Respuesta :
Answer:
The required sample size is 35.
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].
That is z with a pvalue of [tex]1 - 0.025 = 0.975[/tex], so Z = 1.96.
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.
Determine the required sample size.
The required sample size is n, which is found when [tex]M = 8[/tex]
The standard deviation of 24 means that [tex]\sigma = 24[/tex]. So
[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]
[tex]8 = 1.96\frac{24}{\sqrt{n}}[/tex]
[tex]8\sqrt{n} = 1.96*24[/tex]
Simplifying both sides by 8
[tex]\sqrt{n} = 1.96*3[/tex]
[tex](\sqrt{n})^2 = (1.96*3)^2[/tex]
[tex]n = 34.6[/tex]
Rounding up:
The required sample size is 35.
