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Introduction To Statistical Quality Control 7th Edition Douglas C Montgomery - Solutions

The tensile strength of a paper product is related to the amount of hardwood in the pulp. Ten samples are produced in the pilot plant, and the data obtained are shown in Table 4E.9.(a) Fit a linear regression model relating strength to percentage hardwood.(b) Test the model in part (a) for
A plant distills liquid air to produce oxygen, nitrogen, and argon. The percentage of impurity in the oxygen is thought to be linearly related to the amount of impurities in the air as measured by the €œpollution count€ in parts per million (ppm). A sample of plant operating data is shown
Plot the residuals from Problem 4.43 and comment on model adequacy.
Plot the residuals from Problem 4.44 and comment on model adequacy.
The brake horsepower developed by an automobile engine on a dynamometer is thought to be a function of the engine speed in revolutions per minute (rpm), the road octane number of the fuel, and the engine compression. An experiment is run in the laboratory and the data are drawn in Table 4E.10:(a)
Analyze the residuals from the regression model in Exercise 4.47. Comment on model adequacy.
Table 4E.11 contains the data from a patient satisfaction survey for a group of 25 randomly selected patients at a hospital. In addition to satisfaction, data were collected on patient age and an index that measured the severity of illness.(a) Fit a linear regression model relating satisfaction to
Suppose that you are testing the following hypotheses where the variance is unknown: H0 :  100 H1 :  100 The sample size is n = 12. Find bounds on the P-value for the following values of the test statistic. (a) t0 = 2.55 (b) t0 = 1.87 (c) t0 = 2.05 (d) t0 = 2.80
Analyze the residuals from the regression model on the patient satisfaction data from Exercise 4.49. Comment on the adequacy of the regression model.
Reconsider the patient satisfaction data in Table 4E.11. Fit a multiple regression model using both patient age and severity as the regressors. (a) Test for significance of regression. (b) Test for the individual contribution of the two regressors. Are both regressor variables needed in the
Analyze the residuals from the multiple regression model on the patient satisfaction data from Exercise 4.51. Comment on the adequacy of the regression model.
Consider the Minitab output below.(a) Fill in the missing values. What conclusions would you draw? (b) Is this a one-sided or two-sided test? (c) Use the output and a normal table to find a 95% CI on the mean. (d) How was the SE mean calculated? (e) What is the P-value if the alternative
Suppose that you are testing H0 : 1  2 versus H1 : 1  2 with a n1 = n2 = 15. Use the table of the t distribution percentage points to find lower and upper bounds on the P-value for the following observed values of the test statistic: (a) t0 = 2.30 (b) t0 = 1.98 (c) t0 = 3.41
Suppose that you are testing H0 : 1  2 versus H1 : 1  2 with a n1 = n2 = 10. Use the table of the t distribution percentage points of find lower and upper bounds on the P-value of the following observed values of the test statistic:
Consider the Minitab output below.(a) Fill in the missing values. Can the null hypothesis be rejected at the 0.05 level? Why? (b) Is this a one-sided or two-sided test? (c) How many degrees of freedom are there on the t-test statistic? (d) Use the output and a normal table to find a 95% CI on the
Consider the Minitab output below.(a) Is this a one-sided or two-sided test? (b) Can the null hypothesis be rejected at the 0.05 level? (c) Construct an approximate 90% CI for p. (d) What is the P-value if the alternative hypothesis is H1: p > 0.3?
Consider the Minitab output below.(a) Fill in the missing values. (b) Can the null hypothesis be rejected at the 0.05 level? Why? (c) Use the output and the t-table to find a 99% CI on the difference in means. (d) Suppose that the alternative hypothesis was H1: ï1 =
Consider the Minitab output below.(a) Fill in the missing values. (b) Is this a one-sided or two-sided test? (c) What is the P-value if the alternative hypothesis is H1: p1 = p2 versus H0: p1 > p2? (d) Construct an approximate 90% CI for the difference in the two proportions.
