Respuesta :
Answer:
[tex]n=(\frac{1.640(12)}{4})^2 =24.206 \approx 25[/tex]
So the answer for this case would be n=25 rounded up to the nearest integer
Step-by-step explanation:
Previous concepts
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
Solution to the problem
The margin of error is given by this formula:
[tex] ME=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex] (a)
And on this case we have that ME =4 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 90% of confidence interval now can be founded using the normal distribution. And in excel we can use this formla to find it:"=-NORM.INV(0.05;0;1)", and we got [tex]z_{\alpha/2}=1.640[/tex], replacing into formula (b) we got:
[tex]n=(\frac{1.640(12)}{4})^2 =24.206 \approx 25[/tex]
So the answer for this case would be n=25 rounded up to the nearest integer
Answer:
The number of commuters to be randomly selected to estimate the mean driving time of Chicago commuters is 25
Step-by-step explanation:
At 90% confidence
we have
Minimum sample size, n given by
[tex]n = \left (\frac{z_{c}\sigma }{E} \right )^{2}[/tex]
Where:
c = Confidence level = 90%
σ = Standard deviation = 12 minutes
E = Maximum error in estimate = minutes
-[tex]z_c[/tex] = 1.64 and [tex]z_c[/tex] = 1.64
Therefore, [tex]z_c[/tex] = 0.51994 and therefore, n is given by
[tex]n = \left (\frac{1.64\times 12 }{4} \right )^{2}[/tex] = 24.2064
As we are talking of population, we round to the next whole number, thus;
n = 25 commuters.
