Respuesta :
Answer:
c. 139
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 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.
How large a sample should be taken to estimate the average age of out-of-state guests with a margin of error no larger than 5 and with a 95% level of confidence?
We need a sample size of n.
n is found when [tex]M = 5, \sigma = 30[/tex]
So
[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]
[tex]5 = 1.96*\frac{30}{\sqrt{n}}[/tex]
[tex]5\sqrt{n} = 1.96*30[/tex]
Simplifying by 5
[tex]\sqrt{n} = 1.96*6[/tex]
[tex](\sqrt{n})^{2} = (1.96*6)^{2}[/tex]
[tex]n = 138.30[/tex]
We round up,
So the correct answer is:
c. 139
Answer:
[tex]n=(\frac{1.960(30)}{5})^2 =138.30 \approx 139[/tex]
And if we round up to the nearest integer we got n =139, and the best answer for this case is:
c. 139
Step-by-step explanation:
For this case we have this previous info:
[tex]\sigma = 30[/tex] represent the previous estimation for the population deviation
[tex]Confidence =0.95[/tex] represent the confidence level
The margin of error for the true mean is given by this formula:
[tex] ME=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex] (a)
The desired margin of error is ME =5 and we are interested in order to find the value of n, if we solve n from equation (a) we got:
[tex]n=(\frac{z_{\alpha/2} \sigma}{ME})^2[/tex] (b)
The critical value for a 95% of confidence interval given now can be founded using the normal distribution. For this case the critical value would be given by [tex]z_{\alpha/2}=1.960[/tex], replacing into formula (b) we got:
[tex]n=(\frac{1.960(30)}{5})^2 =138.30 \approx 139[/tex]
And if we round up to the nearest integer we got n =139, and the best answer for this case is:
c. 139