Suppose that you are testing the following hypotheses where the variance is unknown: H0 :  100 H1 :  100 The sample size is n = 25. Find bounds on the P-value for the following values of the test statistic. (a) t0 = −2.80 (b) t0 = −1.75 (c) t0 = −2.54 (d) t0 = −2.05
Consider a one-way or single-factor ANOVA with four treatments and five replications. Use the table of the F distribution percentage points to find lower and upper bounds on the P-value for the following observed values of the test statistic: (a) F0 = 2.50 (b) F0 = 3.75 (c) F0 = 5.98 (d) F0 = 1.70
Consider the Minitab ANOVA output below. Fill in the blanks. You may give bounds on the P-value. What conclusions can you draw based on the information in this display?
The inside diameters of bearings used in an aircraft landing gear assembly are known to have a standard deviation of  = 0.002 cm. A random sample of 15 bearings has an average inside diameter of 8.2535 cm.(a) Test the hypothesis that the mean inside bearing diameter is 8.25 cm. Use a two-sided
The tensile strength of a fiber used in manufacturing cloth is of interest to the purchaser. Previous experience indicates that the standard deviation of tensile strength is 2 psi. A random sample of eight fiber specimens is selected, and the average tensile strength is found to be 127 psi.(a) Test
The service life of a battery used in a cardiac pacemaker is assumed to be normally distributed. A random sample of ten batteries is subjected to an accelerated life test by running them continuously at an elevated temperature until failure, and the following lifetimes (in hours) are obtained:
A molding process uses a five-cavity mold for a part used in an automotive assembly. The wall thickness of the part is the critical quality characteristic. It has been suggested to use X and R charts to monitor this process, and to use as the subgroup or sample all five parts that result from a
A manufacturing process produces 500 parts per hour. A sample part is selected about every half hour, and after five parts are obtained, the average of these five measurements is plotted on an x control chart. (a) Is this an appropriate sampling scheme if the assignable cause in the process results
Consider the sampling scheme proposed in Exercise 5.11. Is this scheme appropriate if the assignable cause results in a slow, prolonged upward drift in the mean? If your answer is no, propose an alternative procedure.
What information is provided by the operating characteristic curve of a control chart?
Is the average run length performance of a control chart a more meaningful measure of performance than the type I and type II error probabilities? What information does ARL convey that the statistical error probabilities do not?
Consider the control chart shown in Exercise 5.17. Would the use of warning limits reveal any potential out-of-control conditions?
Apply the Western Electric rules to the control chart in Exercise 5.17. Are any of the criteria for declaring the process out of control satisfied?
Sketch warning limits on the control chart in Exercise 5.19. Do these limits indicate any potential out-of-control conditions?
Apply the Western Electric rules to the control chart presented in Exercise 5.19. Would these rules result in any out-of-control signals?
Consider the time-varying process behavior shown below and on the next page. Match each of these several patterns of process performance to the corresponding and R charts shown in figures (a) to (e) below.
A car has gone out of control during a snowstorm and struck a tree. Construct a cause-and-effect diagram that identifies and outlines the possible causes of the accident.
Laboratory glassware shipped from the manufacturer to your plant via an overnight package service has arrived damaged. Develop a cause-and-effect diagram that identifies and outlines the possible causes of this event.
Construct a cause-and-effect diagram that identifies the possible causes of consistently bad coffee from a large-capacity office coffee pot. Discuss.
Develop a flowchart for the process that you follow every morning from the time you awake until you arrive at your workplace (or school). Identify the value-added and non-value-added activities.
The magnificent seven can be used in our personal lives. Develop a check sheet to record “defects” you have in your personal life (such as overeating, being rude, not meeting commitments, missing class, etc.). Use the check sheet to keep a record of these “defects” for one month. Use a
A process is normally distributed and in control, with known mean and variance, and the usual three-sigma limits are used on the x control chart, so that the probability of a single point plotting outside the control limits when the process is in control is 0.0027. Suppose that this chart is being
Reconsider the situation in Exercise 5.32. Suppose that the process mean and variance were unknown and had to be estimated from the data available from the m subgroups. What complications would this introduce in the calculations that you performed in Exercise 5.32?
What is meant by a process that is in a state of statistical control?
