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Chapter 31: Problem 34

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 \(\mathrm{LCL}=0\).)

Short Answer

Expert verified
Center line: 0.00284; UCL: 0.007895; LCL: 0.

Step by step solution

01

Calculate Proportion of Mishandled Baggage

Based on the report given by the U.S. Department of Transportation, for every 352 passengers, there is one mishandled baggage report. Therefore, the proportion \( \widehat{p} \) is given by: \[ \widehat{p} = \frac{1}{352} \approx 0.00284 \]
02

Determine Initial Center Line

The center line for the control chart is the proportion \( \widehat{p} \). Thus, the center line (CL) is: \[ CL = \widehat{p} = 0.00284 \]
03

Calculate Standard Error

The standard error (SE) of the proportion is given by: \[ SE = \sqrt{\frac{\widehat{p}(1 - \widehat{p})}{n}} \] Where \( n = 1000 \) is the sample size. Plugging in the values: \[ SE = \sqrt{\frac{0.00284(1 - 0.00284)}{1000}} \approx 0.001685 \]
04

Calculate Upper Control Limit (UCL)

The upper control limit is calculated using: \[ UCL = \widehat{p} + 3 \times SE \] Substituting the values: \[ UCL = 0.00284 + 3 \times 0.001685 \approx 0.007895 \]
05

Calculate Lower Control Limit (LCL)

The lower control limit is calculated using: \[ LCL = \widehat{p} - 3 \times SE \] Substituting the values: \[ LCL = 0.00284 - 3 \times 0.001685 \approx -0.002215 \] Since the LCL cannot be negative, we set: \[ LCL = 0 \]

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Key Concepts

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

Proportion Calculation
The first step in creating a control chart involves calculating the proportion of interest. In this case, we're focusing on the proportion of mishandled baggage reports. The U.S. Department of Transportation provided data indicating that for every 352 passengers, there is one mishandled baggage report. This forms the basis for our proportion calculation:
  • Proportion (\( \widehat{p} \)) = Reports / Passengers = \( \frac{1}{352} \)
  • This calculation results in \( \widehat{p} \approx 0.00284 \)
These proportions are essential as they help in setting the expected average level of mishandled baggage reports, which we'll use as the center line in the subsequent steps of the control chart.
Standard Error
After determining the proportion, the next step is to calculate the standard error (SE). The standard error gives us an idea of the variability or spread of our sample proportion. It helps in setting the proper control limits. The formula for standard error in this context is:
  • SE = \( \sqrt{\frac{\widehat{p}(1 - \widehat{p})}{n}} \)
  • Here, \( n = 1000 \) is the sample size
  • Inserting our values: SE = \( \sqrt{\frac{0.00284 (1 - 0.00284)}{1000}} \)
  • This gives us an SE of approximately 0.001685
The standard error plays a critical role in adjusting the range in which our daily mishandled baggage reports should fall if the process is stable.
Upper and Lower Control Limits
Armed with our standard error, we can calculate the control limits which indicate the acceptable range of variation in our data. These limits are set at three standard deviations from the center line, both upwards and downwards.
  • Upper Control Limit (UCL) is calculated as:\( UCL = \widehat{p} + 3 \times SE \)
  • Substituting the values gives:\( UCL = 0.00284 + 3 \times 0.001685 = 0.007895 \)
  • The Lower Control Limit (LCL) is given by:\( LCL = \widehat{p} - 3 \times SE \)
  • Plugging in the values:\( LCL = 0.00284 - 3 \times 0.001685 \approx -0.002215 \)
Since negative values don't make practical sense in this context, the LCL is adjusted to 0. These limits define the normalized range of daily baggage mishandling rates, ensuring any reported rates outside this range are investigated further.
Mishandled Baggage Analysis
With control charts, the examination of mishandled baggage reports becomes systematic and data-driven. By regularly monitoring the daily proportion through control limits, significant trends or outliers can be detected, prompting timely interventions if needed.
  • When the proportion lies between the UCL and LCL, this suggests that mishandled baggage rates are within expected variability.
  • If the rate consistently touches or breaches the UCL, this might indicate systemic issues requiring deeper investigations and corrective actions.
  • Conversely, consistently lower rates than the norm could indicate genuine improvement or measurement errors.
Using these insights, airlines and airports can target operational changes, enhancing luggage handling processes, improving user satisfaction, and potentially preventing baggage mishandling before they become persistent issues.

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Most popular questions from this chapter

A manager who knows no statistics asks you, "What does it mean to say that a process is in control? Is being in control a guarantee that the quality of the product is good?" Answer these questions in plain language that the manager can understand.

The net weight (in ounces) of bags of almond flour is monitored by taking samples of four bags during each hour of production. The process mean should be \(\mu=16 \mathrm{oz}\). When the process is properly adjusted, it varies with \(\sigma=0.4 \mathrm{oz}\). The mean weight \(x\) for each hour's sample is plotted on an \(x\) control chart. Calculate the center line and control limits for this chart.

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