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Chapter 31: Problem 37

A manufacturer of ultrasonic parking sensors samples four sensors during each production shift. The expectation is that the sensor will initially alarm if there is an object within 48 inches of the sensor. The sensors are put on a rack and an object is moved toward the sensors at a \(90^{\circ}\) angle until it alarms. The distance from the object to the sensor is recorded. The process mean should be \(\mu=48\) inches. Past experience indicates that the response varies with \(\sigma=0.8\) inch. The mean response distance is plotted on an \(\mathrm{x}^{-} \bar{x}\) control chart. The center line for this chart is (a) \(0.8\) inch. (b) 48 inches. (c) 4 inches.

Short Answer

Expert verified
(b) 48 inches.

Step by step solution

01

Understand the Problem

We have a control chart for ultrasonic parking sensors where each sample mean is plotted, and we need to find the center line. The center line of an \( \bar{x} \) control chart represents the process mean \( \mu \). Given that \( \mu = 48 \) inches, the goal is to identify what this value represents among the options.
02

Identify Center Line Calculation

The center line is determined by the process mean \( \mu \). Since the process mean given is \( 48 \) inches, we include this information as the center line of the \( \bar{x} \) control chart.
03

Review Options

The options are (a) \(0.8\) inch, (b) \(48\) inches, and (c) \(4\) inches. The center line of the chart corresponds to the process mean, which is \(48\) inches as per the problem statement.
04

Choosing the Correct Option

Since the process mean \( \mu = 48 \) inches aligns with option (b), it is the correct answer for the center line of the control chart.

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Key Concepts

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

Process Mean
The process mean is a crucial aspect of analyzing and controlling quality in manufacturing processes. In this context, it represents the average value that a set of measurements is expected to reach. For the ultrasonic parking sensors, the process mean, denoted by \( \mu \), is expected to be 48 inches. This value indicates that, on average, the sensors should alarm when an object is 48 inches away.

Setting an accurate process mean helps:
  • Ensure consistency across all produced sensors.
  • Identify deviations from expected performance.
  • Maintain customer satisfaction by delivering reliable products.
Understanding the process mean allows manufacturers to maintain control over their processes, ensuring that products meet specifications and safety standards.
Standard Deviation
Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. In the context of the ultrasonic sensors, the standard deviation, \( \sigma \), is given as 0.8 inches. This means that the distance at which the sensors trigger can vary by around 0.8 inches from the process mean of 48 inches.

Key aspects of standard deviation include:
  • Indicating the reliability of the sensor's performance.
  • Providing insights into the consistency of manufacturing processes.
  • Helping identify when a process is out of control or requires adjustment.
A smaller standard deviation implies that the process is tightly controlled, while a larger deviation suggests more variability and potential issues.
Ultrasonic Sensor
Ultrasonic sensors are devices used to detect the presence and distance of an object through the use of sound waves. In this scenario, the sensors are designed to alarm when an object is within 48 inches. These sensors work by emitting ultrasonic waves and measuring the time it takes for the waves to bounce back from an object.

Advantages of using ultrasonic sensors include:
  • High accuracy in measuring distances.
  • Non-contact measurement, which preserves sensor and object integrity.
  • Flexibility in detecting a wide range of materials and surfaces.
Ultrasonic sensors are integral to applications where precision and safety are paramount, such as in automated parking systems.
Quality Control
Quality control involves monitoring and managing manufacturing processes to ensure products meet specified standards. In the case of the ultrasonic sensors, quality control ensures that each sensor operates within the desired 48-inch range.

Key elements of quality control include:
  • Utilizing control charts, such as the \( \bar{x} \) chart, to track process performance.
  • Identifying any deviations from the process mean and addressing issues promptly.
  • Implementing continuous improvements to enhance product quality.
By applying robust quality control measures, manufacturers can minimize defects and maintain high levels of reliability and customer satisfaction.

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Most popular questions from this chapter

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.

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.

A manufacturer of consumer electronic equipment makes full use not only of statistical process control, but also of automated testing equipment that efficiently tests all completed products. Data from the testing equipment show that finished products have only \(3.0\) defects per million opportunities. (a) What is \(\mathrm{p}^{-\bar{p}}\) for the manufacturing process? If the process turns out 4000 pieces per day, how many defects do you expect to see per day? In a typical month of 24 working days, how many defects do you expect to see? (b) What are the center line and control limits for a \(p\) chart for plotting daily defect proportions? (c) Explain why a \(p\) chart is of no use at such high levels of quality.

If the mesh tension of individual monitors follows a Normal distribution, we can describe capability by giving the percent of monitors that meet specifications. The old specifications for mesh tension are \(100-400 \mathrm{mV}\). The new specifications are \(150-350 \mathrm{mV}\). 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 percent of monitors meet the old specifications? (b) What percent meet the new specifications?

You manage the customer service operation for a maker of electronic equipment sold to business customers. Traditionally, the most common complaint is that equipment does not operate properly when installed, but attention to manufacturing and installation quality will reduce these complaints. You hire an outside firm to conduct a sample survey of your customers. Here are the percents of customers with each of several kinds of complaints: $$ \begin{array}{lc} \hline \text { Category } & \text { Percent } \\ \hline \text { Accuracy of invoices } & 27 \\ \hline \text { Clarity of operating manual } & 6 \\ \hline \text { Complete invoice } & 25 \\ \hline \text { Complete shipment } & 16 \\ \hline \text { Correct equipment shipped } & 15 \\ \hline \text { Ease of obtaining invoice adjustments/credits } & 34 \\ \hline \text { Equipment operates when installed } & 5 \\ \hline \text { Meeting promised delivery date } & 11 \\ \hline \text { Sales rep returns calls } & 3 \\ \hline \text { Technical competence of sales rep } & 12 \\ \hline \end{array} $$ (a) Why do the percents not add to \(100 \%\) ? (b) Make a Pareto chart. What area would you choose as a target for improvement?
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