Discuss the logic underlying the use of three-sigma limits on Shewhart control charts. How will the chart respond if narrower limits are chosen? How will it respond if wider limits are chosen?
What are warning limits on a control chart? How can they be used?
A manufacturer of component for automobile transmissions wants to use control charts to monitor a process producing a shaft. The resulting data from 20 samples of 4 shaft diameters that have been measured are:(a) Find the control limits that should be (b) Assume that the 20 preliminary samples plot
The fill volume of soft-drink beverage bottles is an important quality characteristic. The colume is measured (approximately) by placing a gauge over the crown and comparing the height of the liquid in the neck of the bottle against a coded scale. On this scale a reading fo zero corresponds to the
The net weight (in oz) of a dry bleach product is to be monitored by x and R control charts using a sample size of n = 5. Data for 20 preliminary samples are shown in Table 6E.7.(b) Estimate the process mean and standard deviation. (c) Does the fill weight seem to follow a normal distribution? (d)
Rework Exercise 6.8 using the s chart.In Exercise 6.8
Rework Exercise 6.9 using the s chart.In Exercise 6.
Consider the piston ring data shown in Table 6.3. Assume that the specifications on this component are 74.000 ï‚±ï€ 0.05 mm.(a) Set up x and R control charts on this process. Is the process in statistical control? (b) The control limits on the x chart in part (a) are
Table 6E.8 shows 15 additional samples for the piston ring process (Table 6.3), taken after the initial control charts were established. Plot these data on the x and R chart developed in Exercise 6.15. Is the process in control?
Control charts on and s are to be maintained on the torque readings of a earing used in a wingflap actuator assembly. Samples of size n = 10 are to be used, and we know from past experience that when the process is in control, bearing torque has a normal distribution with mean µ = 80 inch-pounds
Samples of n = 6 items each are taken from a process at regular intervals. A quality characteristic is measured, and x and R values are calculated for each sample. After 50 samples, we haveAssume that the quality characteristic is normally distributed. (a) Compute control limits for the x and R
Samples of n = 4 items are taken from a process at regular intervals. A normally distributed quality characteristic is measured and x and s values are calculated for each sample. After 50 subgroups have been analyzed, we have(a) Compute the control limit for the x and s control charts (b) Assume
A company manufacturing oil seals wants to establish control charts on the process. There are 25 preliminary samples of size 5 on the internal diameter of the seal. The summary data (in mm) are as follows: and x R(a) Find the control limits that should be used on the x and R control charts. (b)
Table 6E.9 presents 20 subgroups of five measurements on the critical dimension of a part produced by a machining process.(a) Set up x and R control charts on this process. Verify that the process is in statistical control. (b) Following the establishment of control charts in part (a) above, 10 new
Parts manufactured by an injection molding process are subjected to a compressive strength test. Twenty samples of five parts each are collected, and the compressive strengths (in psi) are shown in Table 6E.12.(a) Establish x and R control charts for compressive strength using these data. Is the
Reconsider the data presented in Exercise 6.21.(a) Rework both parts (a) and (b) of Exercise 6.21 using the x and s charts.(b) Does the s chart detect the shift in process variability more quickly than the R chart did originally in part (b) of Exercise 6.21?In Exercise 6.21
Consider the x and R charts you established in Exercise 6.7 using n = 5.(a) Suppose that you wished to continue charting this quality characteristics using x and R charts based on a sample size of n =3. What limits would be used on the x and R charts? for n new = 3 (b) What would be the impact of
Consider the x and R chart that you established in Exercise 6.15 for the piston ring process. Suppose that you want to continue control charting piston ring diameter using n = 3. What limits would be used on the x and R chart?
Control charts for x and R are maintained for an important quality characteristic. The sample size is n= 7; x and R are computed for each sample. After 35 samples we have found that(a) Set up x and R charts using these data. (b) Assuming that both charts exhibit control, estimate the process mean
Samples of size n = 5 are taken from a manufacturing process every hour. A quality characteristic is measured, and x and R are computed for each sample. After 2 samples have been analyzed, we haveThe quality characteristic is normally distributed. (a) Find the control limits for the x and R
Samples of size n = 5 are collected from a process every half hour. After 50 samples have been collected, we calculate. x = 20.0 and s 1.5. Assume that both charts exhibit control and that the quality characteristic is normally distributed. (a) Estimate the process standard deviation. (b) Find the
Control chart for x and R are maintained on a process. After 20 preliminary subgroups each of size 3 are evaluated, you have the following data:(a) Set up control charts using these data. (b) Assume that the process exhibits statistical control. Estimate the process mean and standard deviation.
