/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 The unique colors of the cashmer... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 31: Problem 15

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.

Short Answer

Expert verified
s chart center line = 0.147, LCL = 0, UCL = 0.310. x虅 chart center line = 5.21, LCL = 5.125, UCL = 5.295.

Step by step solution

01

Determine the Center Line for the s Chart

The center line for the s chart is the average standard deviation, denoted as \( \bar{s} \). The process standard deviation \( \sigma \) is given as 0.147. The center line for the s chart is \( \bar{s} = \sigma = 0.147 \), as samples of size 5 are used and each sample covers the same process variability.
02

Calculate the Control Limits for the s Chart

For a sample size of 5, the control limits for the s chart are determined using the formula: \( LCL = B_3 \cdot \bar{s} \) and \( UCL = B_4 \cdot \bar{s} \). Given \( B_3 = 0 \) and \( B_4 = 2.114 \) for \( n = 5 \) (values from statistical tables), the limits are: \( LCL = 0 \cdot 0.147 = 0 \) and \( UCL = 2.114 \cdot 0.147 = 0.310 \). Therefore, the control limits are: LCL = 0 and UCL = 0.310.
03

Determine the Center Line for the x虅 Chart

The center line for the \( \bar{x} \) chart is the process mean \( \mu \), which is given as \( 5.21 \). Therefore, the center line for the \( \bar{x} \) chart is \( \mu = 5.21 \).
04

Calculate the Control Limits for the x虅 Chart

The control limits for the \( \bar{x} \) chart are calculated using the following formula:\[ LCL = \mu - A_2 \cdot \bar{s} \] and \[ UCL = \mu + A_2 \cdot \bar{s} \]For a sample size of 5, \( A_2 = 0.577 \). We use \( \bar{s} = 0.147 \) (from Step 1). The control limits are: \( LCL = 5.21 - 0.577 \times 0.147 = 5.125 \) and \( UCL = 5.21 + 0.577 \times 0.147 = 5.295 \). Thus, the control limits for the \( \bar{x} \) chart are LCL = 5.125 and UCL = 5.295.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Control Limits
Control limits are vital statistical tools used in process control charts to determine if a process is in acceptable control. They help us identify variations that significantly deviate from normal, suggesting the presence of special causes.
  • Upper Control Limit (UCL): The highest point of acceptable performance.
  • Lower Control Limit (LCL): The lowest point of acceptable performance.
  • Anything beyond these limits signals potential issues requiring investigation.
By using control limits, we can distinguish natural process variability from significant deviations. This helps in maintaining quality standards and ensuring that processes remain efficient and effective.
s Chart
The s chart, short for standard deviation chart, is used to monitor the variability of a process. This type of control chart helps to keep the process variation under check over time. For the s chart:
  • The center line is represented by the average of the standard deviations, denoted as \( \bar{s} \).
  • In our example, since \( \sigma = 0.147 \), the center line is simply \( \bar{s} = 0.147 \).
  • Control limits are determined by constants deriving from the sample size.
  • For a sample size of 5, the control limits are calculated as \( LCL = 0 \) and \( UCL = 0.310 \).
Thus, the s chart monitors small shifts in variability, allowing businesses to intervene before minor issues escalate.
x虅 Chart
The \( \bar{x} \) chart, known as the "average" chart, is a type of control chart that focuses on the monitoring of the process mean over time. It is particularly useful for spotting shifts in the process mean that could indicate underlying problems.Here's how it works:
  • The center line is the process mean \( \mu \).
  • In the given example, \( \mu = 5.21 \) serves as the center line.
  • Control limits are defined using \( A_2 \) values from standard statistical tables, intertwined with sample standard deviation \( \bar{s} \).
  • Calculated as \( LCL = 5.125 \) and \( UCL = 5.295 \) for our scenario.
The \( \bar{x} \) chart is crucial in spotting trends or shifts in average outcome, guiding decisions for corrective actions.
Process Mean
The process mean is a fundamental metric in statistical process control. It represents the central tendency or average of a dataset, illustrating the "typical" value expected from a process over time.
  • Understanding the process mean helps in setting benchmarks and standards for quality.
  • The process mean here is \( \mu = 5.21 \), serving as a reference point for both the s chart and \( \bar{x} \) chart calculations.
  • Deviations from this mean are closely monitored through the control charts to identify abnormalities.
Analyzing the process mean allows companies to evaluate their production consistency and make necessary adjustments.
Standard Deviation
Standard deviation is a critical measure in statistics that indicates the amount of variation or dispersion in a set of data values. Understanding this concept is key to controlling process quality.In this exercise:
  • The standard deviation \( \sigma \) is 0.147, showing that the process variability is relatively low.
  • It provides the basis for calculating control limits, particularly in the s chart.
  • A smaller standard deviation indicates a more consistent process, while a larger value may suggest more variability.
By controlling standard deviation, businesses can ensure that their processes produce items consistently, maintaining quality and reliability.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

John recently lost weight, and during this time, he charted the number of calories consumed each day. His calorie consumption varied each day but was generally stable. There were some days when his calorie count was unusual. Sometimes his calorie intake was much higher and sometimes it was much lower than expected. Give several examples of special causes that might significantly increase or decrease John's calorie consumption on a given day.

A large chain of coffee shops records a number of performance measures. Among them is the time required to complete an order for a cappuccino, measured from the time the order is placed. Suggest some plausible examples of each of the following. (a) Reasons for common cause variation in response time. (b) s-type special causes. (c) \(\mathrm{x}^{-} \bar{x}\)-type special causes.

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?

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.

Choose a process that you know well. If you lack experience with actual business or manufacturing processes, choose a personal process such as ordering something over the Internet, paying a bill online, or recording a TV show on a DVR. Make a flowchart of the process. Make a causeand-effect diagram that presents the factors that lead to successful completion of the process.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.