"a pollster wishes to estimate the number of left-handed scientists. how large of a sample is needed in order to be 98% confident that the sample proportion will no differ from the mean by more than 4%. also suppose we know a previous study concluded that the proportion of left-handed scientists is 8%"

The sample size should be 250.

Our margin of error is 4%, or 0.04.  We use the formula



To find the z-score:
Convert 98% to a decimal:  0.98
Subtract from 1:  1-0.98 = 0.02
Divide both sides by 2:  0.02/2 = 0.01
Subtract from 1:  1-0.01 = 0.99

Using a z-table (http://www.z-table.com) we see that this value has a z-score of approximately 2.33.  Using this, our margin of error and our proportion, we have:



Divide both sides by 2.33:


Square both sides:


Multiply both sides by n:


Divide both sides to isolate n:

The answer is 60.9774