Respuesta :
Answer:
The minimum sample size that should be taken is 72.
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 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.
If we want to be 95% confident that the sample mean is within 3 inches of the true population mean, what is the minimum sample size that should be taken
This is n when M = 3. So
[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]
[tex]3 = 1.96*\frac{13}{\sqrt{n}}[/tex]
[tex]3\sqrt{n} = 25.48[/tex]
[tex]\sqrt{n} = \frac{25.48}{3}[/tex]
[tex]\sqrt{n} = 8.49[/tex]
[tex]\sqrt{n}^{2} = (8.49)^{2}[/tex]
[tex]n = 72[/tex]
The minimum sample size that should be taken is 72.
