Step 1 :The mean of the sampling distribution of the proportion (p-hat) is equal to the population proportion (p). So, \(\mu_{\hat{p}} = p = 0.2\).
Step 2 :The standard deviation of the sampling distribution of the proportion is given by the formula \(\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}\). Substituting the given values, we get \(\sigma_{\hat{p}} = \sqrt{\frac{0.2(1-0.2)}{200}} = 0.028284\).
Step 3 :To find the probability of obtaining 42 or more individuals with the characteristic, we first need to convert 42 to a proportion by dividing by the sample size. So, 42/200 = 0.21. We then need to find the z-score for this proportion, which is given by the formula \(z = \frac{\hat{p} - p}{\sigma_{\hat{p}}}\). Substituting the given values, we get \(z = \frac{0.21 - 0.2}{0.028284} = 0.3536\). Looking up this z-score in the standard normal distribution table, we find that the area to the left of this z-score is 0.6382. Since we want the probability of obtaining 0.21 or more, we need to find the area to the right of the z-score, which is 1 - 0.6382 = 0.3618.
Step 4 :Similarly, to find the probability of obtaining 34 or fewer individuals with the characteristic, we first convert 34 to a proportion, which is 34/200 = 0.17. The z-score for this proportion is \(z = \frac{0.17 - 0.2}{0.028284} = -1.0615\). Looking up this z-score in the standard normal distribution table, we find that the area to the left of this z-score is 0.1444, which is the probability we are looking for.
Step 5 :So, the mean of the sampling distribution of the proportion is \(\boxed{0.2}\), the standard deviation is \(\boxed{0.028284}\), the probability of obtaining 0.21 or more is \(\boxed{0.3618}\), and the probability of obtaining 0.17 or less is \(\boxed{0.1444}\).