Final answer:
To estimate the population proportion without prior knowledge of sample proportion, we let p hat = 0.50, which ensures a conservative sample size. The Central Limit Theorem for proportions is used to calculate the necessary sample size for achieving a desired confidence level and margin of error.
Step-by-step explanation:
In selecting the sample size to estimate the population proportion p, if we have no knowledge of even the approximate values of the sample proportion p hat, we should let p hat = 0.50. This is the most conservative assumption we can make, as it maximises the sample size required ensuring the Central Limit Theorem requirements are met for the normal approximation to hold. This is due to the fact that the product of p and q (which is 1 - p) will be maximised when p is 0.5, since any other value for p would result in a smaller product, and consequently, a smaller required sample size.
Regarding the sample size for estimating a population proportion with a specific level of confidence and margin of error, the formula used is based on the Central Limit Theorem for proportions. For instance, if a researcher wants 95% confidence with a margin of error of 0.03 (3 percentage points) and assumes the population proportion, p, is 0.5, then the required sample size can be calculated using the formula for the sample size of proportions.