91Ó°ÊÓ

The thickness of a metal part is an important quality parameter. Data on thickness (in inches) are given in the following table, for 25 samples of five parts each. $$\begin{array}{cccccc}\hline \begin{array}{l}\text { Sample } \\\\\text { Number }\end{array} & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} \\\\\hline 1 & 0.0629 & 0.0636 & 0.0640 & 0.0635 & 0.0640 \\\2 & 0.0630 & 0.0631 & 0.0622 & 0.0625 & 0.0627 \\\3 & 0.0628 & 0.0631 & 0.0633 & 0.0633 & 0.0630 \\\4 & 0.0634 & 0.0630 & 0.0631 & 0.0632 & 0.0633 \\\5 & 0.0619 & 0.0628 & 0.0630 & 0.0619 & 0.0625 \\\6& 0.0613 & 0.0629 & 0.0634 & 0.0625 & 0.0628 \\\7 & 0.0630 & 0.0639 & 0.0625 & 0.0629 & 0.0627 \\\8 & 0.0628 & 0.0627 & 0.0622 & 0.0625 & 0.0627 \\\9 & 0.0623 & 0.0626 & 0.0633 & 0.0630 & 0.0624 \\\10 & 0.0631 & 0.0631 & 0.0633 & 0.0631 & 0.0630 \\\11 & 0.0635 & 0.0630 & 0.0638 & 0.0635 & 0.0633 \\\12 & 0.0623 & 0.0630 & 0.0630 & 0.0627 & 0.0629 \\\13 & 0.0635 & 0.0631 & 0.0630 & 0.0630 & 0.0630 \\\14 & 0.0645 & 0.0640 & 0.0631 & 0.0640 & 0.0642 \\\15 & 0.0619 & 0.0644 & 0.0632 & 0.0622 & 0.0635 \\\16 & 0.0631 & 0.0627 & 0.0630 & 0.0628 & 0.0629 \\\17 & 0.0616 & 0.0623 & 0.0631 & 0.0620 & 0.0625 \\\18 & 0.0630 & 0.0630 & 0.0626 & 0.0629 & 0.0628 \\\19 & 0.0636 & 0.0631 & 0.0629 & 0.0635 & 0.0634 \\\20 & 0.0640 & 0.0635 & 0.0629 & 0.0635 & 0.0634 \\\21 & 0.0628 & 0.0625 & 0.0616 & 0.0620 & 0.0623 \\\22 & 0.0615 & 0.0625 & 0.0619 & 0.0619 & 0.0622 \\\23 & 0.0630 & 0.0632 & 0.0630 & 0.0631 & 0.0630 \\\24 & 0.0635 & 0.0629 & 0.0635 & 0.0631 & 0.0633 \\\25 & 0.0623 & 0.0629 & 0.0630 & 0.0626 & 0.0628 \\\\\hline\end{array}$$ (a) Using all the data, find trial control limits for \(\bar{X}\) and \(R\) charts, construct the chart, and plot the data. Is the process in statistical control? (b) Use the trial control limits from part (a) to identify outof-control points. If necessary, revise your control limits assuming that any samples that plot outside the control limits can be eliminated. (c) Repeat parts (a) and (b) for \(\bar{X}\) and \(S\) charts.

Step by step solution

01

Calculate the Average (") for Each Sample

For each sample, calculate the average thickness, denoted by \( \bar{x} \). For example, for Sample 1: \( \bar{x}_1 = \frac{0.0629 + 0.0636 + 0.0640 + 0.0635 + 0.0640}{5} = 0.0636 \). Repeat this calculation for all 25 samples.

02

Calculate the Range (R) for Each Sample

The range \( R \) is the difference between the maximum thickness and the minimum thickness within each sample. For Sample 1, \( R_1 = 0.0640 - 0.0629 = 0.0011 \). Repeat for all samples.

03

Calculate Overall Average of the Sample Means (\( \bar{\bar{x}} \))

Find the average of all the sample means calculated in Step 1. \( \bar{\bar{x}} = \frac{\sum \bar{x}}{25} \).

04

Calculate the Average of the Ranges (\( \bar{R} \))

Compute the average of all the ranges calculated in Step 2. \( \bar{R} = \frac{\sum R}{25} \).