Control chart x and s for are maintained on a process. After 25 preliminary subgroups each of size 3 are evaluated, you have the following data:(a) Set up control charts using these data. (b) Assume that the process exhibits statistical control. Estimate the process mean and standard deviation. (c)
Reconsider the situation described in Exercise 6.1. Suppose that several of the preliminary 20 samples plot out of control on the R chart. Does this have any impact on the reliability of the control limits on the x chart?
An x chart is used to control the mean of a normally distributed quality characteristic. It is known that σ = 6.0 and n = 4. The center line = 200, UCL = 209, and LCL = 191. If the process mean shifts to 188, find the probability that this shift is detect on the first subsequent sample.
A critical dimension of a machined part has specifications 100 + –10. Control chart analysis indicates tht the process is in control with x=104 and R = 9.30. The control charts use samples of size n = 5. If we assume that the characteristic is normally distributed, can the mean be located (by
A process is to be monitored with standard values µ = 10 and σ= 2.5. The sample size is n = 2. (a) Find the center line and control limits for the x chart. b) Find the center line and control limits for the R chart. (c) Find the center line and control limits for the s chart.
Samples f n= 5 units are taken from a process every hour. The x and R values for a particular quality characteristic are determined. After 25 sample have been collected, we calculate x =20 and R = 4.56. (a) What are the three-sigma control limits for the x and R? (b) Both charts exhibit control.
A TiW layer is deposited on a substrate using a sputtering tool. Table 6E.14 contains layer thickness measurements (in angstroms) on 20 subgroups of four substrates.(a) Setup x and R control charts on this process. Is the process in control? Revise the control limits as necessary. (b) Estimate the
Table 6E.15 contains 10 new subgroups of thickness data. Plot this data on the control charts constructed in Exercise 6.26(a). Is the process in statistical control?
Suppose that following the construction of the x and R control charts in Exercise 6.34, the process engineers decided to change the subgroup size to n = 2. Table 6E.16 contains 10 new subgroups of thickness data. Plot this data on the control charts from Exercise 6.34(a) based on the new subgroup
Rework Exercises 6.34 and 6.35 using x and s control charts. The process is out of control on the x chart at subgroup 18. After finding assignable cause, exclude subgroup 18 from control limits calculations:
Control charts for x and R are to be established to control the tensile strength of a metal part. Assume that tensile strength is normally distributed. Thirty samples of size n= 6 parts are collected over a period of time with the following results:(a) Calculate the control limits for x and R . (b)
An x chart has a center line of 100, uses three-sigma control limits, and is based on a sample size of four. The process standard deviation is known to be six. If the process mean shifts from 100 to 92, what is the probability of detecting this shift on the first sample following the shift?
Discuss why it is important to establish control on the R chart first when using x and R control charts to bring a process into statistical control
The data in Table 6E.17 were collected form a process manufacturing power supplies. The variable of interest is output voltage, and n = 5.(a) Compute center lines and control limits suitable for controlling future production. (b) Assume that the quality characteristic is normally distributed.
