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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

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