Step 1 :The researcher wants to estimate the population proportion with a 95% confidence level. This means that the z-score corresponding to this confidence level is 1.96. The margin of error is 3% or 0.03. Since no preliminary estimate is available, we use the most conservative estimate for the population proportion, which is 0.5. This gives the maximum possible sample size.
Step 2 :We can use the formula for sample size in estimating a population proportion: \(n = \frac{{Z^2 * p * (1-p)}}{{E^2}}\) where Z is the z-score, p is the estimated proportion of the population, and E is the margin of error.
Step 3 :Substituting the given values into the formula, we get \(n = \frac{{(1.96)^2 * 0.5 * (1-0.5)}}{{(0.03)^2}}\)
Step 4 :Solving the equation gives us \(n = 1068\)
Step 5 :So, the minimum sample size needed, assuming that no prior information is available, is 1068.
Step 6 :Final Answer: The minimum sample size needed is \(\boxed{1068}\)