The Normal Distribution Make sure you answer all questions in context and print out any outputs and graphs that you perform in RStudio. These printouts will serve as justification for your work. For ALL probabilities round your answers to 4 decimal places. When computing probabilities, please include an RStudio plot for the Normal distribution. Introduction: (Part I) Beverage cans can be made from a very thin sheet of aluminum, only 1/8 inch thick. Yet they must withstand pressure up to 90 lbs/in2 (approximately 3 times the pressure of an automobile tire). Beverage companies often purchase can in large shipments. To ensure that can failures are rare, quality control inspectors sample several cans from each shipment and test their strength by placing them in testing machines that apply force until the can fails (is punctured or crushed). The testing process destroys the cans, so there's a limited number of cans tested. Assume that a can is considered defective if it fails at a pressure of 90 lbs/in2 or less. The quality control inspectors want the proportion of defective cans to be no more than 0.001, or 1 in 1000. They test 10 cans, with the following results: Can 1 2 3 4 5 6 7 8 9 10 Pressure at Failure 89 90 92 93 93 94 95 95 97 98 Two of the cans in the sample were defective; in other words two of them failed at a pressure of 90 lbs/in2 or less. The quality control inspectors want to use these data to estimate the proportion of defective cans in the shipment. If the estimate is to be no more than 0.001, they will accept the shipment; otherwise they will return for a full refund. The following project will guide you through the process used by quality control inspectors. Assume the failure pressures are normally distributed. 1. Compute the sample mean and sample standard deviation for the pressures. 2. Estimate the population mean with and the population standard deviation with .. Under this data, what percentage of the cans will fail at a pressure of 90 lbs/in2 or less? Include R-generated normal distribution graph and probability output. Will this shipment be accepted with a cutoff of 0.001? Why or why not? A second shipment of cans is received. Ten randomly selected cans are tested with the following results: Can 1 2 3 4 5 6 7 8 9 10 Pressure at Failure 90 91 92 93 94 95 95 96 98 115 3. A visual inspection of the data should provide evidence of a stronger shipment. Explain why this sample is stronger than the first sample. 4. Compute the sample mean and sample standard deviation of the pressures for the second sample. 5. Using the results from #4, estimate, now, the proportion of cans that fail at a pressure of 90 lbs/in2 or less. Include R-generated normal distribution graph and probability output. 6. Will this shipment be accepted considering the same pressure threshold? Why or why not? 7. Use R to make a boxplot for the second sample. What stands out to you from this shipment regarding this boxplot? Any surprises considering your answer to 3 above? 8. A third shipment has been received and this time you are asked to select a sample of 10 cans and run the quality assessment using the criteria of 90 lbs/in2 or less. Below is the code in R to randomly select a sample of 10 cans with pressures between 88 and 95; write down your sample in the table below: > sample(88:95, 10, replace=TRUE) Can 1 2 3 4 5 6 7 8 9 10 Pressure at Failure Estimate the proportion of cans that fail at a pressure of 90 lbs/in2 or less. Will this shipment be accepted? Include R-generated normal distribution graph and probability output. 9. A fourth shipment has been received and this time you are asked to select a sample of 10 cans and run the quality assessment using the criteria of 90 lbs/in2 or less. Use R to randomly select a sample of 10 cans with pressures between 90 and 103; write down your sample in the table below: Can 1 2 3 4 5 6 7 8 9 10 Pressure at Failure a. By inspecting the numbers you have on the table above, why do you think that this sample of 10 cans will be accepted? b. Estimate the proportion of cans that fail at a pressure of 90 lbs/in2 or less. Will this shipment be accepted? Does this estimate agree with your response in part a? Include R-generated normal distribution graph and probability output. 10. Use R to make boxplots for the third and fourth shipment. Label them accordingly. What do you see most influential in the acceptance or rejection of the shipments? (Part II) Use the third and fourth shipments above to answer the following questions. 11. Instead of inspecting ten cans from the shipments, quality control inspectors have decided to pick 3 of those 10 cans at random and find their averages to evaluate their received shipments. Your task is to randomly select 3 cans from the third and fourth shipments and determine if the shipment would be accepted. Provide evidence of your cans selected in the tables below and attach all outputs and graphs. Third Shipment: Can 1 2 3 Pressure at Failure Fourth Shipment: Can 1 2 3 Pressure at Failure How do these sub-samples compare to the original samples above? Do you change the acceptance level for either? Why do you suppose things are different here