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Use the following data. Five AC adaptors that are used to charge batteries of a cellular phone are taken from the production line each 15 minutes and tested for their direct- current output voltage. The output voltages for 24 sample subgroups are as follows: $$\begin{array}{c|ccccc} \text {Subgroup} & \multicolumn{3}{|c} {\text {Output Voltages of Five Adaptors}} \\ \hline 1 & 9.03 & 9.08 & 8.85 & 8.92 & 8.90 \\ 2 & 9.05 & 8.98 & 9.20 & 9.04 & 9.12 \\ 3 & 8.93 & 8.96 & 9.14 & 9.06 & 9.00 \\ 4 & 9.16 & 9.08 & 9.04 & 9.07 & 8.97 \\ 5 & 9.03 & 9.08 & 8.93 & 8.88 & 8.95 \\ 6 & 8.92 & 9.07 & 8.86 & 8.96 & 9.04 \\ 7 & 9.00 & 9.05 & 8.90 & 8.94 & 8.93 \\ 8 & 8.87 & 8.99 & 8.96 & 9.02 & 9.03 \\ 9 & 8.89 & 8.92 & 9.05 & 9.10 & 8.93 \\ 10 & 9.01 & 9.00 & 9.09 & 8.96 & 8.98 \\ 11 & 8.90 & 8.97 & 8.92 & 8.98 & 9.03 \\ 12 & 9.04 & 9.06 & 8.94 & 8.93 & 8.92 \\ 13 & 8.94 & 8.99 & 8.93 & 9.05 & 9.10 \\ 14 & 9.07 & 9.01 & 9.05 & 8.96 & 9.02 \\ 15 & 9.01 & 8.82 & 8.95 & 8.99 & 9.04 \\ 16 & 8.93 & 8.91 & 9.04 & 9.05 & 8.90 \\ 17 & 9.08 & 9.03 & 8.91 & 8.92 & 8.96 \\ 18 & 8.94 & 8.90 & 9.05 & 8.93 & 9.01 \\ 19 & 8.88 & 8.82 & 8.89 & 8.94 & 8.88 \\ 20 & 9.04 & 9.00 & 8.98 & 8.93 & 9.05 \\ 21 & 9.00 & 9.03 & 8.94 & 8.92 & 9.05 \\ 22 & 8.95 & 8.95 & 8.91 & 8.90 & 9.03 \\ 23 & 9.12 & 9.04 & 9.01 & 8.94 & 9.02 \\ 24 & 8.94 & 8.99 & 8.93 & 9.05 & 9.07 \end{array}$$ Plot an \(R\) chart.

Step by step solution

01

Calculate the Range for Each Subgroup

The range for each subgroup is the difference between the maximum and minimum output voltages. For Subgroup 1, calculate the range as follows: \( R_1 = \max(9.03, 9.08, 8.85, 8.92, 8.90) - \min(9.03, 9.08, 8.85, 8.92, 8.90) = 0.23 \). Repeat this for all subgroups.

02

Calculate the Average Range

Compute the average range \( \bar{R} \) by summing the ranges of all subgroups and dividing by the number of subgroups. If \( R_i \) is the range of the \( i \)-th subgroup, then \( \bar{R} = \frac{1}{24} \sum_{i=1}^{24} R_i \).

03

Determine the Control Limits

Using the average range \( \bar{R} \), calculate the upper control limit (UCL) and lower control limit (LCL) for the \( R \) chart. These limits are given by \( UCL = D_4 \times \bar{R} \) and \( LCL = D_3 \times \bar{R} \), where \( D_3 \) and \( D_4 \) are factors based on the subgroup size (5 in this case). Look up \( D_3 \) and \( D_4 \) values from statistical tables.

04

Plot the R Chart

Plot each subgroup range on a graph, with the subgroup number on the x-axis and the range values on the y-axis. Draw the average range \( \bar{R} \), upper control limit (UCL), and lower control limit (LCL) as horizontal lines across the plot to judge if the process is in control or not.

05

Analyze the R Chart

Examine the plotted \( R \) chart to check if any subgroup ranges are outside the control limits. If all points lie within the bounds and show a random pattern, the process variation is under control.

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Key Concepts

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

R chart

An R chart, also known as a range chart, is a type of control chart used to monitor the variability within a process, especially in manufacturing. Here, we are using it to examine the output voltage variation of AC adaptors. The R chart's primary purpose is to assess the stability of a process based on the range of data samples.

Each subgroup, in this case, consists of five AC adaptors tested every 15 minutes. The range is calculated by subtracting the smallest value from the largest value in each subgroup. The plotted data points on the R chart show these range values.

This chart is especially useful because it provides a straightforward way to spot inconsistencies in process variability. When the process is under control, the range should stay relatively constant, with fluctuations remaining between established control limits.

Process Control

Process control, in the context of the R chart, refers to the monitoring and adjustment of processes to ensure they produce consistent and quality outputs. The goal is to identify variations that could indicate problems before they cause faulty products.

In this AC adapter example, process control involves tracking voltage outputs. By examining these outputs through statistical methods, we can determine whether the process produces consistent results over time.

Effective process control usually involves regular sampling and analysis, such as calculating ranges and comparing them to historical data and control limits. As you plot the R chart and analyze its patterns, you ensure the process remains in control by detecting variation early and making necessary corrections.

Quality Control

Quality control is the overarching process of ensuring that a product meets the required specifications and standards. In this exercise, quality control revolves around checking if the AC adaptors' voltage output is consistent and within acceptable variations.

An R chart is one tool within this broader quality control framework. It helps detect early signs of process issues that might affect product quality. By maintaining process stability, your finished products are more likely to meet quality standards.

Quality control entails continuous observation, recording, and adjustments to processes. With effective quality control processes like using R charts, manufacturers can minimize defects, reduce waste, and enhance customer satisfaction by delivering superior products.

Statistical Tables

Statistical tables play a crucial role when setting up control limits for an R chart. These tables provide necessary factors, like the values of \( D_3 \) and \( D_4 \), which are used to calculate the upper and lower control limits of a process.

For an R chart, \( D_3 \) and \( D_4 \) values correspond to subgroup sizes and are used as multipliers for the average range to determine control limits. In this case, with a subgroup size of 5, you would refer to a table to obtain these constants to accurately compute the control limits.

Knowing how to use statistical tables effectively is vital in process control and quality management. Consistent application of these predetermined factors ensures accurate assessments of process stability and allows for valid conclusions regarding the control of a process.