Applied Statistics and Probability for Engineers 6th Edition solutions

Applied Statistics and Probability for Engineers 6
Consider a process whose specifications on a quality characteristic are \(100 \pm 15\). You know that the standard deviation of this normally distributed quality characteristic is 5. Where should you center the process to minimize the fraction defective produced? Now suppose that the mean shifts to 1

Applied Statistics and Probability for Engineers 6
The diameter of fuse pins used in an aircraft engine application is an important quality characteristic. Twenty-five samples of three pins each are shown as follows: (a) Set up \(\bar{X} \text { and } R\) charts for this process. If necessary, revise limits so that no observations are out of cont

Applied Statistics and Probability for Engineers 6
Rework Exercise 15-91 with \(\bar{X} \text { and } S\) charts. Equation Transcription: Text Transcription: \bar X and S

Applied Statistics and Probability for Engineers 6
Plastic bottles for liquid laundry detergent are formed by blow molding. Twenty samples of n = 100 bottles are inspected in time order of production, and the fraction defective in each sample is reported. The data are as follows: (a) Set up a P chart for this process. Is the process in st

Applied Statistics and Probability for Engineers 6
Cover cases for a personal computer are manufactured by injection molding. Samples of five cases are taken from the process periodically, and the number of defects is noted. Twenty-five samples follow: (a) Using all the data, find trial control limits for a \(U\) chart for the process. (b) Use th

Applied Statistics and Probability for Engineers 6
An article in Quality Engineering [Is the Process Capable? Tables and Graphs in Assessing \(\mathrm{C}_{\mathrm{pm}}\) (1992, Vol. 4(4)]. Considered manufacturing data. Specifications for the outer diameter of the hubs were \(60.3265 \pm 0.001 \mathrm{\ \ mm}\). A random sample with size n = 20 was

Applied Statistics and Probability for Engineers 6
The following data from the U.S. Department of Energy Web site (www.eia.doe.gov) reported the total U.S. renewable energy consumption by year (trillion BTU) from 1973 to 2004. (a) Using all the data, find calculate control limits for a control chart for individual measurements, construct the char

Applied Statistics and Probability for Engineers 6
The following dataset was considered in Quality Engineering [Analytic Examination of Variance Components (19941995, Vol. 7(2)]. A quality characteristic for cement mortar briquettes was monitored. Samples of size \(n=6\) were taken from the process, and 25 samples from the process are shown in the fo

Applied Statistics and Probability for Engineers 6
Suppose that an \(\bar{X}\) control chart with 2-sigma limits is used to control a process. Find the probability that a false out-of-control signal is produced on the next sample. Compare this with the corresponding probability for the chart with 3-sigma limits and discuss. Comment on when you would

Applied Statistics and Probability for Engineers 6
Consider the diameter data in Exercise 15-91. (a) Construct an EWMA control chart with \(\lambda=0.2\) and \(L=3\). Comment on process control. (b) Construct an EWMA control chart with \(\lambda=0.5\) and \(L=3\) and compare your conclusion to part (a).

Applied Statistics and Probability for Engineers 6
Consider the renewable energy data in Exercise 15-96. (a) Construct an EWMA control chart with \(\lambda=0.2\) and L = 3 Do the data appear to be generated from an incontrol process? (b) Construct an EWMA control chart with \(\lambda=0.5\) and L = 3 and compare your conclusion to part (a).

Applied Statistics and Probability for Engineers 6
Consider the hub data in Exercise 15-95. (a) Construct an EWMA control chart with \(\lambda=0.2\) and L = 3 Comment on process control. (b) Construct an EWMA control chart with \(\lambda=0.5\) and L = 3 and compare your conclusion to part (a).

Applied Statistics and Probability for Engineers 6
Consider the data in Exercise 15-20. Set up a CUSUM scheme for this process assuming that \(\mu=16\) is the process target. Explain how you determined your estimate of \(\sigma\) and the CUSUM parameters K and H.

Applied Statistics and Probability for Engineers 6
Consider the hardness measurement data in Exercise 15-19. Set up a CUSUM scheme for this process using \(\mu=50\) and \(\sigma=2\) so that K=1 and H=10. Is the process in control?

Applied Statistics and Probability for Engineers 6
Reconsider the viscosity data in Exercise 15-22. Construct a CUSUM control chart for this process using \(\mu_0=500\) as the process target. Explain how you determined your estimate of \(\sigma\) and the CUSUM parameters H and K.

