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. (a) Describe the sampling distribution of $\hat{p}$. Choose the phrase that best describes the shape of the sampling distribution below. A. Approximately normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p) \geq 10$. B. Approximately normal because $\mathrm{n} \leq 0.05 \mathrm{~N}$ and $n p(1-p)<10$. C. Not normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p)<10$. D. Not normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p) \geq 10$. Determine the mean of the sampling distribution of $\hat{p}$. \[ \mu_{\hat{p}}=\square \text { (Round to one decimal place as needed.) } \]

Step 1 ：Given that the sample size n = 75, the population size N = 25000, and the population proportion p = 0.6.

Step 2 ：First, we need to check if the sample size is less than or equal to 5% of the population size. This is a condition for the sampling distribution of the sample proportion to be approximately normal.

Step 3 ：Calculate 5% of the population size: \(0.05 \times 25000 = 1250\). Since 75 is less than 1250, the condition is satisfied.

Step 4 ：Next, we need to check if the expected number of successes and failures are both at least 10. This is another condition for the sampling distribution to be approximately normal.

Step 5 ：Calculate the expected number of successes: \(n \times p = 75 \times 0.6 = 45\) and the expected number of failures: \(n \times (1-p) = 75 \times 0.4 = 30\). Since both are greater than 10, the condition is satisfied.

Step 6 ：Therefore, the sampling distribution of the sample proportion is approximately normal.

Step 7 ：Finally, we need to find the mean of the sampling distribution of the sample proportion. The mean is equal to the population proportion p.

Step 8 ：So, the mean of the sampling distribution of the sample proportion is \(\boxed{0.6}\).

Step 9 ：In conclusion, the best description for the shape of the sampling distribution is 'Approximately normal because n is less than or equal to 5% of N and np(1-p) is greater than 10'. The mean of the sampling distribution of the sample proportion is 0.6.