Respuesta :
Answer:
The should sample at least 303 components.
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 of [tex]1-\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 zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].
The margin of error is:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
99% confidence level
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].
(A) How many components should they sample at least?
This is n when [tex]M = 0.074[/tex]
We do not know the true proportion, so we use [tex]\pi = 0.5[/tex], which is the case in which we are going to need to sample the largest number of components.
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
[tex]0.074 = 2.575\sqrt{\frac{0.5*0.5}{n}}[/tex]
[tex]0.074\sqrt{n} = 2.575*0.5[/tex]
[tex]\sqrt{n} = \frac{2.575*0.5}{0.074}[/tex]
[tex](\sqrt{n})^{2} = (\frac{2.575*0.5}{0.074})^{2}[/tex]
[tex]n = 302.7[/tex]
Rounding up
The should sample at least 303 components.
