The music industry must adjust to the growing practice of 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.
Why did the scarecrow win an award? Because he was outstanding in his field! Now, let's tackle your question about determining the percentage of downloaded songs with confidence.
To calculate the required sample size, we need to consider the desired level of confidence, the desired margin of error, and the estimated proportion of songs downloaded.
Since you want to be 99% confident and have the sample percentage within two percentage points of the true population percentage, we can use a formula based on the normal distribution.
The formula for calculating the sample size needed is:
n = (Z^2 * P * (1-P)) / E^2
Where:
n = sample size
Z = Z-score corresponding to the desired level of confidence (in this case, 99% -> Z = 2.576)
P = estimated proportion of songs downloaded
E = desired margin of error (in this case, 2 percentage points -> 0.02)
Now, let's use this formula. However, we need to know the estimated proportion of songs that are downloaded to properly determine the sample size. Can you provide that estimate?
To determine the sample size needed to estimate the percentage of songs downloaded with 99% confidence and a margin of error of 2 percentage points, we can use the formula:
n = (Z^2 * p * (1-p)) / E^2
Where:
n = sample size needed
Z = Z-value corresponding to the desired confidence level (99% confidence corresponds to a Z-value of approximately 2.576)
p = estimated proportion (we can use 0.5 as it's the most conservative estimate, assuming equal chances of downloaded and non-downloaded songs)
E = margin of error (2 percentage points as specified)
Plugging in the values into the formula:
n = (2.576^2 * 0.5 * (1-0.5)) / 0.02^2
Calculating:
n = (6.635776 * 0.25) / 0.04
n = 1.658944 / 0.04
n ≈ 41.472
Therefore, we would need to survey at least 42 randomly selected song purchases to determine the percentage of songs downloaded with 99% confidence and a margin of error of 2 percentage points.
To determine the sample size needed to estimate the percentage of songs downloaded with a desired level of confidence, you need to use the formula for sample size determination for proportions.
The formula is as follows:
n = (Z^2 * p * (1-p)) / E^2
Where:
n = required sample size
Z = Z-score corresponding to the desired confidence level (in this case, for 99% confidence level, Z is approximately 2.58)
p = estimated proportion of songs downloaded (unknown, we assume p = 0.5 to get the maximum sample size)
E = desired margin of error, which is two percentage points (E = 0.02)
Now let's plug in the values:
n = (2.58^2 * 0.5 * (1-0.5)) / 0.02^2
Simplifying the equation, we get:
n = (6.6564 * 0.25) / 0.0004
n = 16641
Therefore, you would need to survey approximately 16,641 randomly selected song purchases to determine the percentage of songs that are downloaded with a 99% confidence level and a margin of error of two percentage points.