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Chapter 31: Problem 34

The Department of Transportation reports that about one of every 208 passengers on domestic flights of the 18 largest U.S. airlines files 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 mishandled-baggage reports? (You will find that LCL \(<0\). Because proportions \(\mathrm{p}^{\wedge \hat{p}}\) are always zero or positive, take \(\mathrm{LCL}=0 .)\)

Short Answer

Expert verified
CL = 0.00481, UCL = 0.01132, LCL = 0.

Step by step solution

01

Understand the Problem Statement

We are given that about one out of every 208 passengers files a report of mishandled baggage. We need to use this information to calculate the control chart parameters for 1000 sampled passengers each day.
02

Find the Proportion (\(p\))

The proportion, \( p \), of mishandled baggage reports is the number of reported cases divided by the total number of passengers. Since 1 out of 208 files a report, \( p = \frac{1}{208} \approx 0.00481 \).
03

Calculate the Center Line (CL)

The center line for the control chart is the expected daily proportion of mishandled baggage reports, which is equal to \( p \). Thus, \( CL = p = 0.00481 \).
04

Determine the Standard Deviation of \( \\hat{p} \)

The standard deviation of the sample proportion \( \hat{p} \) is given by the formula: \( \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \), where \( n = 1000 \). So, \( \sigma_{\hat{p}} = \sqrt{\frac{0.00481(1-0.00481)}{1000}} \approx 0.00217 \).
05

Compute the Upper Control Limit (UCL)

The upper control limit is computed using the formula: \( UCL = CL + 3\sigma_{\hat{p}} \). Therefore, \( UCL = 0.00481 + 3 \times 0.00217 \approx 0.01132 \).
06

Compute the Lower Control Limit (LCL) and Adjust

Initially, the lower control limit is calculated using: \( LCL = CL - 3\sigma_{\hat{p}} \). However, since \( LCL = 0.00481 - 3 \times 0.00217 = -0.0017 \), we set \( LCL = 0 \) because the proportion cannot be negative.

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Key Concepts

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

Understanding Control Charts
Control charts are essential tools used in statistical process control. They help us understand variation in processes over time, as well as identify trends or points that require immediate attention. When creating a control chart, we are interested in three main lines: the Center Line (CL), Upper Control Limit (UCL), and Lower Control Limit (LCL). These lines help us distinguish between normal random variation and any unusual variation.

  • Center Line (CL): Represents the average or expected level in the process; for our scenario, it is the expected proportion of mishandled baggage reports in a sample.
  • Upper and Lower Control Limits (UCL and LCL): These denote the boundaries within which the process variation is considered acceptable. Anything outside these limits could indicate abnormal variations.
Control charts are especially effective in continuous improvement efforts as they visually represent process performance over time. By understanding these elements, one can effectively monitor and improve operational activities.
Proportion Calculations in Statistical Process Control
Proportion calculations are fundamental when working with control charts involving categorical data, like mishandled baggage reports. The proportion, denoted as \( p \), represents the fraction of interest out of the total possible outcomes.
To compute the proportion in our example:
  • We start with the given ratio, which is 1 mishandled report per 208 passengers.
  • This converts to \( p = \frac{1}{208} \approx 0.00481 \).
This proportion forms the basis for further calculations on the control chart, such as determining the center line and control limits.
These calculations are significant because they translate qualitative data into a numerical format that can be analyzed statistically. Ensuring accuracy in these calculations is crucial for the reliability of the control chart and subsequent decision-making processes.
Calculating Standard Deviation for Proportion Control
Understanding standard deviation in the context of control charts for proportions is vital. Standard deviation measures the amount of variation or dispersion in a set of values. In our baggage claim example, we need to calculate the standard deviation of the sample proportion \( \hat{p} \).
The formula to calculate this is:
  • \( \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \), where \( n \) is the sample size.
Since we are checking 1000 passengers each day, \( n = 1000 \), and \( p = 0.00481 \), it results in \( \sigma_{\hat{p}} \approx 0.00217 \).
Knowing the standard deviation helps in setting the control limits. It ensures that the control chart reflects an accurate range of process variation. This process is crucial as it establishes a statistical basis for identifying whether changes in data patterns are due to randomness or special causes.

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Most popular questions from this chapter

Is each of the following examples of a special cause most likely to first result in (i) one-point-out on the \(s\) or \(R\) chart, (ii) one-point-out on the \(\mathrm{x}^{-} \bar{x}\) chart, or (iii) a run on the \(\mathrm{x}^{-} \bar{x}\) chart? In each case, briefly explain your reasoning. (a) The time it takes a new coffee barista to complete your order at your favorite coffee shop. (b) The precision of a measurement tool is affected by dirt getting on the sensors and needs to be cleaned when this happens. (c) The accuracy of an inspector starts to degrade after the first six hours of his shift. (d) A person who is training for a \(5 \mathrm{k}\) race created a control chart for her running time on the same route each week. She started running at what she considered a slow pace and is now very happy with her running times.

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 samples of size 5 . The process has been operating in control with \(\mu=5.21\) and \(\sigma=0.147\). (a) Give the center line and control limits for the \(s\) chart. (b) Give the center line and control limits for the \(\mathrm{x}^{-} \bar{x}\) chart.

The net weight (in ounces) of bags of almond flour is monitored by taking samples of five bags during each hour of production. The process mean should be \(\mu=32 \mathrm{oz}\). When the process is properly adjusted, it varies with \(\sigma=0.5\) oz. The mean weight \(x^{-} \bar{x}\) for each hour's sample is plotted on an \(x^{-} \bar{x}\) control chart. Calculate the center line and control limits for this chart.

What type of control chart or charts would you use as part of efforts to improve each of the following performance measures in a college admissions office? Explain your choices. (a) Time to acknowledge receipt of an application (b) Percent of admission offers accepted (c) Student participation in a healthy meal plan

The quality guru W. Edwards Deming (1900-1993) taught (among much else) that \({ }^{16}\) (a) "People work in the system. Management creates the system." (b) "Putting out fires is not improvement. Finding a point out of control, finding the special cause and removing it, is only putting the process back to where it was in the first place. It is not improvement of the process." (c) "Eliminate slogans, exhortations and targets for the workforce asking for zero defects and new levels of productivity." (d) "No one can guess the future loss of business from a dissatisfied customer. The cost to replace a defective item on the production line is fairly easy to estimate, but the cost of a defective item that goes out to a customer defies measure." Choose one of these sayings. Explain carefully what facts about improving quality the saying attempts to summarize.
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