Question: In manufacturing, it is very important to control the variability of production in order to maintain the quality of the product. In the production of computer chips, the amount of “dope” laid down on the surface of a silicon wafer is measured at 10 different points on the surface. It is critical that the thickness not vary too much. Denote the 10 realized thickness measurements for one wafer as \[{{x}_{1}},\ldots ,{{x}_{10}}\]. A statistical model is proposed for the measurements: Let the 10 measurements be represented by the random variables \[{{X}_{1}},{{X}_{2}},...,{{X}_{10}}\], assumed to be a normal Random Sample, with common mean \[\mu \] and variance \[{{\sigma }^{2}}\], and the only assumption on the parameters being \[{{\sigma }^{2}}>0\].

a) Give values for a and b such that \[P\left( \frac{\sum\nolimits_{i=1}^{10}{{{({{X}_{i}}-\bar{X})}^{2}}}}{{{\sigma }^{2}}}<a \right)=0.025\] and \[P\left( \frac{\sum\nolimits_{i=1}^{10}{{{({{X}_{i}}-\bar{X})}^{2}}}}{{{\sigma }^{2}}}<b \right)=0.975\]. Discuss how this can be used to make a 95% confidence interval for \[{{\sigma }^{2}}\]. [You will need to use a table or a software function to get a and b. For example, there are tables inside the covers of your textbook, and Excel has the necessary functions – so does SAS.]

b) Suppose the realized measurements are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Provide the 95% confidence interval for \[{{\sigma }^{2}}\].