05

Determine Control Limits for \( \bar{X} \) Chart

For \( \bar{X} \) chart, use the formulas: Upper Control Limit (UCL) = \( \bar{\bar{x}} + A_2 \cdot \bar{R} \), Lower Control Limit (LCL) = \( \bar{\bar{x}} - A_2 \cdot \bar{R} \), where \( A_2 \) is a factor based on the sample size. Here, \( A_2 \) for \( n = 5 \) is 0.577.

06

Determine Control Limits for R Chart

For R chart, the limits are calculated using: \( UCL = D_4 \cdot \bar{R} \) and \( LCL = D_3 \cdot \bar{R} \), where \( D_4 = 2.114 \) and \( D_3 = 0 \) for sample size 5.

07

Plot Data on the \( \bar{X} \) and R Charts

Using the calculated limits, plot each sample's mean and range to the corresponding charts. Identify any points outside the control limits.

08

Identify Out-of-Control Points and Revise Limits

If any points are outside the control limits, they are considered out-of-control. Exclude them and recalculate overall averages and control limits.

09

Calculate Sample Standard Deviation (s)

For each sample, calculate the standard deviation \( s \). For Sample 1, \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \).

10

Calculate Average of Standard Deviations (\( \bar{s} \))

Find the average of the standard deviations from Step 9.

11

Determine Control Limits for \( S \) Chart

Calculate control limits for \( S \) chart using \( UCL_s = B_4 \cdot \bar{s} \) and \( LCL_s = B_3 \cdot \bar{s} \), where \( B_4 = 2.114 \) and \( B_3 = 0.284 \).

12

Repeat \( \bar{X} \) Chart Control Limits using \( S \)

Adjust the \( \bar{X} \) chart limits using the \( \bar{s} \) with new factors: \( UCL = \bar{\bar{x}} + A_3 \cdot \bar{s} \) and \( LCL = \bar{\bar{x}} - A_3 \cdot \bar{s} \). \( A_3 \) for \( n = 5 \) is 1.427.

13

Plot \( \bar{X} \) and \( S \) Chart Data

Plot and check for any data points that fall outside of these limits. Revise the control limits by excluding any outliers.

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Key Concepts

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

Control Charts

Control charts are essential tools in statistical process control (SPC). They help monitor a process over time to detect any changes that may indicate a shift in quality. Imagine a control chart as a graph that displays how the process is behaving.

There are two main types involved in the exercise: the \( \bar{X} \) (or X-bar) chart and the R (range) chart:

• The \( \bar{X} \) chart tracks the mean or average of measurements from each sample. It shows whether the process mean stays consistent over time. This helps in identifying trends that can affect quality.
• The R chart, on the other hand, looks at the variation within each sample by showing the ranges. It highlights the fluctuations or spread in the sample data which might go unnoticed when just looking at the averages.

These charts are crucial in determining whether a process is in a state of statistical control, meaning stable and predictable, or if it needs adjustment. By plotting data points from sample means and ranges, we can see if they lie within calculated control limits. If points are outside these limits, it signals potential quality issues needing investigation.

Quality Control

Quality control is all about ensuring a product or service meets a certain standard. In the context of manufacturing, like with metal part thickness, quality control ensures that every piece produced matches the specifications desired.

In SPC, control charts are used specifically for this kind of quality management where:

• You set control limits (upper and lower) that define acceptable levels of variation in the process.
• Data points falling outside those limits are outliers signaling that something might be wrong with the process, warranting a closer look.
• Continuous monitoring via control charts helps in maintaining consistent quality and improving processes over time.

This discipline prevents defects, reduces waste, and aids in maintaining consistency, promoting customer satisfaction with uniform quality. By being proactive through quality control, entities can predict and prevent issues before they become significant problems.

Engineering Statistics

Engineering statistics plays a pivotal role in the analysis and improvement of manufacturing processes like in our exercise of determining metal thickness reliability. This branch of statistics applies mathematical tools to understand and optimize the production process.

Key statistical methods and terms used in process control include:

• Sample Means (\( \bar{x} \)): Represents the average thickness in each sample, used to identify if the process is on target or needs calibration.
• Range (R): Measures the difference between the highest and lowest values in a sample, providing insights into data variability.
• Standard Deviation (s): Indicates how much the individual measurements in a set differ from the average. Smaller values mean closer conformity to the desired quality.

Tools such as control charts use these statistical calculations and constructs to assess whether a process is stable and predictable. Good engineering statistics enhance decision-making, drawing powerful inferences from data to improve subsequent production cycles.