1 STAT 252 Handout 25 Winter 2010 Quality Control: Control Charts for Location, Variation, Attributes Quality control methods in general, and control charts in particular, have been very effective tools for management and industry. They have played an important role in improving the quality of goods and service produced by manufacturers and service providers. We will examine three of the more widely used types of control charts: • X-Charts for Process Location • R-Charts for Process Variation • p-Charts for Process Attributes Example 1: Airport Waiting Times On each of 20 different days, the waiting time (in minutes) was recorded for five randomly selected customers who entered an airline check-in line at Reno Airport. The airline gathered these data in order to investigate whether their waiting times are consistent from day to day. The data for the first five days appear below, and all of the data appear in the Minitab worksheet RenoWaitTimes.mtw. Day Waiting Times Mean XRange R1 3.2, 6.7, 1.3, 8.4, 2.2 2 5.0, 4.1, 7.9, 8.1, 0.4 3 7.1, 3.2, 2.1, 6.5, 3.7 4 4.2, 1.6, 2.7, 7.2, 1.4 5 1.7, 7.1, 1.6, 0.9, 1.8 … … … … a) Use Minitab (Calc> Row Statistics…) to calculate the sample mean waiting time for each of the 20 days. Store the sample means in c5 and name the column x-bar. Record the first 5 means in the table above. b) Use Minitab (Calc> Row Statistics…) to calculate the sample range of the waiting times for each of the 20 days. Store the sample ranges in c6 and name the column range. Record the first 5 range values in the table above. c) Produce a line chart of the sample means by creating a scatterplot of the means vs. day with the “with connect lines” feature. Does it appear that this process is “in control” with respect to location? d) Calculate the mean of the sample means (MTB> mean c7). Record this value below. (We denote this value with the symbol x .)2 To assess whether the waiting time process is “under control,” we ask whether any of the sample means are farther from the overall mean than would be expected by normal variation. Most control charts are based on the idea of considering a process to be in control if all of the sample means are within 3 standard deviations of the overall mean. You know from the Central Limit Theorem that the standard deviation of a sample mean X is given by ()nXSDσ= . Of course we typically do not know the population standard deviation σ, so we could reasonably use ns . But this is typically not done, because the sample standard deviation s is difficult to compute by hand. Instead we base the control limits on the average of the sample ranges, denoted by R. e) Use Minitab to calculate the average of the sample ranges. Record it below with the appropriate symbol. • The control limits for an X–Chart (of Process Location) are: x ± A2×R, where A2 is the appropriate constant, dependent on the sample size n and tabulated in Table Q, to make A2×R an unbiased estimator of 3× nσ. f) Use Table Q to determine the appropriate value of A2 for these data. g) Calculate the upper and lower control limits for the X–Chart. h) According to these control limits, are any of the 20 sample mean waiting times out of control? i) Use Minitab to produce this control chart (Stat> Control Charts> Variable Charts for Subgroups> Xbar… and choose “Observations for a subgroup are in one row of columns”). Do the upper and lower control limits that you calculated by hand agree with what Minitab displays? [Note: To make them agree, click Xbar Options, then choose Estimate from the tabs across the top, and finally select Rbar.] Does Minitab show any of the 20 sample means to be out of control?3 j) Suppose that every single waiting time in all 20 samples was 30 minutes longer than it actually was. How would this change (if at all) the values of x , R, A2, LCL, and UCL? Justify your answers. Would any of the sample means now be “out of control”? • Notice that “in control” just means that none of the sample means is inconsistent with the pattern of variation shown in the others. Being in control does not necessarily mean that the values are acceptable from a service or manufacturing perspective. k) Now create a line chart of the sample ranges by creating a scatterplot of the ranges vs. day with the “with connect lines” feature. Does it appear that this process is “in control” with respect to variation? • The control limits for an R-Chart (of Process Variation) are: D3×R and D4×R, where D3 and D4 are the appropriate constants, again dependent on the sample size n and tabulated in Table Q. l) Use Table Q to determine the appropriate values of D3 and D4 for these data. m) Calculate the upper and lower control limits for the R–Chart. n) According to these control limits, are any of the 20 sample ranges (of waiting times) out of control?4 o) Use Minitab to produce this control chart (Stat> Control Charts> Variable Charts for Subgroups> R… and choose “Observations for a subgroup are in one row of columns”). Do the upper and lower control limits that you calculated by hand agree with what Minitab displays? Does Minitab show any of the 20 sample ranges to be out of control? p) Now use Minitab to produce both the X–Chart and R-Chart at the same time (Stat> Control Charts> Variable Charts for Subgroups> Xbar-R). To see if you understand what the X–Chart and R-Chart are monitoring, try to change the waiting times for only the day 1 sample in order to make the following happen: q) Make the process out of control with respect to location but remain in control with respect to variation. Record new values for day 1 waiting times that make this happen, and explain your reasoning process. r) Make the process out of control with respect to variation but remain in control with respect to location. Record new values for day 1 waiting times that make this happen, and explain your reasoning process. s) Make the process out of control with respect to both location and variation. Record new values for day 1 waiting times that make this happen, and explain your reasoning process.5 Now we turn to control charts for attributes, in other words for a (binary) categorical variable. We will learn how to construct and interpret a p-chart. Example 2: Basketball Shooting Suppose a group of
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