A simple random sample of size $n=41$ is obtained from a population with $\mu=66$ and $\sigma=16$. (a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample mean? Assuming that this condition is true, describe the sampling distribution of $\bar{x}$. (b) Assuming the normal model can be used, determine $P(\bar{x}<70.2)$ (c) Assuming the normal model can be used, determine $P(\bar{x} \geq 68.4)$.

Step 1 ：The sample size is 41, which is greater than 30, so we can use the normal model according to the Central Limit Theorem.

Step 2 ：The mean of the sampling distribution is \(\mu = 66\) and the standard deviation is \(\sigma/\sqrt{n} = 16/\sqrt{41} \approx 2.5\).

Step 3 ：To find \(P(\bar{x}<70.2)\), we first standardize the value 70.2 to get a z-score: \(Z = (70.2 - 66) / 2.5 = 1.68\).

Step 4 ：Looking up this z-score in a standard normal distribution table or using a calculator, we find that \(P(Z < 1.68) \approx 0.9535\). So, \(P(\bar{x}<70.2) \approx 0.9535\).

Step 5 ：\(\boxed{P(\bar{x}<70.2) \approx 0.9535}\)

Step 6 ：To find \(P(\bar{x} \geq 68.4)\), we again standardize to get a z-score: \(Z = (68.4 - 66) / 2.5 = 0.96\).

Step 7 ：Looking up this z-score, we find that \(P(Z < 0.96) \approx 0.8315\). However, we want the probability that Z is greater than or equal to 0.96, so we subtract the value we found from 1: \(P(\bar{x} \geq 68.4) = 1 - P(Z < 0.96) = 1 - 0.8315 = 0.1685\).

Step 8 ：\(\boxed{P(\bar{x} \geq 68.4) = 0.1685}\)