Respuesta :
Answer:
c.A 90% confidence level and a sample size of 300 subjects.
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 [tex]1-\alpha[/tex], we have the confidence interval with a margin of error of:
[tex]M = 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].
In this problem
The proportions are the same for all the options, so we are going to write our margins of error as functions of [tex]\sqrt{\pi(1-\pi)}[/tex]
So
a.A 99% confidence level and a sample size of 50 subjects.
[tex]n = 50[/tex]
99% confidence interval
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 margin of error is
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}} = \frac{2.575}{\sqrt{50}}\sqrt{\pi(1-\pi)} = 0.3642\sqrt{\pi(1-\pi)}[/tex]
b.A 90% confidence level and a sample size of 50 subjects.
[tex]n = 50[/tex]
90% confidence interval
So [tex]\alpha = 0.1[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.1}{2} = 0.95[/tex], so [tex]Z = 1.645[/tex].
The margin of error is
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}} = \frac{1.645}{\sqrt{50}}\sqrt{\pi(1-\pi)} = 0.2623\sqrt{\pi(1-\pi)}[/tex]
c.A 90% confidence level and a sample size of 300 subjects.
[tex]n = 300[/tex]
90% confidence interval
So [tex]\alpha = 0.1[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.1}{2} = 0.95[/tex], so [tex]Z = 1.645[/tex].
The margin of error is
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}} = \frac{1.645}{\sqrt{300}}\sqrt{\pi(1-\pi)} = 0.0950\sqrt{\pi(1-\pi)}[/tex]
This produces smallest margin of error.
d.A 99% confidence level and a sample size of 300 subjects.
[tex]n = 300[/tex]
99% confidence interval
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 margin of error is
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}} = \frac{2.575}{\sqrt{300}}\sqrt{\pi(1-\pi)} = 0.1487\sqrt{\pi(1-\pi)}[/tex]
