Step 1 :Given that the sample size is \(n = 75\), the population size is \(N = 15000\), and the population proportion with a specified characteristic is \(p = 0.4\).
Step 2 :The sampling distribution of \(\hat{p}\) is approximately normal if \(n \leq 0.05 N\) and \(n p(1-p) \geq 10\).
Step 3 :The mean of the sampling distribution of \(\hat{p}\) is equal to the population proportion \(p\), which is \(0.4\).
Step 4 :The standard deviation of the sampling distribution of \(\hat{p}\) can be calculated using the formula \(\sqrt{\frac{p(1-p)}{n}}\).
Step 5 :Substituting the given values into the formula, we get the standard deviation as approximately \(0.056568\).
Step 6 :Final Answer: The shape of the sampling distribution of \(\hat{p}\) is approximately normal. The mean of the sampling distribution of \(\hat{p}\) is \(\boxed{0.4}\). The standard deviation of the sampling distribution of \(\hat{p}\) is \(\boxed{0.056568}\).