A music industry researcher wants to estimate, with a 90% confidence level, the proportion of young urban people (ages 21 to 35 years) who go to at least 3 concerts a year. Previous studies show that 21% of those people (21 to 35 year olds) interviewed go to at least 3 concerts a year. The researcher wants to be accurate within 1% of the true proportion. Find the minimum sample size necessary.
To find the minimum sample size necessary, we can use the formula for calculating sample size in a proportion estimation:
n = (Z^2 * p * (1-p)) / E^2
Where:
n = sample size
Z = Z-score (corresponding to the desired confidence level)
p = estimated proportion
E = margin of error
Given information:
- Confidence level: 90% (which corresponds to a Z-score of 1.645)
- Estimated proportion: 21% (or 0.21)
- Margin of error: 1% (or 0.01)
Plugging these values into the formula:
n = (1.645^2 * 0.21 * (1-0.21)) / 0.01^2
Simplifying the equation:
n = (2.708 * 0.21 * 0.79) / 0.0001
n ≈ 3,868.34
Since we cannot have a fraction of a person in a sample, we need to round up to the next whole number. Therefore, the minimum sample size necessary is 3,869.
Therefore, the music industry researcher will need a minimum sample size of 3,869 young urban people (ages 21 to 35 years) to estimate the proportion of those who go to at least 3 concerts a year with a 90% confidence level and an accuracy of within 1%.