Respuesta :
Answer:
We need a sample size of at least 97.
Step-by-step explanation:
We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:
[tex]\alpha = \frac{1-0.95}{2} = 0.025[/tex]
Now, we have to find z in the Ztable as such z has a pvalue of [tex]1-\alpha[/tex].
So it is z with a pvalue of [tex]1-0.025 = 0.975[/tex], so [tex]z = 1.96[/tex]
Now, find the margin of error M as such
[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]
In which [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.
Find the sample size needed to assure with 95 percent confidence that the sample mean will not differ from the population mean by more than 2 dollars.
We need a sample size of at least n.
n is found when [tex]M = 2, \sigma = 10[/tex].
So
[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]
[tex]2 = 1.96*\frac{10}{\sqrt{n}}[/tex]
[tex]2\sqrt{n} = 1.96*10[/tex]
[tex]\sqrt{n} = \frac{1.96*10}{2}[/tex]
[tex](\sqrt{n})^{2} = (\frac{1.96*10}{2})^{2}[/tex]
[tex]n = 96.04[/tex]
Rouding up
We need a sample size of at least 97.
