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Chapter 31: Problem 24

The Boston Marathon has been run each year since 1897. Winning times for men were highly variable in the early years, but control improved as the best runners became more professional. A clear downward trend continued until the 1980 s. Rick plans to make a control chart for the winning times from 1950 to the present. The first few times are \(153,148,152,139,141\), and 138 minutes. Calculation from the winning times from 1950 to 2019 gives $$ x=133.615 \text { minutes and } s=6.210 \text { minutes } $$ Rick draws a center line at \(x\) and control limits at \(x \pm 3 s\) for a plot of individual winning times. Explain carefully why these control limits are too wide to effectively signal unusually fast or slow times.

Short Answer

Expert verified
The control limits are too wide to effectively signal unusual winning times due to the naturally decreased variability among top runners.

Step by step solution

01

Understand the Control Chart

The control chart is used to monitor a process over time, with a center line (often the mean) indicating the process average and control limits that signal if the process is outside expected variability. Rick uses the mean winning time, \(x = 133.615\) minutes, as the center line and calculates control limits using \(x \pm 3s\), where \(s = 6.210\) minutes.
02

Calculate Control Limits

For the control limits, apply the formula \(x \pm 3s\). Compute the upper control limit (UCL) and lower control limit (LCL):- Upper Control Limit: \(133.615 + 3 \times 6.210 = 152.245\) minutes- Lower Control Limit: \(133.615 - 3 \times 6.210 = 114.985\) minutesThus, the control limits are 114.985 to 152.245 minutes.
03

Evaluate Control Limits for Effectiveness

The wide range between the lower and upper control limits (114.985 to 152.245 minutes) makes it difficult to detect significant improvements or deteriorations in winning times. Effective control limits should be narrow enough to signal unusual variations promptly. Given the typical variability in human performance, these limits may not flag faster times as unusual.
04

Assess Variability Impact

The historical winning times鈥� variability was reduced as runners became more professional. Therefore, using a control limit based on 3 standard deviations may not reflect current competitive standards, which require smaller deviations to signal anomaly. A \(3s\) range could encompass too much usual variance, especially in a high-performance field like marathon running.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statistical Process Control
Statistical Process Control (SPC) is a method used to monitor and control a process to ensure it operates at its full potential. This is achieved by using statistical metrics to observe the behavior of the process over time. The central element of SPC is the control chart, which displays data in a time-ordered sequence. The central line, often the mean, represents a typical process level, while the control limits around the line show the statistically expected range of variation.

When variations in data points fall outside these limits, it signals that a process may be out of control. This calls for investigation into the cause of variation so adjustments can be made. It's crucial in processes where consistency is key, such as manufacturing or quality assurance in services, because it aids in maintaining a desired level of quality and efficiency. In the context of marathon winning times, properly set control limits can indicate unusually fast or slow wins, helping to identify years with exceptional performances.
Winning Times Analysis
Analyzing marathon winning times, like those of the Boston Marathon, is crucial to understanding trends and performance improvements over time. By using control charts, analysts can visualize variations and trends in winning times from year to year.

- Historical data shows that the winning times of marathons have decreased over time, reflecting improvements in training, nutrition, and knowledge.
- This decreasing trend has significantly slowed since the 1980s, indicating that further substantial improvements are harder to achieve.
- By charting these times and analyzing their statistical properties, one can determine how often runners can achieve extraordinarily fast wins compared to the average. This analysis helps set realistic standards and expectations in competitive racing.
Standard Deviation
The standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. A small standard deviation indicates that the values tend to be close to the mean, while a large standard deviation implies a broader range of values.

In the context of marathon winning times, the standard deviation helps to understand variability in runners' performances. For the years 1950 to 2019, we have a standard deviation of 6.210 minutes around the mean winning time of 133.615 minutes. This statistic is crucial for setting control limits on a control chart.

The choice of applying a range of three standard deviations for control limits (indicating approximately 99.7% of occurrence under normal distribution) may not always be suitable for all processes. Too broad ranges may fail to detect significant changes, such as improvements or declines in marathon times. For highly competitive fields, narrower deviations might be necessary to capture meaningful performance changes and improvements.

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Most popular questions from this chapter

A manufacturer of ultrasonic parking sensors samples four sensors during each production shift. The expectation is that a sensor will initially alarm when an object comes within 60 inches of the sensor. The sensors are put on a rack, and an object is moved toward each sensor, one at a time, at a 90 -degree angle until the sensor alarms. The distance from the object to the sensor at that point is recorded. This results in four measurements, one for each sensor on the rack, and the mean of these four measurements is recorded. The process mean should be \(\mu=60\) inches. Past experience indicates that the response varies with \(\sigma=1.0\) inches. The mean response distance is plotted on an \(x\) control chart. The center line for this chart is a. \(1.0\) inches. b. 4 inches. c. 60 inches.

A process produces rubber fan belts for automobiles. The process is in control, and 100 belts are inspected each day for a period of 10 days. The proportion of nonconforming belts found over this 10 -day period is \(p=0.08\). Based on these data, a \(p\) chart for future samples of size 100 would have center line a. \(0.08\). b. \(8.0\). C. \(0.92\).

The U.S. Department of Transportation reports that in 2018 about one of every 352 passengers on domestic flights of the 12 largest U.S. airlines filed a report of mishandled baggage. Starting with this information, you plan to sample records for 1000 passengers per day at a large airport to monitor the effects of efforts to reduce mishandled baggage. What are the initial center line and control limits for a chart of the daily proportion of mishandledbaggage reports? (You will find that LCL \(<0\). Because proportions \(\widehat{p}\) are always zero or positive, take \(\mathrm{LCL}=0\).)

If the mesh tension of individual monit ors follows a Normal distribution, we can describe capability by giving the percentage of monitors that meet specifications. The old specifications for mesh tension are 100 to \(400 \mathrm{mV}\). The new specifications are 150 to \(350 \mathrm{mV}\). Because the process is in control, we can estimate that tension has mean \(275 \mathrm{mV}\) and standard deviation \(38.4 \mathrm{mV}\). a. What percentage of monitors meet the old specifications? b. What percentage meet the new specifications?

Choose a process that you know well. If you lack experience with actual business or manufacturing processes, choose a personal process such as cooking and serving a meal, ordering something online, or uploading a video to YouTube. Make a flowchart of the process. Make a cause-and-effect diagram that presents the factors that lead to successful completion of the process.
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