The time needed for performing a machining operation is to be investigated. Historically, the process has had a standard deviation equal to 0.146 minutes. The means (in minutes) of 39 samples of n = 6 are: sample mean sample mean sample mean 1 3.86 14 3.81 27 3.81 2 3.90 15 3.83 28 3.86 3 3.83 16 3.86 29 3.98 4 3.81 17 3.82 30 3.96 5 3.84 18 3.86 31 3.88 6 3.83 19 3.84 32 3.76 7 3.87 20 3.87 33 3.83 8 3.88 21 3.84 34 3.77 9 3.84 22 3.82 35 3.86 10 3.80 23 3.89 36 3.80 11 3.88 24 3.86 37 3.84 12 3.86 25 3.88 38 3.79 13 3.88 26 3.90 39 3.85   the averagee of the 39-sample means is equal to 3.85. construct an xbar-chart for this process with two-sigma limits. Is the process in control?

At the beginning of each week, a machine is in one of four conditions: 1 = excellent; 2 = good; 3 = average; 4 = bad. The weekly revenue earned by a machine in state 1, 2, 3, or 4 is 100, 90, 50, or 10, respectively. After observing the condition of the machine at the beginning of the week, the company has the option, for a cost of 200, of instantaneously replacing the machine with an excellent machine. The quality of the machine deteriorates over time, as shown in the file P10 41.xlsx. Four maintenance policies are under consideration: Policy 1: Never replace a machine. Policy 2: Immediately replace a bad machine. Policy 3: Immediately replace a bad or average machine. Policy 4: Immediately replace a bad, average, or good machine. Simulate each of these policies for 50 weeks (using at least 250 iterations each) to determine the policy that maximizes expected weekly profit. Assume that the machine at the beginning of week 1 is excellent.

Use Excels functions (not @RISK) to generate 1000 random numbers from a normal distribution with mean 100 and standard deviation 10. Then freeze these random numbers. a. Calculate the mean and standard deviation of these random numbers. Are they approximately what you would expect? b. What fraction of these random numbers are within k standard deviations of the mean? Answer for k = 1; for k = 2; for k = 3. Are the answers close to what they should be (about 68% for k = 1, about 95% for k = 2, and over 99% for k = 3)? c. Create a histogram of the random numbers using about 10 bins of your choice. Does this histogram have approximately the shape you would expect?