Suppose a simple random sample of size $n=75$ is obtained from a population whose size is $\mathrm{N}=25,000$ and whose population proportion with a specified characteristic is $p=0.6$. Complete parts (a) through (c) below. $\mu_{\hat{p}}=0.6$ (Round to one decimal place as needed.) Determine the standard deviation of the sampling distribution of $\hat{\mathrm{p}}$. $\sigma_{\hat{p}}=0.056569$ " (Round to six decimal places as needed.) (b) What is the probability of obtaining $\mathrm{x}=48$ or more individuals with the characteristic? That is, what is $\mathrm{P}(\hat{p} \geq 0.64)$ ? \[ P(\hat{p} \geq 0.64)=\square \text { (Round to four decimal places as needed.) } \]

Step 1 ：Given a simple random sample of size \(n=75\) from a population of size \(N=25,000\) with a population proportion \(p=0.6\).

Step 2 ：The mean of the sampling distribution of the proportion, \(\mu_{\hat{p}}\), is equal to the population proportion, \(p\), which is 0.6.

Step 3 ：The standard deviation of the sampling distribution of the proportion, \(\sigma_{\hat{p}}\), is calculated to be 0.056569.

Step 4 ：We are asked to find the probability of obtaining 48 or more individuals with the characteristic, which is equivalent to the probability of the sample proportion being greater than or equal to 0.64.

Step 5 ：We can standardize the sample proportion using the Z-score formula: \[Z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}}\] where \(\hat{p}\) is the sample proportion, \(\mu_{\hat{p}}\) is the population proportion, and \(\sigma_{\hat{p}}\) is the standard deviation of the sampling distribution of the proportion.

Step 6 ：Substituting the given values into the formula, we get \(Z = \frac{0.64 - 0.6}{0.056569} = 0.7071010624193469\).

Step 7 ：We then find the probability of Z being greater than or equal to the calculated Z-score. This is equivalent to finding the area to the right of the calculated Z-score in the standard normal distribution, which is 1 minus the cumulative distribution function (CDF) of the calculated Z-score.

Step 8 ：Calculating this, we get \(P = 1 - \text{CDF}(Z) = 0.23975183789836807\).

Step 9 ：Rounding to four decimal places, the final answer is \(\boxed{0.2398}\).