Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
Using the z-distribution, it is found that the needed sample sizes are:
a) 242
b) 1842
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].
The margin of error is:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
99% confidence level, hence[tex]\alpha = 0.99[/tex], z is the value of Z that has a p-value of [tex]\frac{1+0.99}{2} = 0.995[/tex], so [tex]z = 2.575[/tex].
Item a:
The estimate is:
[tex]\pi = 0.223 - 0.189 = 0.034[/tex]
The sample size is n for which M = 0.03, then:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
[tex]0.03 = 2.575\sqrt{\frac{0.034(0.966)}{n}}[/tex]
[tex]0.03\sqrt{n} = 2.575\sqrt{0.034(0.966)}[/tex]
[tex]\sqrt{n} = \frac{2.575\sqrt{0.034(0.966)}}{0.03}[/tex]
[tex](\sqrt{n})^2 = \left(\frac{2.575\sqrt{0.034(0.966)}}{0.03}\right)^2[/tex]
[tex]n = 241.97[/tex]
Rounding up, a sample of 242 is needed.
Item b:
No prior estimates, hence [tex]\pi = 0.5[/tex] is used.
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
[tex]0.03 = 2.575\sqrt{\frac{0.5(0.5)}{n}}[/tex]
[tex]0.03\sqrt{n} = 2.575\sqrt{0.5(0.5)}[/tex]
[tex]\sqrt{n} = \frac{2.575\sqrt{0.5(0.5)}}{0.03}[/tex]
[tex](\sqrt{n})^2 = \left(\frac{2.575\sqrt{0.5(0.5)}}{0.03}\right)^2[/tex]
[tex]n = 1841.8[/tex]
Rounding up, a sample of 1842 is needed.
For more on the z-distribution, you can check https://brainly.com/question/25404151
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
