Respuesta :
Answer:
We would need at least 385 adult Americans.
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.
How many adult Americans would need to be surveyed to estimate the mean amount of sleep per night within 0.12 hour with 95% confidence?
You would need at least n adults, in which n is found when M = 0.12, [tex]\sigma = 1.2[/tex]. So
[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]
[tex]0.12 = 1.96*\frac{1.2}{\sqrt{n}}[/tex]
[tex]0.12\sqrt{n} = 1.96*1.2[/tex]
[tex]\sqrt{n} = \frac{1.96*1.2}{0.12}[/tex]
[tex]\sqrt{n} = 19.6[/tex]
[tex](\sqrt{n})^{2} = (19.6)^{2}[/tex]
[tex]n = 384.16[/tex]
Rounding up, 385
We would need at least 385 adult Americans.
