To determine the required sample size, we need to use a formula for sample size calculation in proportion estimation.
The formula is:
n = (Z^2 * p * q) / E^2
Where:
n = required sample size
Z = Z-score corresponding to the desired confidence level (99% confidence level corresponds to a Z-score of approximately 2.58)
p = estimated proportion of songs downloaded (for now, let's assume it to be 0.5, indicating an equal chance of a song being downloaded or bought on CD)
q = 1 - p (complementary probability of p)
E = desired margin of error (in this case, 2 percentage points, so E = 0.02)
Plugging in the values:
n = (2.58^2 * 0.5 * 0.5) / 0.02^2
n = (6.6564 * 0.25) / 0.0004
n = 1.6641 / 0.0004
n ≈ 4160.25
So, you would need to survey approximately 4161 randomly selected song purchases to determine the percentage of songs downloaded with 99% confidence and a margin of error of 2 percentage points.
It's important to note that the assumption of p = 0.5 might not be accurate. If you have any preliminary data on the proportion of songs downloaded, it is advisable to use that instead to improve the accuracy of the calculation.