12.22 Dyeing yarn.

The unique colors of the cashmere sweaters your firm makes result from heating undyed yarn in a kettle with a dye liquor. The pH (acidity) of the liquor is critical for regulating dye uptake and hence the final color. There are five kettles, all of which receive dye liquor from a common source. Twice each day, the pH of the liquor in each kettle is measured, giving a sample of size 5. The process has been operating in control with and . Give the center line and control limits for the chart.

12.23 Probability out?

An chart plots the means of samples of size 4 against center line and control limits and . The process has been in control. Now the process is disrupted in a way that changes the mean to and the standard deviation to . What is the probability that the first sample after the disruption gives a point beyond the control limits of the chart?

12.24 Alternative control limits.

American and Japanese practice uses three-sigma control charts. That is, the control limits are three standard deviations on either side of the mean. When the statistic being plotted has a Normal distribution, the probability of a point outside the limits is about 0.003 (or about 0.0015 in each direction) by the 68-95-99.7 rule. European practice uses control limits placed so that the probability of a point out is 0.001 in each direction. For a Normally distributed statistic, how many standard deviations on either side of the mean do these alternative control limits lie?

12.25 Monitoring packaged products.

To control the fill amount of its cereal products, a cereal manufacturer monitors the net weight of the product with and charts using a subgroup size of . One of its brands, Organic Bran Squares, has a target of 10.6 ounces. Suppose that 20 preliminary subgroups were gathered, and the following summary statistics were found for the 20 subgroups:

Assume that the process is stable in both variation and level. Compute the control limits for the and charts.

12.26 Measuring bone density.

Loss of bone density is a serious health problem for many people, especially older women. Conventional X-rays often fail to detect loss of bone density until the loss reaches 25% or more. New equipment such as the Lunar bone densitometer is much more sensitive. A health clinic installs one of these machines. The manufacturer supplies a "phantom," an aluminum piece of known density that can be used to keep the machine calibrated. Each morning, the clinic makes two measurements on the phantom before measuring the first patient. Control charts based on these measurements alert the operators if the machine has lost calibration. Table 12.6 contains data for the first 30 days of operation.⁹ The units are grams per square centimeter (for technical reasons, area rather than volume is measured).

12.27 Additional out-of-control signals.

A single extreme point outside of three-sigma limits represents one possible statistical signal of unusual process behavior. As we saw with Figure 12.5(a) (page 608), process change can also give rise to unusual variation within control limits. A variety of statistical rules, known as runs rules, have been developed to supplement the three-sigma rule in an effort to more quickly detect special cause variation. A commonly used runs rule for the detection of smaller shifts of gradual process drifts is to signal if nine consecutive points all fall on one side of the center line. We have learned that for an in-control process and the assumption of Normality, the false alarm rate for the three-sigma rule is about three in 1000. Assuming Normality of the control chart statistics, what is the false alarm rate for the nine-in-a-row rule if the process is in control?

12.28 Alloy composition—retrospective control.

Die casts are used to make molds for molten metal to produce a wide variety of products ranging from kitchen and bathroom fittings to toys, doorknobs, and a variety of auto and electronic components. Die casts themselves are made out of an alloy of metals including zinc, copper, and aluminum. For one particular die cast, the manufacturer must maintain the percent of aluminum between 3.8% and 4.2%. To monitor the percent of aluminum in the casts, three casts are periodically sampled, and their aluminum content is measured. The first 20 rows of Table 12.7 give the data for 20 preliminary subgroups.

12.29 Alloy composition—prospective control.

Project the and chart limits found in the previous exercise for prospective control of aluminum content. The last 15 rows of Table 12.7 give data on the next 15 future subgroups. Refer to Exercise 12.27, and apply the nine-in-a-row rule along with the standard three-sigma rule to the new subgroups. Is the process maintaining control? If not, describe the nature of the process change and indicate the subgroups affected.

12.30 Deming speaks.

The quality guru W. Edwards Deming (1900-1993) taught (among much else) that

Choose one of Deming's sayings. Explain carefully what facts about improving quality the saying attempts to summarize.

12.31 Accounts receivable.

In an attempt to understand the bill-paying behavior of its distributors, a manufacturer samples bills and records the number of days between the issuing of the bill and the receipt of payment. The manufacturer formed subgroups of 10 randomly chosen bills per week over the course of 30 weeks. It found an overall mean of 30.6833 days and an average standard deviation of 7.50638 days.

12.32 Patient monitoring.

There is an increasing interest in the use of control charts in health care. Many physicians are directly involving patients in proactive monitoring of health measurements such as blood pressure, glucose, and expiratory flow rate. Patients are asked to record measurements for a certain number of days. The patient then brings the measurements to the physician who, in turn, will use software to generate control limits. The patient is then asked to plot future measurements on a chart with the limits. Consider data on a patient with hypertension. The data are 30 consecutive self-recorded home systolic measurements.

12.33 Control charting your reaction times.

Consider the following personal data-generating experiment. Obtain a stopwatch, a capability that many electronic watches offer. Alternatively, you can use one of many web-based stopwatches easily found with a Google search (make sure to use a site that reports to at least 0.01 second). Attempt to start and stop your stopwatch as close as possible to 5 seconds. Record the result to as many decimal places as your stopwatch shows. Repeat the experiment 50 times. Input your results into a statistical software package.

12.34 Estimating nonconformance rate.

Suppose a Normally distributed process is centered on target with the target being halfway between specification limits. If , what is the estimated rate of nonconformance of the process to the specifications?

12.35 Measuring capability.

You are in charge of a process that makes metal clips. The critical dimension is the opening of a clip, which has specifications millimeters (mm). The process is monitored by and charts based on samples of five consecutive clips each hour. Control has recently been excellent. The past week's 40 samples have

A Normal quantile plot shows no important deviations from Normality.

12.36 Hospital losses again.

Table 12.4 (page 615) gives data on a hospital's losses for 120 joint replacement patients, collected as 15 monthly samples of eight patients each. The process has been in control, and losses have a roughly Normal distribution. The sample standard deviation () for the individual measurements is 811.53. The hospital decides that suitable specification limit for its loss in treating one such patient is .

12.37 Measuring your personal capability.

Refer to Exercise 12.33 in which you collected 50 sequential observations on your ability to measure 5 seconds. Suppose we define acceptable performance as seconds.

12.38 Alloy composition process capability.

Refer to Exercise 12.28 as it relates to the 20 preliminary subgroups on percents of aluminum content. The acceptable range for the percents of aluminum is 3.8% to 4.2%.

12.39 Six-Sigma quality.

A process with is sometimes said to have “Six-Sigma quality.” Sketch the specification limits and a Normal distribution of individual measurements for such a process when it is properly centered. Explain from your sketch why this is called Six-Sigma quality.

12.40 More on Six-Sigma quality.

The originators of the Six-Sigma quality standard reasoned as follows. Short-term process variation is described by . In the long term, the process mean will also vary. Studies show that in most manufacturing processes, is adequate to allow for changes in . The Six-Sigma standard is intended to allow the mean to be as much as away from the center of the specifications and still meet high standards for percent of output lying outside the specifications.