In order to qualify for federal job training funds, a neighborhood group must provide an estimate of the mean of education level of unemployed youths less than or equal to 21 years old. The group plans to survey youth in its service area. The population of unemployed youth in the service area is estimated at 2,000. Previous studies have found that the standard deviation of years of education for this population is 10 years. If the neighborhood group must be 99% certain that their estimate is within one year of the actual mean, for approximately how many youths must the group obtain responses.
To determine the sample size needed, we can use the formula:
n = (Z * σ / E)²
Where:
n = sample size
Z = Z-score corresponding to the desired confidence level (99% corresponds to a Z-score of 2.575)
σ = standard deviation of the population
E = desired margin of error (in this case, 1 year)
Substituting the given values:
n = (2.575 * 10 / 1)²
Calculating:
n = (25.75)²
n ≈ 664
Therefore, the neighborhood group must obtain responses from approximately 664 youths in order to be 99% certain that their estimate is within one year of the actual mean.
To determine the sample size needed to estimate the mean with a certain level of confidence, we can use the formula:
n = ((Z * σ) / E)^2
where:
n = sample size
Z = Z-score for the desired level of confidence
σ = standard deviation of the population
E = margin of error (desired precision)
In this case, the group wants to be 99% confident that their estimate is within one year of the actual mean. We can calculate the corresponding Z-score using a standard normal distribution table. For a 99% confidence level, the Z-score is approximately 2.576.
n = ((2.576 * 10) / 1)^2
n = (25.76)^2
n ≈ 664
Therefore, the neighborhood group must obtain responses from approximately 664 youths in order to be 99% confident that their estimate is within one year of the actual mean.