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278 Statistical Process Control Chapter 13 - Statistical Process Control 4

This is file Q8IM13D - The fourth of five files for solutions to this chapter.

23.The sample means and standard deviations gathered in a calibration project by the Weighrite Corporation were observed for sample sizes of 10, as show in worksheet Prob. 13-23. Construct - and s-charts and discuss the results.Answer23.See data and control charts below and spreadsheet Prob.13-23XSWeigh.xls for details.

For the Weighrite Corporation, the center line, CL: = 8.659; CLs : = 7.474a. Control limits for the - s charts are:

A3 = 8.659 0.975 (7.474) = 1.372 to 15.946

For the s-chart: UCLs = B4 = 1.716 (7.474) = 12.825

LCLs = B3 = 0

b. The process appears to be in statistical control, because values are distributed randomly about the mean, all values lie within control limits, and there are no unusual patterns on the or s-charts.24.The temperature in a computer lab at Coyote University is very important for proper functioning of the computer equipment. The data in the worksheet Prob.13-24 shows the results of 30 samples of 5 each, taken at random at different times of day over a three month time period. Construct - and s charts and discuss the results.Answer24.See data and control charts below and spreadsheet Prob.13-24XSCoyote.xls for details.

For the Coyote University computer lab, the center line, CL: = 72.071; CLs : = 1.159a. Control limits for the - s charts are:

A3 = 72.071 1.43 (1.159) = 70.414 to 73.728

For the s-chart: UCLs = B4 = 2.09 (1.159) = 2.422

LCLs = B3 = 0

b. The process appears to be in statistical control, although there seems to be a tendency for values to hug the center lines on both the and s-charts. However, all values lie within control limits.25.Calculate the process capability statistics for the outside diameters of the bottles made on the injection molding machine at the Moby Molding Co. (from Prob. 13-22). Use 0.12 as the upper tolerance limit and -0.10 as the lower tolerance limit for this important measure of process performance. What recommendation would you make to management concerning the process, based on these findings?Answer25.The process is not capable at this point. None of the values of Cp, Cpu, Cpl, Cpk are greater than 1.0

Note that the spreadsheet uses an actual standard deviation of ( = 0.047 calculated from all of the sample values. Although this numeric value happens to be the same, this is the overall standard deviation for all sample values, and is not the same average statistic as .Process Capability

Upper specification0.12Cp0.789

Lower specification 0.10Cpu0.795

Cpl0.784

Cpk0.784

Management should be advised that, the capability is not good, so they should work to reduce variability, so as to approach a Cp = 2.0.26.Calculate the process capability statistics for the temperature in the computer lab at Coyote University (from Prob. 13-24). Use 76 as the upper tolerance limit and 68 as the lower tolerance limit for this important measure. What recommendation would you make to management concerning the process, based on these findings?Answer26.The process is capable at this point. All of the values of Cp, Cpu, Cpl, Cpk are greater than 1.0, although the process is slightly skewed to the left.

Note that the spreadsheet uses an actual standard deviation of ( = 1.268 calculated from all of the sample values. This numeric value is the overall standard deviation for all sample values, and is not the same average statistic as .Process Capability

Upper specification76.0Cp1.051

Lower specification68.0Cpu1.033

Cpl1.070

Cpk1.033

Management should be advised that, while the capability is minimally acceptable, they should work to reduce variability, so as to approach a Cp = 2.0.27.Chief Henry Batter of the Gotham City Police Department is trying to reduce the time required to answer the phone at police headquarters (in fractions of a minute). The data in the worksheet for Prob. 13-27 represent time in fractions of minutes for 3 individual readings taken at random for 25 days.

a. Compute control limits for an x-chart (chart for individuals) using the statistic / d2 with a 3-period moving range, as an estimate of the standard deviation.

b. Construct an x - chart for individuals, using the data. Interpret the results.

Answer

27.See spreadsheet Prob.13-27IVGotham.xls for details. Values, below, for and were obtained from the spreadsheet calculations.a) Preparing a chart for individuals: Center Lines, CLx : = 0.759; CLR : = 0.037

Estimated ( = / d2 = 0.037 / 1.693 = 0.022; actual (= 0.021

- 3 ( est = 0.759 3 (0.021) = 0.696 to 0.822;

- 3 ( actual = 0.759 3 (0.022) = 0.693 to 0.825

These limits apply to individual items only. In this case, the estimated standard deviation was based on a three period moving range, and was only 0.001 different from the actual standard deviation. This small variation will not happen in every case, because ( = / d2 is only an estimator for the population standard deviation. Individual items can only be plotted on x-charts, as shown below.

Control limits - R: UCLR = D4

= 2.574 (0.037) = 0.095

LCLR = D3

= 0 (0.037) = 0

b) The individual values should be plotted on x-charts, but students need to understand how - chart and R-chart results differ in comparison with the above. The chart for individuals and the moving range chart seem to be under control.28.Charlie Plato owns Charlies China Emporium, which sells inexpensive cups, dishes, and bric-a-brack in a seaside resort. She has three checkout stations, which she would like to test to see if they are under control and capable. She considers sales of $36.50 per hour, per station, to be a representative average. Consider the data for 75 individual results of sales dollars per hour, per unit, shown in the worksheet Prob.13-28.

a. Compute control limits for an x-chart (chart for individuals) using the statistic / d2 as an estimate of the standard deviation with a 3-period moving range.

b. Construct an x - chart for individuals, using the data. Interpret the results.

Answer

28.See spreadsheet Prob13-28IVCharlie.xls for details. Values, below, for and were obtained from the spreadsheet calculations. a) Preparing a chart for individuals: Center Lines, CLx : = 36.58; CLR : = 20.64

3 ( est = 36.58 3 (12.19) = 0.01 to 73.15;

3 ( actual = 36.58 3 (11.38) = 2.44 to 70.72

These limits apply to individual items only.

Control limits - R: UCLR = D4

= 2.574 (20.64) = 53.13

LCLR = D3

= 0 (20.64) = 0

b) The individual values should be plotted on x-charts, but students need to understand how - chart and R-chart results differ in comparison with the above. The chart for individuals and the moving range chart seem to be under control.

29. Ricardos Widgets makes a critical part for a popular brand of cell phones. Consider the data for 60 observations of a key dimension for the part, shown in the worksheet Prob.13-29.

a. Compute control limits for an x-chart using the statistic /d2 as an estimate of the standard deviation, with a 4-period moving average for the range calculation.

b. Construct an x- chart for individuals using the data. Interpret the results.

Ans29. a) See spreadsheet Prob.13-29IVRick.xls (X and MR chart template) for details. Values, below, for and were obtained from the spreadsheet calculations.

CLx : = 0.112; CLR : = 0.011 (with a 4-period moving average)

Estimated ( = / d2 = 0.011 / 2.059 = 0.0053

3 ( est = 0.112 3 (0.0053) = 0.096 to 0.128;

for 3 ( = 0.112 3 (0.0055) = 0.096 to 0.129

The limits above apply to individual items only. Individual items can only be plotted on x-charts. See the chart on individuals, below.

Control limits - R: UCLR = D4

= 2.282 (0.011) = 0.025

LCLR = D3

= 0 (0.011) = 0

b) The individual values should be plotted on x-charts, but students need to understand how - chart and R-chart results differ in comparison with the above. The chart for individuals and the moving range chart seem to be under control.

See file Q8IMC13E.DOC - the last file, for more solutions to end-of-chapter problems for Chapter 13.

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