The average production cost for major movies is 68 million dollars and the standard deviation is 22 million dollars. Assume the production cost distribution is normal. Suppose that 6 randomly selected major movies are researched. Answer the following questions. Give your answers in millions of dollars, not dollars. Round all answers to 4 decimal places where possible. a. What is the distribution of b. What is the distribution of $\bar{x} ? \bar{x}-N(68$ c. For a single randomly selected movie, find the probability that this movie's production cost is between 63 and 69 million dollars. d. For the group of 6 movies, find the probability that the average production cost is between 63 and 69 million dollars.

Step 1 ：Given that the average production cost for major movies is 68 million dollars and the standard deviation is 22 million dollars. We are assuming the production cost distribution is normal. We are asked to find the probability for a single movie's production cost being between 63 and 69 million dollars, and the probability of the average production cost of 6 movies being between 63 and 69 million dollars.

Step 2 ：We can use the properties of the normal distribution to solve this. The probability of a random variable falling within a certain range in a normal distribution can be found by calculating the z-scores for the range limits and looking up these z-scores in a standard normal distribution table.

Step 3 ：The z-score is calculated as follows: \(z = \frac{X - \mu}{\sigma}\) where X is the value of the random variable, \(\mu\) is the mean of the distribution, and \(\sigma\) is the standard deviation of the distribution.

Step 4 ：For the first part of the question, we need to calculate the z-scores for 63 and 69 million dollars and find the probability that a random variable from a normal distribution with mean 68 and standard deviation 22 falls within this range. The z-scores are approximately -0.2273 and 0.0455 respectively. The probability is approximately 0.1080.

Step 5 ：For the second part of the question, we need to adjust the standard deviation for the average of 6 movies. The standard deviation of the average of n random variables from a normal distribution is \(\frac{\sigma}{\sqrt{n}}\). We then calculate the z-scores for 63 and 69 million dollars and find the probability that a random variable from a normal distribution with mean 68 and adjusted standard deviation falls within this range. The z-scores are approximately -0.5567 and 0.1113 respectively. The probability is approximately 0.2555.

Step 6 ：Final Answer: The probability that a single randomly selected movie's production cost is between 63 and 69 million dollars is approximately \(\boxed{0.1080}\). The probability that the average production cost of 6 randomly selected movies is between 63 and 69 million dollars is approximately \(\boxed{0.2555}\).