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Chapter 15: Problem 134

Suppose that a \(P\) chart with center line at \(\bar{p}\) with \(k\) -sigma control limits is used to control a process. There is a critical fraction defective \(p_{c}\) that must be detected with probability 0.50 on the first sample following the shift to this state. Derive a general formula for the sample size that should be used on this chart.

Short Answer

Expert verified
The sample size formula is \( n = \frac{k^2 (\bar{p}(1-\bar{p}))}{(p_c - \bar{p})^2} \).

Step by step solution

01

Understand the Problem

We need to derive a formula to calculate the sample size needed for the first sample to have a 50% probability of detecting a critical fraction defective \( p_c \) using a \( P \) chart. The \( P \) chart uses a center line at \( \bar{p} \) and \( k \)-sigma control limits.
02

Review the Structure of a P-Chart

A \( P \) chart plots the proportion of defective items. The center line is the average proportion \( \bar{p} \), and the control limits are set at \( \bar{p} \pm k \sqrt{\frac{\bar{p}(1-\bar{p})}{n}} \), where \( n \) is the sample size.
03

Set Up Probability of Detection

We want to find \( n \) such that the probability of detecting \( p_c \) equals 0.5. This means the probability that the sample proportion exceeds the control limits when \( p = p_c \) is 0.5.
04

Use Normal Approximation

Assume the binomial distribution of the sample proportion \( \hat{p} \) is approximately normal due to the central limit theorem. This means \( \hat{p} \) is normally distributed with mean \( p_c \) and variance \( \frac{p_c(1-p_c)}{n} \).
05

Set Up the Normal Distribution Equation

To find the sample size \( n \), we equate the standardized control limits with a normal distribution. Set up the equation: \( P(\hat{p} > \bar{p} + k \sqrt{\frac{\bar{p}(1-\bar{p})}{n}}) = 0.5 \), where \( \hat{p} \sim N(p_c, \sqrt{\frac{p_c(1-p_c)}{n}}) \).
06

Solve for Sample Size n

Calculate the sample size \( n \) from the normal condition, knowing that a 50% probability corresponds to a \( Z \)-score of 0. Thus, set \( \bar{p} + k \sqrt{\frac{\bar{p}(1-\bar{p})}{n}} = p_c \) and solve for \( n \):\[ n = \frac{k^2 (\bar{p}(1-\bar{p}))}{(p_c - \bar{p})^2} \]
07

Confirm Understanding and Formula

The derived formula \( n = \frac{k^2 (\bar{p}(1-\bar{p}))}{(p_c - \bar{p})^2} \) gives the required sample size to detect \( p_c \) with a probability of 0.5 using a \( P \) chart.

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Key Concepts

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

Sample Size Calculation
Calculating the sample size for a process control chart, like a P chart, is crucial to ensure that shifts or changes in the process are detected accurately. The sample size () determines how many data points we need to sample in order to consistently detect a critical fraction defective, denoted as \( p_c \).
  • The sample size must be large enough to ensure that any significant deviation from the expected defect rate is noticeable.
  • A large sample size increases reliability but also increases the cost and time involved in sampling.
  • The sample size calculation involves understanding the fraction defective and the variability expected in the process.
To derive the sample size formula mentioned in our problem, we use the fact that detection must occur with a probability of 50% when the process has shifted to the state defined by \( p_c \). Solving mathematical equations that incorporate the principles of normal distribution allows us to understand the required sample size for accurate detection.
Control Limits
Control limits are essential features of a P chart, as they help to determine whether the process is under control or whether it has deviated significantly. The center line represents the average proportion of defectives (\( \bar{p} \)), while the upper and lower control limits are calculated using standards deviations (also known as sigma level).
  • Upper control limit (UCL) and lower control limit (LCL) are calculated as: \( \bar{p} \pm k \sqrt{\frac{\bar{p}(1-\bar{p})}{n}} \).
  • The constant \( k \) represents the number of standard deviations from the average proportion.
  • Control limits help in differentiating between common cause variation and special cause variation.
Any point that falls outside of this range can indicate a potential issue that requires investigation. Understanding these limits is crucial for setting the boundaries within which process variations are acceptable.
Critical Fraction Defective
The critical fraction defective \( (p_c) \) is a key concept in quality control processes. It defines the threshold for what is considered an unacceptable defect rate in a production process.
  • It's a predetermined value considered too high and unacceptable, thus warranting corrective actions.
  • Detecting \( p_c \) is crucial because it marks a shift from the expected production quality level.
Understanding \( p_c \) helps teams set clear criteria for monitoring quality and taking timely action to rectify production issues before they become substantial problems. Calculating and identifying \( p_c \) is part of quality management and impacts how sample size and control limits are determined.
Normal Approximation
Normal approximation is a statistical technique used to simplify processes, especially when dealing with binomial distributions. In the context of P charts, it's often used to approximate the distribution of sample proportions.
  • It allows the application of z-scores to help analyze data by transforming the binomial distribution into a normal distribution.
  • This approximation is applicable when the sample size is large enough, thanks to the central limit theorem.
  • With the normal approximation, practitioners can utilize standard statistical methods to analyze and interpret data.
In our specific problem, normal approximation enables us to set up a normal distribution equation for deriving the sample size. By ensuring the detection probability is set to 50% when the fraction defective shifts to \( p_c \), the approximation facilitates simpler calculations and a more interactive analysis of control processes.

