You are given a population with standard deviation of 8.6. Determine the sample size needed to estimate the mean of the population with error of 0.5 at 99 percent confidence.

We are given that,

$E=0.5\\\sigma=8.6\\\alpha=0.01$

To find the required value of the sample size $n$ needed to estimate the mean of the population with error of 0.5, we use the formula below.

$n=({Z_{\alpha\over2}\times \sigma\over E})^2$ where $Z_{\alpha\over2}=Z_{0.001\over2}=Z_{0.005}=2.575$

Replacing for the values above in the formula, we have,

$n=({2.575\times8.6\over0.5})^2=({22.145\over0.5})^2=44.29^2=1961.6041\approx 1962$

Therefore, the sample size needed to estimate the mean of the population with error of 0.5 at 99 percent confidence is $n=1962.$