I want to obtain a sample to estimate a population mean. Researchers believe the population standard deviation is approximately 41.7. We would like to be 99.5% confident that the estimate is within 4...

I want to obtain a sample to estimate a population mean.  Researchers believe the population standard deviation is approximately 41.7.  We would like to be 99.5% confident that the estimate is within 4 of the true population mean.  How large of a sample size is required?

The answer is 857.

How did you get z (2.807) and what is the square root of 'n'.  How did you get n?