Respuesta :
Answer:
She needs a sample of at least 385.
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]
95% confidence level
So [tex]\alpha = 0.05[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].
How large a sample should she take to achieve this?
She needs a sample of size at least n.
n is found when M = 0.04.
We do not know the true population proportion, so we use [tex]\pi = 0.5[/tex], which is the case for which we are going to need the largest sample size.
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
[tex]0.05 = 1.96\sqrt{\frac{0.5*0.5}{n}}[/tex]
[tex]0.05\sqrt{n} = 1.96*0.5[/tex]
Dividing both sides by 0.05
[tex]\sqrt{n} = 19.6[/tex]
[tex](\sqrt{n})^{2} = 19.6^{2}[/tex]
[tex]n = 384.2[/tex]
Rounding up
She needs a sample of at least 385.
