Suppose a simple random sample of size $n=150$ is obtained from a population whose size is $N=15,000$ and whose population proportion with a specified characteristic is $p=0.8$.

Click here to view the standard normal distribution table (page 1).
Click here to view the standard normal distribution table (page 2).
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(b) What is the probability of obtaining $x=123$ or more individuals with the characteristic? That is, what is $P(\hat{p} \geq 0.82)$ ?
$P(\hat{p} \geq 0.82)=\square$ (Round to four decimal places as needed.)

Step 1 ：Given a simple random sample of size \(n=150\) from a population of size \(N=15,000\) with a population proportion \(p=0.8\).

Step 2 ：We are asked to find the probability of obtaining \(x=123\) or more individuals with the characteristic, i.e., \(P(\hat{p} \geq 0.82)\).

Step 3 ：This is a problem of hypothesis testing for proportions. We can solve this problem by first calculating the z-score of the sample proportion and then finding the corresponding probability from the standard normal distribution table.

Step 4 ：The z-score is calculated as follows: \(z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}\), where \(\hat{p}\) is the sample proportion, \(p\) is the population proportion, and \(n\) is the sample size.

Step 5 ：Substituting the given values, we get \(z = \frac{0.82 - 0.8}{\sqrt{\frac{0.8(1-0.8)}{150}}}\), which simplifies to \(z \approx 0.612\).

Step 6 ：This means that the sample proportion of 0.82 is 0.612 standard deviations above the mean.

Step 7 ：Now, we need to find the probability corresponding to this z-score from the standard normal distribution table. However, since we want the probability of obtaining a sample proportion greater than or equal to 0.82, we need to find the area to the right of the z-score, which is \(1 - P(Z < z)\).

Step 8 ：Calculating this probability, we get \(P(\hat{p} \geq 0.82) = 1 - P(Z < 0.612) \approx 0.2701\).

Step 9 ：So, the probability of obtaining a sample proportion greater than or equal to 0.82 is approximately 0.2701.

Step 10 ：Final Answer: \(P(\hat{p} \geq 0.82)=\boxed{0.2701}\) (rounded to four decimal places as needed).