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Answer :
The c-Chart is used to monitor defects per minivan. The center line, upper control limit, lower control limit, and process control are determined. The probability of a Type I Error is calculated.
1. The Center Line of the control chart represents the average number of defects per minivan. It is calculated by finding the average of the sample defects: (12 + 6 + 7 + 15 + 14 + 7) / 6 = 61 / 6 ≈ 10.167.
2. The Upper Control Limit (UCL) is calculated using the formula: UCL = Center Line + 3 * √(Center Line) = 10.167 + 3 * √(10.167) ≈ 24.157.
3. The Lower Control Limit (LCL) is calculated using the formula: LCL = Center Line - 3 * √(Center Line) = 10.167 - 3 * √(10.167) ≈ -3.824. Since the LCL cannot be negative, it is set to zero.
4-6). To determine if the process is "In Control" or "Out of Control" at each time period, compare the number of defects detected to the control limits. If the number of defects is within the control limits, the process is "In Control"; otherwise, it is "Out of Control".
7. The probability of a Type I Error is the alpha risk (9%) specified in the problem. Therefore, the probability that the process is actually "In Control" when it is concluded as "Out of Control" is 0.09, or 9%. This represents the chance of incorrectly rejecting the null hypothesis and concluding that the process is "Out of Control" when it is actually "In Control".
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