Sample size find

Question

You want to obtain a sample to estimate a population proportion. Based on previous evidence, you believe the population proportion is approximately p∗=15%. You would like to be 90% confident that your estimate is within 0.2% of the true population proportion. How large of a sample size is required?

n= _______

You want to obtain a sample to estimate a population proportion. At this point in time, you have no reasonable estimate for the population proportion. You would like to be 90% confident that you esimate is within 1.5% of the true population proportion. How large of a sample size is required?

n= ______________

Answer 1

To determine the required sample size, we can use the formula for sample size calculation for estimating a population proportion:

n = (Z^2 * p * (1 - p)) / E^2

where: n = required sample size Z = z-score corresponding to the desired confidence level p = estimated population proportion E = desired margin of error

In this case, we want to estimate the population proportion with a 90% confidence level and a margin of error of 0.2%. The z-score for a 90% confidence level is approximately 1.645 (obtained from standard normal distribution tables) and the estimated population proportion is p* = 15%.

Substituting these values into the formula:

n = (1.645^2 * 0.15 * (1 - 0.15)) / (0.002^2) n ≈ 86254.546875

Rounding up to the nearest whole number, the required sample size is approximately 86255.

When there is no reasonable estimate for the population proportion, we can use a conservative estimate of p* = 0.5, which maximizes the required sample size.

To calculate the sample size, we can use the formula:

n = (Z^2 * p * (1 - p)) / E^2

where: n = required sample size Z = z-score corresponding to the desired confidence level E = desired margin of error

In this case, we want to be 90% confident with a margin of error of 1.5%, so the z-score for a 90% confidence level is approximately 1.645, and the margin of error is 0.015.

Substituting these values into the formula:

n = (1.645^2 * (0.5)*(1-0.5)) / (0.015^2) n ≈ 3006.69444444

Rounding up to the nearest whole number, the required sample size is 3007.