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. p. $\sigma_{\hat{p}}=0.056569$ (Round to six decimal places as needed.) (b) What is the probability of obtaining $x=48$ or more individuals with the characteristic? That is, what is $P(\hat{p} \geq 0.64)$ ? $P(\hat{p} \geq 0.64)=0.2398$ (Round to four decimal places as needed.) (c) What is the probability of obtaining $x=36$ or fewer individuals with the characteristic? That is, what is $\mathrm{P}(\hat{\mathrm{p}} \leq 0.48)$ ? \[ P(\hat{p} \leq 0.48)=\square \text { (Round to four decimal places as needed.) } \]

Step 1 ：We are given a simple random sample of size \(n=75\) from a population of size \(N=25,000\) with a population proportion of \(p=0.6\). We are asked to find the probability of obtaining 36 or fewer individuals with the characteristic, which is equivalent to finding the probability that the sample proportion is less than or equal to 0.48.

Step 2 ：We can solve this problem using the normal approximation to the binomial distribution. The standard deviation of the sample proportion is given as \(\sigma_{\hat{p}}=0.056569\).

Step 3 ：We standardize the sample proportion to find the corresponding z-score. The z-score is a measure of how many standard deviations an element is from the mean. In this case, we want to find the z-score for the sample proportion of 0.48.

Step 4 ：Using the formula for the z-score, \(z = \frac{p_{hat} - p}{\sigma_{\hat{p}}}\), we find that \(z = -2.1213031872580386\).

Step 5 ：We then use the standard normal distribution to find the probability that a randomly selected element from the distribution is less than or equal to this z-score. This gives us the probability that the sample proportion is less than or equal to 0.48.

Step 6 ：The probability obtained is 0.016948148167193103. This is the probability of obtaining 36 or fewer individuals with the characteristic in the sample.

Step 7 ：Rounding to four decimal places as needed, the final answer is \(\boxed{0.0169}\).