3. A failure density function f(t) is given by: f(t) = (ß/n³) (tß-1) exp [-(t/n) ³] A cumulative failure density function, F(t) is given by: F(t) = 1-exp [-(t/n) ³] What is the reliability function, R(t)? What is the hazard rate (failure rate) function, h(t)? What can be said about the failure rate for ẞ equal to 1, less than 1 and greater than 1?

5. For a constant failure rate, mission reliability is given by: -λt R(t) A system has a failure rate of 5 failures per million hours. = e What is the mission reliability for 5,000 hours? The system survives that 5,000 hours. If the system survives that first 5,000 hours, assuming the failure rate is still constant, what is the probability of surviving the next 5,000 hours? (Hint: Don't forget about the 10-6 factor in the failure rate.)

1. A piece of equipment has a failure rate of 10 failures per million hours, and is run for 1000 hours. What is the probability that the one piece of equipment will survive that 1000 hours? Three (3) of these pieces of equipment are each run for 1000 hours. For the 3 pieces of equipment, use the Binomial equation to calculate: The probability of zero (0) failures. The probability of one (1) failure. The probability more than one (>1) failure.

Problem 1. A company kept a chart on resistance. Xdouble bar was: and UCLX was: 25 MegOhms, with a sample size of: The company adopted a new specificationto keep the resistance at a minimum of 18 MegOhms with no USL . What is: Cpk RR ppm 22 MegOhms 25.

Question 2 Background - "DTF Restaurant" Hiring Process DTF Restaurant (DTF for short) is a famous chain restaurant that offers high-quality dining services in Singapore. DTF has a few branch stores located in different malls in Singapore. Since the government eased most of the Covid-19 pandemic safe management measures in May 2022, DTF was facing growing customer demand. However, its branch stores were facing manpower shortages in daily operations. DTF is also planning to open more branches in Singapore or even overseas. For restaurants, the employee turnover rate is often high. Thus, DTF has to regularly hire staff in different positions to meet the growing demand and further expansion requirement. Recently, many branch store managers complained that the Human Resource (HR) department was not recruiting the staff fast enough to fill the vacant positions. On 1 August 2023, DTF's HR Director - Jennifer Tan investigated the matter and appointed you, who is trained in Lean Six Sigma methodology, to lead a Lean Six Sigma project to improve the hiring process by reducing the average lead time taken to hire staff by at least 30% in three months. (a) (b) Discuss the VOC and define the CTQ (Project Y) for your Lean Six Sigma project. Create a project charter before embarking your project. Develop the SIPOC diagram for a typical hiring process which may not be necessarily the same as the hiring process in the DTF restaurant. Examine three (3) factors that could affect the lead time of the hiring process.

Problem 4: A new contract was written with Company B with the following stipulations: Cpk = 1.1, Specification Nominal (N) = 10 ± 0.3 mils. Company B wants to design their process by setting the process average = 10 mils and the sample size (n) = 50. Find: Xdouble bar: UCLX: LCLX: LCLr: Rbar: Reject Rate: UCLr: ppm

Pls read and summarize Chapter 2-3 from your book. Quality Management for Org. Excellence. • You will read the chapters and make a summary of 2 pages for each chapter.if you include a schema or table it may exceed 2 pages, but do not include all • You will put your comments and Ideas about the topic as my opinions title • It is what you understand • Scores are given for how much you understand and how neat the summary is prepared (the summary needs to be done after reading the whole chapter, not as part by part) Dont forget hat I told you abot the summaries to receive a high score. (Do you agree or not agree which part and why.) Don't copy paste!

3. A reliability test is run for 5000 hours. There are four (4) failures, all happening sometime in the first 4500 hours. Then, the test was run for an additional 500 hours with no more failures. In other words, this was a time-terminated 5000 hour test with 4failures. For a time-terminated test with four (4) failures, how many degrees of freedom (v) does this represent? Using Chi-Squared, calculate the lower confidence level MTBF for a 60% confidence level (40% risk factor) for this test. The same test was continued to 7631 hours (2631 additional hours) with no additional failures. Using Chi-Squared, calculate the lower confidence level MTBF for a 90% confidence level (10% risk factor) for this test. Compare your two MTBF answers, and comment on how this illustrates a way of increasing the confidence level of a test from 60% to 90%

4. Some failure test data was collected, and the results were as follows: 0-1000 hours: 12 failures 1001-2000 hours: 8 failures 2001-3000 hours: 7 failures 3001-4000 hours: 13 failures 4001-5000 hours: 11 failures We wish to test this data to see if it follows a constant failure rate of 10 failures per 1000 hours. Use the Chi-squared goodness of fit test to determine if this data can represent a constant failure rate of 10 failures every 1000 hours. Hint: there are 5 'bins", so use DOF (v) of 5-1 = 4. The Chi-squared table is on pages 453 and 454 of your textbook.

1. A reliability test of 7 samples was run until all samples had failed. As each sample failed, it was taken out of the test (not replaced). The times to failure were (in hours): 150, 400, 500, 1000, 1500, 2000, 2500. Perform a data plot, using median ranking, on 2 cycle Weibull paper. What is the slope (B)? What is the scale factor (n)? What does the slope tell you about the failure rate? What does n represent (besides scale factor) with the data having the slope you just determined?