Respuesta :
Answer:
1) The 95% confidence interval for the mean check-in time is between 2.003 hours and 2.331 hours.
2) The sample size that needs to be taken is 273.
Step-by-step explanation:
1)
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.
[tex]M = 1.96*\frac{0.48}{\sqrt{33}} = 0.164[/tex]
The lower end of the interval is the sample mean subtracted by M. So it is 2.167 – 0.164 = 2.003 hours
The upper end of the interval is the sample mean added to M. So it is 2.167 + 0.164 = 2.331 hours.
The 95% confidence interval for the mean check-in time is between 2.003 hours and 2.331 hours.
2)
In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].
The margin of error is given by:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
For this problem, we have that:
[tex]\pi = 0.05[/tex]
95% confidence level
So [tex]\alpha = 0.05[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].
What is the sample size that needs to be taken for 0.03 desired margin of error?
This is n when M = 0.03. So
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
[tex]0.03 = 1.96\sqrt{\frac{0.05*0.95}{n}}[/tex]
[tex]0.03\sqrt{n} = 1.96\sqrt{0.05*0.95}[/tex]
[tex]\sqrt{n} = \frac{1.96\sqrt{0.05*0.95}}{0.03}[/tex]
[tex](\sqrt{n})^{2} = (\frac{1.96\sqrt{0.05*0.95}}{0.03})^{2}[/tex]
[tex]n = 203[/tex]
The sample size that needs to be taken is 273.
