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```50288-21-12E AID: 59493 | 14/11/2017
Refer to Example 21.5 and Example 21.6 in textbook.
Consider the new pairs of measurement.
There are 5 subgroups with 1 sample per subgroup. There are 2 quality characteristics per sample.
Consider the formula to determine Hotelling’s statistic.
…… (1)
Where,
n denotes subgroup of measurements
denotes the subgroup sample means
k denotes the sampling instant
denotes the grand mean
S denotes the covariance matrix
Consider the formula to determine the covariance matrix.
…… (2)
Where,
N denotes the number of subgroups
is the mean of
i, j represents the BOD and Solids
Determine .
The mean is 26.02.
The mean is 1422.8.
Substitute the value of and in equation (2).
Solve for S.

Substitute 1 for k in equation (1)
Similarly determine for different samples.
Using excel a graph is drawn between and k by taking the sample k along the x-axis and along y axis. The chart is shown in Figure 1. The 99% control chart limit (UCL) is .
From Figure 1, the straight red line represents the UCL limit. The dots represent the values for different samples.
From Figure 1, a chart violation occurs at . The value exceeds the UCL limit. To overcome this behavior the measurement must be further investigated. Also, extra measurement is taken before the process operation is abnormal.
Thus, the value of Hotelling’s for each pair is calculated and the values are plotted. A chart violation occurs at since the value exceeds the UCL limit.```