Applied Statistics and Probability for Engineers 6
The following data were considered in Quality Progress [Digidot Plots for Process Surveillance (1990, May, pp. 6668)]. Measurements of center thickness (in mils) from 25 contact lenses sampled from the production process at regular intervals are shown in the following table. (a) Construct a CUSUM

Applied Statistics and Probability for Engineers 6
Suppose that a process is in control and an \(\bar{X}\) chart is used with a sample size of 4 to monitor the process. Suddenly there is a mean shift of \(1.5 \sigma\). (a) If 3-sigma control limits are used on the \(\bar{X}\) chart, what is the probability that this shift remains undetected for thr

Applied Statistics and Probability for Engineers 6
Problem 107SE Consider the control chart for individuals with 3-sigma limits. (a) Suppose that a shift in the process mean of magnitude ? occurs. Verify that the ARL for detecting the shift is ARL = 43.9. (b) Find the ARL for detecting a shift of magnitude 2? in the process mean. (c) Find the ARL for

Applied Statistics and Probability for Engineers 6
Consider a control chart for individuals applied to a continuous 24-hour chemical process with observations taken every hour. (a) If the chart has 3-sigma limits, verify that the in-control ARL is 370. How many false alarms would occur each 30-day month, on the average, with this chart? (b) Suppo

Applied Statistics and Probability for Engineers 6
The depth of a keyway is an important part quality characteristic. Samples of size \(\mathrm{n}=5\) are taken every four hours from the process, and 20 samples are summarized in the following table. (a) Using all the data, ind trial control limits for \(\bar{X} \text { and } \mathrm{R}\) charts. I

Applied Statistics and Probability for Engineers 6
Consider a control chart for individuals with 3-sigma limits. What is the probability that there is not a signal in 3 samples? In 6 samples? In 10 samples?

Applied Statistics and Probability for Engineers 6
Problem 111SE Suppose that a process has a PCR = 2, but the mean is exactly 3 standard deviations above the upper specification limit. What is the probability of making a product outside the specification limits?

Applied Statistics and Probability for Engineers 6
Problem 112SE process is controlled by a P chart using samples of size 100. The center line on the chart is 0.05. (a) What is the probability that the control chart detects a shift to 0.08 on the first sample following the shift? (b) What is the probability that the control chart does not detect a sh

Applied Statistics and Probability for Engineers 6
Problem 113SE Suppose that the average number of defects in a unit is known to be 8. If the mean number of defects in a unit shifts to 16, what is the probability that it is detected by a U chart on the first sample following the shift (a) if the sample size is n = 4? (b) if the sample size is n = 10

Applied Statistics and Probability for Engineers 6
Problem 114SE Suppose that the average number of defects in a unit is known to be 10. If the mean number of defects in a unit shifts to 14, what is the probability that it is detected by a U chart on the first sample following the shift (a) if the sample size is n = 1? (b) if the sample size is n = 4

Applied Statistics and Probability for Engineers 6
Problem 115SE An EWMA chart with ?= . 0 5 and L= . 3 07 is to be used to monitor a process. Suppose that the process mean is ?0 = 10 and ? = 2. (a) Assume that n = 1. What is the ARL without any shift in the process mean? What is the ARL to detect a shift to ? = 12. (b) Assume that n= . 4 Repeat part

Applied Statistics and Probability for Engineers 6
The following table provides the costs for gasoline by month in the United States over recent years and the percentage of the cost due to reining, distribution and marketing, taxes, and crude oil. The table is from the U.S. Department of Energy Web site (http://tonto.eia.doe.gov/oog/info/gdu/gaspump

Applied Statistics and Probability for Engineers 6
The following table shows the number of e-mails a student received each hour from 8:00 a.m. to 6:00 p.m. The samples are collected for five days from Monday to Friday. (a) Use the rational subgrouping principle to comment on why an \(\bar{X}\) chart that plots one

Applied Statistics and Probability for Engineers 6
The following are the number of defects observed on 15 samples of transmission units in an automotive manufacturing company. Each lot contains five transmission units. (a) Using all the data, compute trial control limits for a \(U\) control chart, construct the chart, and plot the data. (b) Determine

Applied Statistics and Probability for Engineers 6
Consider an \(\bar{X}\) control chart with \(U C L=32.802, U C L=24.642\), and \(\mathrm{n}=5\). Suppose that the mean shifts to 30. (a) What is the probability that this shift is detected on the next sample? (b) What is the ARL to detect the shift? Equation Transcription:

Applied Statistics and Probability for Engineers 6
The number of visits (in millions) on a Web site is recorded every day. The following table shows the samples for 25 consecutive days. (a) Estimate the process standard estimation. (b) Set up a CUSUM control chart for this process, assuming the target is 10. Does the process appear to be in control?

Applied Statistics and Probability for Engineers 6
An article in Microelectronics Reliability [Advanced Electronic Prognostics through System Telemetry and Pattern Recognition Methods, (2007, 47(12), pp. 18651873)] presented an example of electronic prognostics (a technique to detect faults in order to decrease the system downtime and the number of u

Applied Statistics and Probability for Engineers 6
A article in the Journal of Quality in Clinical Practice [The Application of Statistical Process Control Charts to the Detection and Monitoring of Hospital-Acquired Infections, (2001, Vol. 21, pp. 112117)] reported the use of SPC methods to monitor hospital-acquired infections. The authors appl

Applied Statistics and Probability for Engineers 6
The control limits for an \(\bar{X}\) chart with n=4 are 12.8 and 24.8, and the PCR for a measurement is 1.33. (a) Estimate the process standard deviation \(\sigma\). (b) Calculate the specification limits. Assume that they are centered around the process mean.

