1. Suppose the estimate of a proportion of a normal population is to be within 0.05 using the 95% confidence level. How large a sample is required if:
a. It is known from previous studies that p = 0.3?
b. There is no prior knowledge of p?
To determine the sample size required for estimating a proportion within a specified margin of error and confidence level, you 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
p = estimated proportion (if available)
E = margin of error
a. When it is known from previous studies that p = 0.3:
In this case, you can substitute the known value of p into the formula. Let's assume a 95% confidence level, which corresponds to a Z-score of approximately 1.96 (from the standard normal distribution).
n = (1.96^2 * 0.3 * (1-0.3)) / 0.05^2
n = (3.8416 * 0.21) / 0.0025
n = 81.5232 / 0.0025
n ≈ 32,609
So, you would need a sample size of approximately 32,609 when the estimated proportion is known to be 0.3.
b. When there is no prior knowledge of p:
When you don't have an estimate for the proportion, you can assume a conservative estimate of p = 0.5. This value of p will result in the largest sample size, ensuring that you are more likely to capture the true proportion within the desired margin of error.
Using the same formula as above with p = 0.5 and a 95% confidence level (corresponding to a Z-score of 1.96):
n = (1.96^2 * 0.5 * (1-0.5)) / 0.05^2
n = (3.8416 * 0.25) / 0.0025
n = 0.9604 / 0.0025
n ≈ 384
In this case, you would need a sample size of approximately 384 when there is no prior knowledge of the proportion.