The Z-value for a 90 percent confidence is 1.645
To find the sample size use the formula:
Sample size = Z-value x SD / error)^2
Using the provided information:
SD = 4,000 miles
Error = 500 miles
Sample size = ((1.645) (4000)/ 500)^2 = 173.18 = 174
Answer:
The required sample size is n=8,660.
Step-by-step explanation:
We have to calculate the minimum sample size that will give us a maximum margin of error of 500 miles, with a 90% confidence.
A pilot sample of n=50 give a sample standard deviation of 4,000 miles.
With the pilot sample we can calculate the population standard deviation as:
[tex]\sigma_M=\sigma/\sqrt{n }\\\\\sigma=\sqrt{n}\sigma_M=\sqrt{50}*4,000=7.071*4,000=28,284[/tex]
The equation for the margin of error is:
[tex]E=z\cdot \sigma/\sqrt{n}[/tex]
The z-value for a 90% confidence interval is z=1.645.
Then, we can estimate the sample size as:
[tex]n=\left(\dfrac{z\cdot \sigma}{E}\right)^2=\left(\dfrac{1.645\cdot 28,284}{500}\right)^2=93.06^2=8,659.28\approx8,660[/tex]
