Question

How many students must be randomly selected to estimate the mean weekly earnings of students at one college? we want 95% confidence that the sample mean is within $2 of the population mean, and the population standard deviation is known to be $10?

Answer 1

The sample size needs to be 96.

First we find the z-score for this confidence interval:
Convert 95% to a decimal:  95/100 = 0.95
Subtract from 1:  1 - 0.95 = 0.05
Divide by 2:  0.05/2 = 0.025
Subtract from 1:  1 = 0.025 = 0.975

Using a z-table (http://www.z-table.com) we look up this value in the chart; the z-score that goes with this value is 1.96.

Now we use the formula

$n=(\frac{z_{\alpha/2}*\sigma}{E})^2 \\ \\=(\frac{1.96*10}{2})^2=\frac{19.6}{2}^2=96.04\approx96$