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# 统计代写|非参数统计代写Nonparametric Statistics代考|ST505 Charts for the Mean

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## 统计代写|非参数统计代写Nonparametric Statistics代考|Charts for the Mean

Example 3.9 A Phase II Shewhart $\bar{X}$ Control Chart for the Mean When Both $\mu$ And $\sigma$ Are Unknown

Column (a) of Table $3.13$ presents some simulated data from a normal distribution, which represent measurements taken from $m=25$ independent random samples, each of size $n=5$ on a type of wafer. Suppose that these are the reference data from an IC process that were obtained after a careful Phase I analysis. The mean and the standard deviation of each reference sample are shown in Columns (b) and (c) of Table $3.13$, respectively.

The estimator of the mean $\mu$ is the mean of the Phase I sample means or the grand mean
$$\overline{\bar{X}}=\frac{1.5119+1.4951+\cdots+1.5264}{25}=1.5056 .$$
As noted above, the estimators $\hat{\sigma}$ used in the control limits are based on either (i) the average of the sample ranges, $\bar{R}=0.3256$ or (ii) the average of sample standard deviations, $\bar{S}=0.1316$ or the pooled estimator $S_{p}=$ $\sqrt{\frac{\sum_{i=1}^{m} S_{i}^{2}}{m}}=0.1391$. Note that all three estimators of $\sigma$ are close to each other.
The Phase II control charts for the mean are given below.
\begin{aligned} U C L &=\overline{\bar{X}}+k \frac{\hat{\sigma}}{\sqrt{n}} \ C L &=\overline{\bar{X}} \ L C L &=\overline{\bar{X}}-k \frac{\hat{\sigma}}{\sqrt{n}} \end{aligned}

## 统计代写|非参数统计代写Nonparametric Statistics代考|Charts for the Standard Deviation

Here we illustrate the $R$ and the $S$ charts which can be used to monitor the spread and the standard deviation, respectively. We start with an example of the Shewhart $R$ chart.

Example 3.10 A Phase II Shewhart $R$ Control Chart for the Standard Deviation in the Unknown Parameter Case

The same data that were used to illustrate the Shewhart $\bar{X}$ chart are now used to illustrate the Shewhart $R$ chart in Case U. Recall that Column (a) of Table $3.15$ presents the measurements taken from 25 independent Phase I samples on wafers that are each of size $(n=5)$ from a normal distribution. The range of each sample is shown in Column (d) of Table 3.15. For these data, $\bar{R}=0.325$. First, for $n=5$, we find from Table $\mathrm{C}$ in Appendix A that $D_{3}=0$ and $D_{4}=2.114$. Thus, with $\bar{R}=0.325$, the 3 -sigma Phase II $R$ chart control limits in Case $\mathrm{U}$ for $\sigma$ are given by $L C L=0$ and $U C L=2.114 \times 0.325=0.688$ with $C L=0.325$. However, these control limits do not properly account for parameter estimation and are not expected to be accurate unless one has a large number of Phase I data. Conversely, from Table $\mathrm{H}$ in Appendix $\mathrm{A}$, for $m=25, n=5$ and $A R L_{0}=370$, we find that $D_{3}^{}(25,5)=0.16603$ and $D_{4}^{}(25,5)=2.32788$. Hence the probability limits-based Phase II Shewhart $R$ control limits for $\sigma$, using the estimator $\bar{R} / d_{2}$, are given by $L C L=D_{3}^{}(m, n) \bar{R}=(0.16603)(0.325)=0.054$ and $U C L=D_{4}^{}(m, n) \bar{R}=(2.32788)(0.325)=0.757$, with $C L=\bar{R}=0.325$. Note that, while the $L C L$ for the 3 -sigma $R$ chart is to be reset to 0 as it is negative, the $L C L$ for the probability limits chart is positive and no such adjustment is necessary. Figure $3.25$ displays the sample ranges, $R_{i}$, of Column (b), which are plotted on a Shewhart $R$ control chart together with the control limits.

## 统计代写|非参数统计代写NONPARAMETRIC STATISTICS代 考|CHARTS FOR THE STANDARD DEVIATION

$L C L=D_{3}(m, n) \bar{R}=(0.16603)(0.325)=0.054$ 和 $U C L=D_{4}(m, n) \bar{R}=(2.32788)(0.325)=0.757$ ，和 $C L=\bar{R}=0.325$. 请注意，虽然 $L C L$ 对于 3 .
sigma $R$ 图表将被重置为 0 ，因为它是负数， $L C L$ 因为概率限制图是正数，不需要这样的调整。数字 $3.25$ 显示样本范围， $R_{i}$, 列 $b$, 绘制在 Shewhart 上 $R$ 控制图和控 制限。

$U C L=\overline{\bar{X}}+k \frac{\hat{\sigma}}{\sqrt{n}} C L \quad=\overline{\bar{X}} L C L=\overline{\bar{X}}-k \frac{\hat{\sigma}}{\sqrt{n}}$

## 统计代写|非参数统计代写NONPARAMETRIC STATISTICS代 考|CHARTS FOR THE STANDARD DEVIATION

$L C L=D_{3}(m, n) \bar{R}=(0.16603)(0.325)=0.054$ 和 $U C L=D_{4}(m, n) \bar{R}=(2.32788)(0.325)=0.757$ ，和 $C L=\bar{R}=0.325$. 请注意，虽然 $L C L$ 对于 3 .
sigma $R$ 图表将被重置为 0 ，因为它是负数， $L C L$ 因为概率限制图是正数，不需要这样的调整。数字 $3.25$ 显示样本范围， $R_{i}$, 列 $b$, 绘制在 Shewhart 上 $R$ 控制图和控 制限。

## Matlab代写

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