Step 1 :We are given a population with proportion \(P = 0.30\) and we are considering random samples of size \(n = 50\).
Step 2 :We are asked to find the standard error of the distribution of sample proportions.
Step 3 :The formula for the standard error of the distribution of sample proportions is given by \(SE = \sqrt{\frac{P \cdot (1 - P)}{n}}\), where \(P\) is the population proportion and \(n\) is the sample size.
Step 4 :Substituting the given values into the formula, we get \(SE = \sqrt{\frac{0.3 \cdot (1 - 0.3)}{50}}\).
Step 5 :Calculating the above expression, we find that \(SE = 0.0648074069840786\).
Step 6 :Rounding this to three decimal places, we get \(SE = 0.065\).
Step 7 :Final Answer: The standard error of the distribution of sample proportions is \(\boxed{0.065}\).