1. To find the minimum sample size needed with a prior estimate available, we can use the formula for sample size calculation for proportions:
n = (Z^2 * p * (1-p)) / E^2
Where:
- n is the required sample size
- Z is the z-score corresponding to the desired level of confidence (in this case, 95%), which is approximately 1.96
- p is the prior estimate (48% or 0.48 in decimal form)
- E is the desired margin of error (4% or 0.04 in decimal form)
Substituting the given values into the formula, we have:
n = (1.96^2 * 0.48 * (1-0.48)) / 0.04^2
n = (3.8416 * 0.48 * 0.52) / 0.0016
n = 0.9867 / 0.0016
n ≈ 616.69
So, the researcher would need a minimum sample size of approximately 617 participants to estimate the proportion of adults with high-speed internet access with a 95% confidence interval and a margin of error of 4% based on the prior estimate of 48%.
2. Without a preliminary estimate available, we can assume the most conservative estimate, which is p = 0.5 (50% in decimal form). This is because when no information about the proportion is available, assuming p = 0.5 provides the largest sample size requirement, giving the most conservative estimate.
Using the same formula as above:
n = (Z^2 * p * (1-p)) / E^2
Substituting the values:
n = (1.96^2 * 0.5 * (1-0.5)) / 0.04^2
n = (3.8416 * 0.5 * 0.5) / 0.0016
n = 0.9604 / 0.0016
n ≈ 600.25
Therefore, a minimum sample size of approximately 601 participants would be needed to estimate the proportion of adults with high-speed internet access with a 95% confidence interval and a margin of error of 4% without a preliminary estimate available.