The music industry must adjust to the growing practice o consumers downloading songs instead of buying CDs. It therefore becomes important to estimate the proportion of songs that are currently downloaded. How many randomly selected song purchases must be surveyed to determine the percentage that were obtained by downloading? Assume that we want to be 99% confident that the sample percentage is within two percentage points of the true population percentage of songs that are downloaded.
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.