You want to obtain a sample to estimate a population mean.

Question

You want to obtain a sample to estimate a population mean. Based on previous evidence, you believe the population standard deviation is approximately σ=61.8. You would like to be 90% confident that your estimate is within 10 of the true population mean. How large of a sample size is required? 

n =  

Answer 1

For a Normal Distribution, a 90% Confidence Level corresponds to a

Critical Z Value of z_c = 1.645.

Guide on the relation z_c = (X-bar − μ) ÷ σ_X-barand restate as z_cσ_X-bar = (X-bar − μ).

This last translates to z_c[σ/√n] = (X-bar − μ); σ_X-bar is replaced by [σ/√n] by presuming

that the sample size is less than 5% of the population size (or n < 0.05N).

Enumerate z_c[σ/√n] = (X-bar − μ) to 1.645[61.8 ÷ √n] = 10 which goes to

0.1645 = √n ÷ 61.8.

Finally √n = 10.1661 gives n as 103.3495892; round to the highest integer

to obtain 104 as the sample size sought.