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Most popular questions from this chapter

The \(P C R\) for a measurement is 1.5 and the control limits for an \(\bar{X}\) chart with \(n=4\) are 24.6 and 32.6 . (a) Estimate the process standard deviation \(\sigma\). (b) Assume that the specification limits are centered around the process mean. Calculate the specification limits.

The purity of a chemical product is measured every two hours. The results of 20 consecutive measurements are as follows: $$\begin{array}{cccc}\text { Sample } & \text { Purity } & \text { Sample } & \text { Purity } \\\\\hline 1 & 89.11 & 11 & 88.55 \\\2 & 90.59 & 12 & 90.43 \\\3 & 91.03 & 13 & 91.04 \\\4 & 89.46 & 14 & 88.17 \\\5 & 89.78 & 15 & 91.23 \\\6 & 90.05 & 16 & 90.92 \\\7 & 90.63 & 17 & 88.86 \\\8 & 90.75 & 18 & 90.87 \\\9 & 89.65 & 19 & 90.73 \\\10 & 90.15 & 20 & 89.78\end{array}$$ (a) Set up a CUSUM control chart for this process. Use \(\sigma=\) 0.8 in setting up the procedure, and assume that the desired process target is \(90 .\) Does the process appear to be in control? (b) Suppose that the next five observations are 90.75,90.00 , \(91.15,90.95,\) and \(90.86 .\) Apply the CUSUM in part (a) to these new observations. Is there any evidence that the process has shifted out of control?

Consider an \(\bar{X}\) control chart with k-sigma control limits and subgroup size \(n\). Develop a general expression for the probability that a point plots outside the control limits when the process mean has shifted by \(\delta\) units from the center line.

The following data are the number of spelling errors detected for every 1000 words on a news Web site over 20 weeks. $$\begin{array}{cccc}\hline \text { Week } & \begin{array}{l}\text { No. of Spelling } \\ \text { Errors }\end{array} & \text { Week } & \begin{array}{l}\text { No. of Spelling } \\\\\text { Errors }\end{array} \\\\\hline 1 & 3 & 11 & 1 \\\2 & 6 & 12 & 6 \\\3 & 0 & 13 & 9 \\\4 & 5 & 14 & 8 \\\5 & 9 & 15 & 6 \\\6 & 5 & 16 & 4 \\\7 & 2 & 17 & 13 \\\8 & 2 & 18 & 3 \\\9 & 3 & 19 & 0 \\\10 & 2 & 20 & 7 \\\\\hline\end{array}$$ (a) What control chart is most appropriate for these data? (b) Using all the data, compute trial control limits for the chart in part (a), construct the chart, and plot the data. (c) Determine whether the process is in statistical control. If not, assume that assignable causes can be found and outof-control points eliminated. Revise the control limits.

The following table of data was analyzed in \(Q u a l\) ity Engineering [1991-1992, Vol. 4(1)]. The average particle size of raw material was obtained from 25 successive samples. $$\begin{array}{crcl}\hline \text { Observation } & \text { Size } & \text { Observation } & \text { Size } \\\\\hline 1 & 96.1 & 14 & 100.5 \\\2 & 94.4 & 15 & 103.1 \\\3 & 116.2 & 16 & 93.1 \\\4 & 98.8 & 17 & 93.7 \\\5 & 95.0 & 18 & 72.4 \\\6 & 120.3 & 19 & 87.4 \\\7 & 104.8 & 20 & 96.1 \\\8 & 88.4 & 21 & 97.1 \\\9 & 106.8 & 22 & 95.7 \\\10 & 96.8 & 23 & 94.2 \\\11 & 100.9 & 24 & 102.4 \\\12 & 117.7 & 25 & 131.9 \\\13 & 115.6 & &\end{array}$$ (a) Using all the data, compute trial control limits for individual observations and moving-range charts. Construct the chart and plot the data. Determine whether the process is in statistical control. If not, assume that assignable causes can be found to eliminate these samples and revise the control limits. (b) Estimate the process mean and standard deviation for the in-control process.
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