Question 1 Suppose that a particular production process fills detergent in boxes of a given sizes. Specifically, this process fills the boxes with an amount of detergent (in ounces) that is normally distributed with mean μ=50 ounces and standard deviation σ= 0.5 ounces. a. What is the fraction of boxes will contain less than 50 ounces of detergent? b. A box of detergent is rejected by quality control personnel if it is found to contain less than 49 ounces or more that 51 ounces of detergent. Given this quality standards, what proportion of all boxes are rejected? c. One percent of all boxes will weigh more than what value? d. Ten percent of all boxes will weigh less than what value? Question 2 ) continued form Question 1 Simulation this production process for the filling of 50 boxes of detergent. a. What will the expected value of the average weight of the 50 simulation boxes be? (i.e. the mean of the sampling distribution) b. What will the standard deviation of the average weight of the 50 simulation boxes be? (i.e. the standard deviation of the sampling distribution or the standard error of the sample mean) c. How many boxes should be simulated if we need to have a standard deviation of 0.05 ounces? Question 3 A company analyzed demand data for the past two years and found that daily demands for a company’s production are independent and identically distributed with mean 20 and standard deviation 5. Based on this data analysis, answer the following questions. a. What is the probability distribution of the total demand for next 30 days? Specify the mean and the standard deviation of the total demand. b. Suppose that the company currently has 650 units in stock, and it will not be receiving anymore shipments from its supplier for at least 30 days. What is the probability that stock will run out during this thirty-days period? Question 4 Kate is a security analyst and has identified a gold ETF (ticker symbol GEF) as a particularly attractive investment. Currently, GEF is trading at $100 per share. Kate notes that gold has underperformed in the last decade and believes that the time is ripe for a large increase in gold prices. Kate has constructed seven scenarios for the price of GEF one month from now. These scenarios and corresponding probabilities are shown below: Question 4 Data Scen. 1 Scen. 2 Scen. 3 Scen. 4 Scen. 5 Scen. 6 Scen. 7 Probability 0.05 0.10 0.20 0.30 0.20 0.10 0.05 GEF price 150 130 110 100 90 80 70 a. What is the probability that the price of GEF is higher than $100 one month from now? b. What is the expected return of GEF one month from now? Note: Stock Return = (Ending stock price - Initial stock price) / Initial stock price c. What is the standard deviation of the return of GEF one month from now?