Estimating population proportion HELP - Prob/stats

Question

1) You want to obtain a sample to estimate a population proportion. Based on previous evidence, you believe the population proportion is approximately p∗=30%. You would like to be 95% confident that your estimate is within 1% of the true population proportion. How large of a sample size is required?

n = _____________________

2) 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 99.5% confident that you estimate is within 2.5% of the true population proportion. How large of a sample size is required?

n = _____________________

Answer 1

1) To determine the required sample size to estimate a population proportion with a specified margin of error and confidence level, we can use the formula:

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

Where: n = sample size Z = Z-score corresponding to the desired confidence level (for a 95% confidence level, Z ≈ 1.96) p = estimated population proportion (in decimal form, p = 0.30) E = margin of error (in decimal form, E = 0.01)

Substituting the values into the formula:

n = (1.96^2 * 0.30 * (1-0.30)) / 0.01^2

Calculating the value:

n ≈ 8067.36

Therefore, a sample size of approximately 8068 is required to estimate the population proportion with a 95% confidence level and a margin of error of 1%.

2) When no reasonable estimate for the population proportion is available, we can use a conservative estimate of p = 0.5 to calculate the sample size. This approach ensures that we obtain the largest possible sample size, which provides the highest level of precision for the estimation.

To determine the required sample size, we can use the formula:

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

Where: n = sample size Z = Z-score corresponding to the desired confidence level (for a 99.5% confidence level, Z ≈ 2.807) p = estimated population proportion (p = 0.5) E = margin of error (in decimal form, E = 0.025)

Substituting the values into the formula:

n = (2.807^2 * 0.5 * (1-0.5)) / 0.025^2

Calculating the value:

n ≈ 3151.6996

Therefore, a sample size of approximately 3152 is required to estimate the population proportion with a 99.5% confidence level and a margin of error of 2.5%.