Control charts on x and R for samples of size n- 5 are to be maintained on the tensile strength in pounds of a yarn. To start the charts, 30 samples were selected, and the mean and range of each computed. This yields(a) Compute the center line and control limits for the x and R control charts. (b)
Specifications on a cigar lighter detent are 0.3220 and 0.3200 in. Samples of size 5 are taken every 45 minutes with the results shown in Table 6E.18 (measured as deviations from 0.3210 in 0.0001 in.).(a) Set up an R chart and examine the process for statistical control. (b) What parameters would
Reconsider the data from Exercise 6.42 and establish x and R charts with appropriate trial control limits. Revise these trial limits as necessary to produce a set of control charts for monitoring future production. Suppose that the new data in Table 6E.19 are observed.(a) Plot these new
Two parts are assembled as shown in Figure 6.28. Assume that the dimensions x and y are normally distributed with means ïx and ïy and standard deviations ï³x and ï³y, respectively. The parts are produced on different machines and are
Control charts for x and R are maintained on the tensile strength of a metal fastener. After 30 samples of size n= 6 are analyzed, we find that(a) Compute control limits on the R chart. (b) Assuming that the R chart exhibits control, estimate the parameters µ and Ïƒ. (c) If the
Control charts for and s are maintained on a quality characteristic. The sample size is n = 4. After 30 samples, we obtain(a) Find the three-sigma limits for the s chart. (b) Assuming that both charts exhibit control, estimate the parameters µ and Ïƒ.
An chart on a normally distributed quality characteristic is to be established with the standard values µ= 100, σ = 8, and n = 4. Find the following: (a) The two-sigma control limits. (b) The 0.005 probability limits.
An chart with three-sigma limits has parameters as follows: UCL = 104 Center line = 100 LCL = 96 n = 5 Suppose the process quality characteristic being controlled is normally distributed with a true mean of 98 and a standard deviation of8. What is the probability that the control chart would
Consider the chart defined in Exercise 6.48. Find the ARL1 for the chart.
A hospital emergency department is monitoring the time required to admit a patient using x and R charts.Table 6E.1 presents summary data for 20 subgroups of two patients each (time is in minutes).
Control charts for with n = 4 are maintained on a quality characteristic. The parameters of these charts are as follows and x sBoth charts exhibit control. Specifications on the quality characteristic are 197.50 and 202.50. What can be said about the ability of the process to produce product that
Statistical monitoring of a quality characteristic uses both an x and s chart. The charts are to be based on the standard values m = 200 and sx = 10, with n = 4. (a) Find three-sigma control limits for the s chart. (b) Find a center line and control limits for the x chart such that the probability
Specifications on a normally distributed dimension are 600 ± 20. x and R charts are maintained on this dimension and have been in control over a long period of time. The parameters of these control charts are as follows (n = 9)(a) What are your conclusions regarding the capability of the
Thirty samples each of size 7 have been collected to establish control over a process. The following data were collected:(a) Calculate trial control limits for the two charts. (b) On the assumption that the R chart is in control, estimate the process standard deviation. (c) Suppose an s chart were
An chart is to be established based on the standard values µ = 600 and σ = 12, with n= 9. The control limits are to be based on an a-risk of 0.01. What are the appropriate control limits?
and R charts with n = 4 are used to monitor a normally distributed quality characteristic. The control chart parameters areBoth charts exhibit control. What is the probability that a shift in the process mean to 790 will be detected on the first sample following the shift?
Consider the chart in Exercise 6.55. Find the average run length for the chart.
Control charts for and R are in use with the following parameters:The sample size is n = 9. Both charts exhibit control. The quality characteristic is normally distributed. (a) What is the a-risk associated with the chart? (b) Specifications on this quality characteristic are 358 ± 6. What
A normally distributed quality characteristic is monitored through use of anBoth charts exhibit control. (a) What is the estimated standard deviation of the process? (b) Suppose an s chart were to be substituted for the R chart. What would be the appropriate parameters of the s chart? (c) If
Control charts for have been maintained on a process and have exhibited statistical control. The sample size is n = 6. The control chart parameters are as follows:(a) Estimate the mean and standard deviation of the process. (b) Estimate the natural tolerance limits for the process. (c) Assume that
Components used in a cellular telephone are manufactured with nominal dimension of 0.3 mm and lower and upper specification limits of 0.295 mm and 0.305 mm respectively. The x and R control charts for this process are based on subgroups of size 3 and they exhibit statistical control, with the
The following x and s charts based on n = 4 have shown statistical control:(a) Estimate the process parameters µ and Ïƒ. (b) If the specifications are at 705 ± 15, and the process output is normally distributed, estimate the fraction nonconforming. (c) For the chart, find the

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