Applied Statistics and Probability for Engineers 6
Consider the turnaround time \(\text { (TAT) }\) for complete blood counts in Exercise 15-18. Suppose that the specifications for \(\text { TAT }\) are set at 20 and 80 minutes. Use the control chart summary statistics for the following. (a) Estimate the process standard deviation. (b) Calculate

Applied Statistics and Probability for Engineers 6
Consider the inventory accuracy in Exercise 15-27. Because lower values are better, only the \(U C L=0.3\) is specified. Use the revised control chart to calculate \(\mathrm{PCR}_{\mathrm{k}}\). Equation Transcription: Text Transcription: UCL=0.3 PCR_k

Applied Statistics and Probability for Engineers 6
Consider the TAT data in Exercise 15-18. (a) Construct a CUSUM control chart with the target equal to the estimated process mean, k=0.5 and h=4. Does the process appear to be in control at the target? (b) If the mean increases by 5 minutes, approximate the chart's ARL.

Applied Statistics and Probability for Engineers 6
An article in Electric Power Systems Research [On the Self-Scheduling of a Power Producer in Uncertain Trading Environments (2008, 78(3), pp. 311317)] considered a self scheduling approach for a power producer. The following table shows the forecasted prices of energy for a 24-hour time period accord

Applied Statistics and Probability for Engineers 6
Suppose that a process is in control, and 3-sigma control limits are in use on an \(\bar{X}\) chart. The subgroup size is 4. Let the mean shift by \(1.5 \sigma\). What is the probability that this shift remains undetected for three consecutive samples? What would its probability be if 2-sigma control

Applied Statistics and Probability for Engineers 6
Consider an \(\bar{X}\) control chart with k-sigma control limits and subgroup size n. Develop a general expression for the probability that a point plots outside the control limits when the process mean has shifted by \(\delta\) units from the center line. Equation Tran

Applied Statistics and Probability for Engineers 6
Suppose that an \(\bar{X}\) chart is used to control a normally distributed process and that samples of size n are taken every n hours and plotted on the chart, which has k-sigma limits. (a) Find a general expression for the expected number of samples and time that is taken until a false signal is

Applied Statistics and Probability for Engineers 6
Suppose that a \(\mathrm{P}\) chart with center line at \(\bar{p}\) with \(\text { k-sigma }\) control limits is used to control a process. There is a critical fraction defective \(p_{c}\) that must be detected with probability 0.50 on the first sample following the shift to this state. Derive a gene

Applied Statistics and Probability for Engineers 6
Suppose that a \(\mathrm{P}\) chart with center line at \(\bar{p}\) and \(\text { k-sigma }\) control limits is used to control a process. What is the smallest sample size that can be used on this control chart to ensure that the lower control limit is positive? Equation Tra

Applied Statistics and Probability for Engineers 6
Problem 136MEE A process is controlled by a P chart using samples of size 100. The center line on the chart is 0.05. What is the probability that the control chart detects a shift to 0.08 on the first sample following the shift? What is the probability that the shift is detected by at least the third

Applied Statistics and Probability for Engineers 6
Problem 137MEE Consider a process whose specifications on a quality characteristic are 100 15. You know that the standard deviation of this normally distributed quality characteristic is 5. Where should you center the process to minimize the fraction defective pro

Applied Statistics and Probability for Engineers 6
NP Control Chart. An alternative to the control chart for fraction defective is a control chart based on the number of defectives or the NP control chart. The chart has centerline at \(n \bar{p}\), the control limits are \(U C L=n \bar{p}+3 \sqrt{n \bar{p}(1-\bar{p})}\) \(L C L=n \bar{p}-3 \sqrt

Applied Statistics and Probability for Engineers 6
C Control Chart. An alternative to the U chart is a chart based on the number of defects. The chart has centerline at \(n \bar{u}\), and the control limits are \(\mathrm{UCL}=n \bar{u}+3 \sqrt{n \bar{u}}\) \(\mathrm{LCL}=n \bar{u}-3 \sqrt{n \bar{u}}\) (a) Verify that the control limits provided

Applied Statistics and Probability for Engineers 6
Standardized Control Chart. Consider the P chart with the usual 3-sigma control limits. Suppose that we define a new variable \(Z_{i}=\frac{\hat{P}_{i}-\bar{P}}{\sqrt{\frac{\bar{P}(1-\bar{P})}{n}}}\) as the quantity to plot on a control chart. It is proposed that this new chart has a center line

Applied Statistics and Probability for Engineers 6
Unequal Sample Sizes. One application of the standardized control chart introduced in Exercise 15-140 is to allow unequal sample sizes on the control chart. Provide details concerning how this procedure would be implemented and illustrate using the following data: