### Implementation of SPC Program and Scrap Rate AnalysisSince the implementation of an SPC (Statistical Process Control) program in the foundry department in January 1990, engineers made changes to their operations. These changes resulted in a reduction of the scrap rate from the 1989 average rate of 28%, with a standard deviation of 2%.#### Weekly Scrap Rate Data for Early 1990The following table summarizes the weekly scrap rate data for the first half of 1990:| i | Xi ||----|-----|| 1 | 26.4|| 2 | 25.3|| 3 | 31.8|| 4 | 27.1|| 5 | 28.5|| 6 | 20.2|| 7 | 19.3|| 8 | 22.7|| 9 | 20.4|| 10 | 19.5|| 11 | 21.7|| 12 | 24 || 13 | 22.6|#### Questions and Analysis(a) **CUSUM Chart Construction**: Construct a CUSUM (Cumulative Sum Control Chart) to analyze the data for verifying the engineer’s claim regarding the scrap reduction. Use the parameters \( h = 5 \) and \( k = 0.25 \).(b) **Process Control**: Determine when the process was last in control.(c) **Magnitude of Mean Shift**: Calculate the magnitude of the mean shift observed. ### Explanation of CUSUM ChartA CUSUM chart is utilized to detect small shifts in the process mean by plotting cumulative sums of deviations from a target value. It is particularly useful in quality control and process monitoring.The parameters \( h \) (decision interval) and \( k \) (reference value) are critical in setting the sensitivity of the chart—lower values can detect smaller shifts more quickly but may also increase false alarms.

Given,ixi126.4225.3331.8427.1528.5620.2719.3822.7920.41019.51121.712241322.6(a)Define…

Answered: (a) Construct a CUSUM chart to analyze…

A First Course in Probability (10th Edition)

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(a) Construct a CUSUM chart to analyze the data for the purpose of verifying the engineer's claim regarding the scrap reduction. (h=5 and k=0.25). (b) When is process last in control? (c) What is the magnitude of the mean shift?