Answer:
n = 133.
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 - 0.9}{2} = 0.05[/tex]
Now, we have to find z in the Z-table as such z has a p-value of [tex]1 - \alpha[/tex].
That is z with a pvalue of [tex]1 - 0.05 = 0.95[/tex], so Z = 1.645.
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.
Standard deviation of the times is 14 hours.
This means that [tex]\sigma = 14[/tex]
How large a sample should be taken to get the desired interval?
Within 2 hours, which means that we want n for which M = 2. So
[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]
[tex]2 = 1.645\frac{14}{\sqrt{n}}[/tex]
[tex]2\sqrt{n} = 1.645*14[/tex]
Dividing both sides by 2
[tex]\sqrt{n} = 1.645*7[/tex]
[tex](\sqrt{n})^2 = (1.645*7)^2[/tex]
[tex]n = 132.6[/tex]
Rounding up, n = 133.
