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统计代写|统计推断代写STATISTICAL INFERENCE代写|ST502 Shewhart Control Charts

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统计代写|统计推断代写STATISTICAL INFERENCE代写|Shewhart Control Charts for Monitoring Process Mean

For monitoring the mean of a process, we typically use the sample mean $(\bar{X})$. An example follows.
Example 3.1 A Parametric Shewhart $\bar{X}$ Control Chart
Shewhart charts are typically applied with subgroup data. Column (a) of Table $3.4$ presents some simulated data from a normal distribution, which represent measurements taken from 25 independent samples, each of size 5 $(n=5)$ on a type of wafer. Suppose that, from engineering considerations, the IC mean dimension of the wafers $\mu_{0}$ is specified to be $1.5 \mathrm{~cm}$ and the IC process standard deviation $\sigma_{0}$ is known to be equal to $0.1 \mathrm{~cm}$. The mean of each sample is shown in Column (b) of Table 3.4. To illustrate the calculations, consider sample number 1 . The first charting statistic, the mean of sample 1 , is calculated as follows
$$\bar{X}{1}=\frac{1.3235+1.4128+1.6744+1.4573+1.6914}{5}=\frac{7.5594}{5}=1.5119$$ To apply the formulas for the control limits in Equation 3.1, note that the expected value of $\bar{X}$ (i.e., $\mu{\bar{X}}$ ) is simply the specified process mean $\mu_{0}$, whereas the standard deviation of $\bar{X}$, namely, $\sigma_{\bar{X}}$, is equal to $\frac{\sigma_{0}}{\sqrt{n}}$. Thus, the $C L$ and the $k$-sigma control limits for a Shewhart $\bar{X}$ control chart are given by
\begin{aligned} U C L &=\mu_{0}+k \frac{\sigma_{0}}{\sqrt{n}} \ C L &=\mu_{0} \ L C L &=\mu_{0}-k \frac{\sigma_{0}}{\sqrt{n}} \end{aligned}

统计代写|统计推断代写STATISTICAL INFERENCE代写|Shewhart Control Charts for Monitoring Process Variation

Variation is an important aspect of any analysis and thus it is necessary to monitor the process variation or spread and ensure that it is IC. Moreover, as we see in Equation 3.1, the Shewhart control limits for the process mean depend on the process standard deviation. Thus, unless the standard deviation remains IC, the control chart for the mean will not be very informative. So, we need to monitor the variance or the standard deviation using a control chart.

There are several possible statistics that can be used to monitor variation. The most popular choices are the sample range $(R)$, the sample standard deviation $(S)$, and the sample variance $\left(S^{2}\right)$.

Typically, we use a control chart to monitor the process mean together with a control chart to monitor the process variation. If the variation is IC, we go ahead and examine the control chart for the mean. For example, a Shewhart $\bar{X}$ chart for the mean is often used together with a Shewhart $R$ chart for the spread. Note that, for illustration, we consider the Shewhart $R$ chart even though recent literature recommends using a different spread chart, such as the Shewhart $S$ chart; see, for instance, Mahmoud et al. (2010). We do this because the Shewhart $R$ chart is simple and continues to be used in the industry.

In Case $\mathrm{K}$, the values of $\mu$ and $\sigma$ are known or are specified so that they can be used to construct the respective control charts. We illustrate the Shewhart $R$ and $S$ charts for the known standard deviation $\sigma_{0}$.

统计代写|统计推断代写STATISTICAL INFERENCE代写|Shewhart Control Charts for Monitoring Process Mean

Shewhart 图通常与子组数据一起应用。表 (a) 栏 $3.4$ 呈现一些来自正态分布的模拟数据，这些数据代表从 25 个独立样本中获取 的测量值，每个样本大小为 $5(n=5)$ 在一种晶圆上。假设，从工程考虑，晶片的 IC 平均尺寸 $\mu_{0}$ 被指定为 $1.5 \mathrm{~cm}$ 和 IC 工艺标准 差 $\sigma_{0}$ 已知等于 $0.1 \mathrm{~cm}$. 每个样本的平均值显示在表 $3.4$ 的 (b) 栏中。为了说明计算，请考虑样本编号 1 。第一个图表统计量，即 样本 1 的平均值，计算如下
$$\bar{X} 1=\frac{1.3235+1.4128+1.6744+1.4573+1.6914}{5}=\frac{7.5594}{5}=1.5119$$

$$U C L=\mu_{0}+k \frac{\sigma_{0}}{\sqrt{n}} C L \quad=\mu_{0} L C L=\mu_{0}-k \frac{\sigma_{0}}{\sqrt{n}}$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。