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Chapter 7: Problem 58

A hardwoods manufacturing plant has a production line designed to produce baseball bats weighing 32 ounces. During a period of time when the production process was known to be in statistical control, the average bat weight was found to be 31.7 ounces. The observed data were gathered from 50 samples, each consisting of 5 measurements. The standard deviation of all samples was found to be \(s=.2064\) ounces. Construct an \(\bar{x}\) -chart to monitor the 32 -ounce bat production process.

Short Answer

Expert verified
Answer: The upper control limit (UCL) is 31.8191 ounces, and the lower control limit (LCL) is 31.5809 ounces.

Step by step solution

01

Calculate the average and sample size

The given average of the bat weights is \(\bar{x} = 31.7\) ounces. Given that there are 50 samples with 5 measurements each, the sample size \(n = 5\) and the total number of samples is \(k = 50\).
02

Calculate the standard deviation

The given standard deviation of all samples is \(s = .2064\) ounces.
03

Calculate the control limits

To calculate the control limits, we first need to determine the average of the standard deviations, which can be expressed as \(\bar{s} =.2064\). Then, we need to find the control chart factors \(A_2\) for a sample size of 5. The \(A_2\) value can be found in statistical quality control books or online tables. For a sample size of 5, \(A_2 = 0.577\). Now, we can calculate the control limits for the \(\bar{x}\)-chart: Upper control limit (UCL): \(UCL = \bar{x} + A_2\bar{s} = 31.7 + 0.577(0.2064) = 31.7 + 0.1191 = 31.8191\). Lower control limit (LCL): \(LCL = \bar{x} - A_2\bar{s} = 31.7 - 0.577(0.2064) = 31.7 - 0.1191 = 31.5809\).
04

Construct the \(\bar{x}\)-chart

To construct the \(\bar{x}\)-chart, plot the average bat weight, the upper control limit, and the lower control limit on the vertical axis, and the sample number on the horizontal axis. 1. Mark the average bat weight of 31.7 ounces as a solid horizontal line across the chart. 2. Mark the upper control limit of 31.8191 ounces as a dashed horizontal line above the average. 3. Mark the lower control limit of 31.5809 ounces as a dashed horizontal line below the average. 4. For each of the 50 samples, plot their average bat weight on the chart, connecting the points with lines. The chart should show the average bat weight of each sample and its variation with the control limits. If all points are within the control limits, the production process can be considered stable. Any points outside the control limits indicate an out-of-control process.

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Key Concepts

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

Control Chart
A Control Chart is a graphical tool used in statistical quality control to monitor a process over time. It helps to determine if a manufacturing or business process is in a state of statistical control.
The core idea is to visualize variations of a process and identify whether they are within acceptable limits, or if they signify an "out of control" condition.
In the context of the hardwood manufacturing plant, a control chart is used to monitor the production of 32-ounce baseball bats. By plotting the average weight of samples, any variations from the expected performance can be quickly identified.
Control charts typically include:
  • A central line (CL), representing the average or target value of a process.
  • Upper and lower control limits (UCL and LCL) to signify the acceptable range of variations.
  • Data points that show measurements over time, which are plotted and analyzed.
This data visualization is crucial for ongoing quality assurance and can guide the troubleshooting and improvements within a process.
Standard Deviation
Standard Deviation is a measure that quantifies the amount of variation or dispersion in a set of values. In statistical quality control, it's used to determine process variability and monitor consistency.
In the exercise, the standard deviation (\( s = 0.2064 \) ounces) reflects the spread in bat weights around the average weight of 31.7 ounces.
A small standard deviation means the data points tend to be close to the mean, implying more consistency in the process. Conversely, a larger standard deviation indicates more variation which may mean a less stable process.
Understanding standard deviation is essential for setting control limits in a chart. It helps define the typical range of variations within which a process is considered to be "in control."
Sample Size
Sample Size (\( n \)) refers to the number of observations in each sample used to monitor a process. It directly affects the reliability of the control chart's indications.
In the exercise, each sample consists of 5 measurements, and there are 50 such samples. This size is crucial, as it influences the calculation of control limits and the overall sensitivity of the control chart.
A larger sample size generally provides more reliable data, minimizing the effect of variability in individual measurements. However, too large of a sample can be costly and may not provide significantly better insights in some cases. Therefore, selecting an optimal sample size is key to obtaining accurate results efficiently.
Upper Control Limit
The Upper Control Limit (UCL) is the highest value that a process can reach and still be considered "in control." It represents the threshold above which variations in a process' output are deemed unacceptable.
For the bat production line, the UCL was calculated to be 31.8191 ounces. This is determined using the mean weight (\( \bar{x} \)) and a factor of standard deviation (\( A_2 \)) specific to the sample size. It is crucial in monitoring as it helps to determine if the process is drifting upwards, indicating possible issues like overfilling or excess material usage.
By keeping outputs below the UCL, quality control is maintained, ensuring consistent product output.
Lower Control Limit
The Lower Control Limit (LCL) defines the lowest boundary within which process variations are considered "in control." It acts as a guardrail to ensure that the process doesn't produce output too low in value.
In the hardwood baseball bat context, the LCL is set at 31.5809 ounces. Calculated similarly to the UCL but marking the acceptable lower range, it prevents issues like underweight products which may be due to inadequate material usage or processing errors.
Adhering to the LCL ensures that the output consistently meets minimum quality and performance expectations. Ensuring processes run within both the upper and lower limits maintains stability and reliability.

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