many states are considering steps that would help them collect sales taxes on items purchased through the internet. how many randomly selected sales transactions must be surveyed to determine the percentage that transpired over the internet? assume that we want to be 99 percent confident that the sample percentage is within 3 percentage points of the true population percentage for all sales transactions.
339
To determine the number of randomly selected sales transactions that must be surveyed, we can use the formula for the sample size of a proportion.
The formula for sample size (n) is:
n = (Z^2 * p * q) / E^2
Where:
- Z is the Z-score corresponding to the desired level of confidence. For a 99% confidence level, the Z-score is approximately 2.576.
- p is the estimated proportion of sales transactions over the internet. Since we don't have an estimated proportion, we can use 0.5 as a conservative value as it will give us the largest sample size.
- q is the complement of p (1 - p).
- E is the margin of error (in decimal form) you want to achieve. In this case, since we want to be within 3 percentage points, the margin of error is 0.03.
Plugging in these values into the formula:
n = (2.576^2 * 0.5 * (1 - 0.5)) / 0.03^2
Simplifying the equation:
n = (6.656 * 0.5 * 0.5) / 0.0009
n = 0.0832 / 0.0009
n ≈ 92,444.44
Since you can't have a fraction of a transaction, you should round the sample size up to the nearest whole number.
Therefore, you would need to survey at least 92,445 randomly selected sales transactions to determine the percentage of sales transactions that transpired over the internet with 99% confidence and a margin of error of 3 percentage points.