EU (European Union) countries report that 46% of their labor force is female. The United Nations wants to determine if the percentage of women in the U.S. labor force is the same. Representatives from the United States Department of Labor plan to check a random sample of more than 10,000 employment records on file to estimate the percentage of women in the U.S. labor force.
The Department of Labor wants to estimate the percentage of women in the U.S. labor force to within ±5% with 90% confidence. How many employment records should be sampled?
EU (European Union) countries report that 46% of their labor force is female. The United Nations wants to determine if the percentage of women in the U.S. labor force is the same. Representatives from the United States Department of Labor plan to check a random sample of more than 10,000 employment records on file to estimate the percentage of women in the U.S. labor force.
Suppose the Department of Labor wants to be 90% confident of estimating the percentage of women in the labor force to within ±2% of the true percentage. To do this they would have to
Select one:
a. decrease the sample size.
b. increase the sample size.
c. increase the sampling error. Incorrect
d. select the same number of employment records.
e. decrease the precision.
To determine the number of employment records that should be sampled, we can use the formula for sample size calculation for proportions.
The formula is given as:
n = (Z^2 * p * q) / E^2
Where:
n = required sample size
Z = Z-value for the desired confidence level
p = proportion estimate (46% or 0.46)
q = 1 - p
E = margin of error (0.05 or 5%)
In this case, the desired confidence level is 90%, which corresponds to a Z-value of 1.645 (obtained from a standard normal distribution table).
Substituting these values into the formula:
n = (1.645^2 * 0.46 * 0.54) / 0.05^2
n = (2.705025 * 0.46 * 0.54) / 0.0025
n = (0.68621191) / 0.0025
n ≈ 274.48
Since we can't have a fraction of a sample, we need to round up the sample size to the nearest whole number:
n = 275
Therefore, the Department of Labor should sample at least 275 employment records to estimate the percentage of women in the U.S. labor force with a margin of error of ±5% and a confidence level of 90%.
To estimate the percentage of women in the U.S. labor force with a specified level of confidence and margin of error, you can use the formula for sample size calculation:
n = (Z^2 * p * (1-p)) / E^2
Where:
- n is the required sample size
- Z is the z-score corresponding to the desired confidence level (in this case, 90% confidence corresponds to a z-score of 1.645)
- p is the estimated proportion of women in the population (given as 46% or 0.46)
- E is the desired margin of error (given as ±5% or 0.05)
Plugging in the values:
n = (1.645^2 * 0.46 * (1-0.46)) / (0.05^2)
Calculating this equation, we get:
n ≈ 1059.16
Therefore, you should sample at least 1,060 employment records to estimate the percentage of women in the U.S. labor force with a margin of error of ±5% and 90% confidence. Keep in mind that since you need more than 10,000 employment records, you might need to consider a larger sample size to maintain the desired level of confidence and margin of error.