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The average production cost for major movies is 6767 million dollars and the standard deviation is 2323 million dollars. Assume the production cost distribution is normal. Suppose that 4646 randomly selected major movies are researched. Answer the following questions. Give your answers in millions of dollars, not dollars. Round all answers to 44 decimal places where possible.\newlinea. What is the distribution of XX? \newlineX∼N(67,232)X \sim N(67, 23^2)\newlineb. What is the distribution of xˉ\bar{x}? \newlinexˉ∼N(67,(2346)2)\bar{x} \sim N\left(67, \left(\frac{23}{\sqrt{46}}\right)^2\right)\newlinec. For a single randomly selected movie, find the probability that this movie's production cost is between 6969 and 7373 million dollars. \newline232300\newlined. For the group of 4646 movies, find the probability that the average production cost is between 6969 and 7373 million dollars. 232344\newlinee. For part d), is the assumption of normal necessary?\newlineNo\newlineYes\newline232355

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Q. The average production cost for major movies is 6767 million dollars and the standard deviation is 2323 million dollars. Assume the production cost distribution is normal. Suppose that 4646 randomly selected major movies are researched. Answer the following questions. Give your answers in millions of dollars, not dollars. Round all answers to 44 decimal places where possible.\newlinea. What is the distribution of XX? \newlineX∼N(67,232)X \sim N(67, 23^2)\newlineb. What is the distribution of xˉ\bar{x}? \newlinexˉ∼N(67,(2346)2)\bar{x} \sim N\left(67, \left(\frac{23}{\sqrt{46}}\right)^2\right)\newlinec. For a single randomly selected movie, find the probability that this movie's production cost is between 6969 and 7373 million dollars. \newline232300\newlined. For the group of 4646 movies, find the probability that the average production cost is between 6969 and 7373 million dollars. 232344\newlinee. For part d), is the assumption of normal necessary?\newlineNo\newlineYes\newline232355
  1. Understand Given Information: To solve this problem, we first need to understand the given information and what is being asked.\newlineWe are given:\newline- The average (mean) production cost for major movies, μ=67\mu = 67 million dollars.\newline- The standard deviation, σ=23\sigma = 23 million dollars.\newline- The number of movies being sampled, n=46n = 46.\newline- We are asked to find the distribution of XX, the distribution of the sample mean xˉ\bar{x}, the probability of a single movie's production cost being between 6969 and 7373 million dollars, and the probability that the average production cost for 4646 movies is between the same range.
  2. Distribution of X: a. The distribution of X, which represents the production cost for a single movie, is given as a normal distribution with the mean (μ\mu) and standard deviation (σ\sigma). Therefore, the distribution of X is:\newlineX∼N(67,23)X \sim N(67, 23)
  3. Distribution of Sample Mean: b. The distribution of the sample mean xˉ\bar{x} is also normally distributed with the same mean as the population mean, but the standard deviation is reduced by the square root of the sample size (nn). The standard deviation of the sample mean is σ/n\sigma/\sqrt{n}. Let's calculate it:\newlineσ/n=23/46≈23/6.7823≈3.3906\sigma/\sqrt{n} = 23/\sqrt{46} \approx 23/6.7823 \approx 3.3906 million dollars\newlineTherefore, the distribution of xˉ\bar{x} is:\newlinexˉ∼N(67,3.3906)\bar{x} \sim N(67, 3.3906)\newlineHowever, the problem statement seems to suggest a standard deviation of 0.50.5 for xˉ\bar{x}, which is likely a mistake. We will use the calculated standard deviation for further calculations.

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