi = 0.653, n = 100[/tex]
95% confidence level
So [tex]\alpha = 0.95[/tex], z is the value of Z that has a p-value of [tex]\frac{1+0.95}{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}} = 0.653 - 1.96\sqrt{\frac{0.653(0.347)}{100}} = 0.56[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.653 + 1.96\sqrt{\frac{0.653(0.347)}{100}} = 0.746[/tex]
We can be 95% confident that the value of the population proportion of people from Vermont who exercised for at least 30 minutes a day 3 days a week is (0.56, 0.746).
A similar problem is given at https://brainly.com/question/16807970
