Question

I need to prove bimultivariate below equation follow chi-square degree of freedom number2

at the sample meun vector and sample covariance matri omesponding popolation quantities; that is /(X)-μ and E(S)-z 514 Chapher 11Multivariate SP We can show th ndcontrolprocedure is the Hot It is a direct analog of the the Hotelling T 11.3 The Hotelling Control Chart The most familiar multivariate process-monitoringa control chart for monitoring the mean vectoro variate Shewhart i chart. We present two versions grouped data, and another for individual observations the ini of the Hotelling 12 chart one 11.3.1 Subgrouped Data uppose that two quality characteristics xi and x2 are jointly distributed accordin that two quality characteristics 시 und ein characteristics and let ơ, and σ be the standard deviations of ri and r2-respectivequality covariance between r, and is denoted by ơ12 We assume that Ơi q, and ơn are y ne variate normal distribution (see Fig. 11.3), Let u and Ho be the mean values of the r t, and x2 are the sample averages of the two quality characteristics computed fro K ple of size n, then the statistic a sam will have a chi-square distribution with 2 degrees of freedom. This equation can be used as the basis of a control chart for the process means μί and μ. If the process means remain at the ues μ1 and μ2, then values ofr, should be less than the upper control limit UCL- val. If at least one of the means shifts to some new (out-of-control) value, then the probability tha the statistic o exceeds the upper control limit increases. The process-monitoring procedure may be represented graphically. Consider the case in which the two random variables r,ad are independent, that is. σ12-0. Ifo,2-0. ten equation 1 1.1 I defines an ellipse centered at(μί, μ2) with principal axes parallel to the x axes, as shown in Figure 11.4. Taking χō in equation 11.11 equal to χα, 2 implies that a pair of sample averages ielding a value of o plotting inside the ellipse indicates that the process is in control, whereas if the corresponding value of χ0 plots outside the ellipse the process is out of control. Figure 11.4 is often called a control ellipse In the case where the two quality characteristics are dependent, then σ1-0, and he corresponding control ellipse is shown in Figure 11.5. When the two variables are dependent, the principal axes of the ellipse are no longer parallel to the T1,x2 axes. Also, note that sam- ple point number I1 plots outside the control ellipse, indicating that an assignable cause is present, yet point 11 is inside the control limits on both of the individual control charts for i and 32. Thus there is nothing apparently unusual about point 11 when the variables are viewed individually, yet the customer who received that shipment of material would quite likel observe very different performance in the product. It is nearly impossible to detect an assign able cause resulting in a point such as this one by maintaining individual control charts.

Answer 1

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