Problem
The average production cost for major movies is 64 million dollars and the standard deviation is 23 million dollars. Assume the production cost distribution is normal. Suppose that 15 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 $x$ ? $x-\mathrm{N}(64$ c. For a single randomly selected movie, find the probability that this movie's production cost is between 58 and 61 million dollars. d. For the group of 15 movies, find the probability that the average production cost is between 58 and 61 million dollars. e. For part d), is the assumption of normal necessary? No Yes
Solution
Step 1 ：Given that the average production cost for major movies is 64 million dollars and the standard deviation is 23 million dollars, and assuming the production cost distribution is normal, the distribution of \(x\) (the production cost of a single movie) would be \(N(64, 23^2)\), where \(N(\mu, \sigma^2)\) denotes a normal distribution with mean \(\mu\) and variance \(\sigma^2\).
Step 2 ：The distribution of \(x\) (the production cost of a single movie) is \(N(64, 23^2)\).
Step 3 ：\(\boxed{N(64, 23^2)}\)
