Measure Phase

To solve this problem, we need to calculate the required standard deviation reduction to achieve Ppk = 1.33.

Given information:
- LSL = 145°C, USL = 175°C, Target = 160°C
- Process mean = 158°C
- Long-term standard deviation (σLT) = 4.8°C
- Current Ppk = 0.90
- Required Ppk = 1.33

Since the mean cannot be adjusted, we need to reduce the long-term standard deviation to improve Ppk.

Ppk is calculated as the minimum of:
- (USL - Mean) / (3 × σLT) = (175 - 158) / (3 × σLT) = 17 / (3 × σLT)
- (Mean - LSL) / (3 × σLT) = (158 - 145) / (3 × σLT) = 13 / (3 × σLT)

The limiting factor is the lower specification (13°C distance), which gives the smaller value and determines Ppk.

To find the required σLT for Ppk = 1.33:
1.33 = 13 / (3 × σLT_new)
3 × σLT_new × 1.33 = 13
σLT_new = 13 / (3 × 1.33) = 13 / 3.99 = 3.26°C

Percentage reduction required:
(4.8 - 3.26) / 4.8 × 100 = 1.54 / 4.8 × 100 ≈ 32%

The correct answer states that the long-term standard deviation must be reduced by approximately 32% from 4.8°C to approximately 3.25°C, which matches our calculation.

The other answers are incorrect because:
- Reducing short-term standard deviation affects Cpk, not Ppk, and Ppk is specifically what needs to meet the 1.33 requirement
- A 48% reduction would be excessive and is based on incorrect reasoning about target offset
- The suggestion about combined standard deviations and rational sampling strategies is not mathematically sound for addressing the Ppk deficiency

The mode is the value that appears most frequently in a dataset.

Let's count the frequency of each value in the dataset: 12, 15, 15, 18, 20, 15, 22

• 12 appears 1 time
• 15 appears 3 times
• 18 appears 1 time
• 20 appears 1 time
• 22 appears 1 time

Since 15 appears three times, which is more than any other value, 15 minutes is the mode of this dataset.

The other answers are incorrect because:

• The answer stating 17 minutes calculated by averaging all values describes the mean, not the mode. The mean would actually be (12+15+15+18+20+15+22)/7 = 117/7 ≈ 16.7 minutes.

• The answer stating 15 minutes because it falls in the middle when arranged in order describes the median, not the mode. While 15 happens to be both the median and mode in this case, the reasoning provided is incorrect for identifying the mode.

• The answer stating 16.7 minutes as the central tendency describes the arithmetic mean (average), not the mode.

The correct answer states that the process shows evidence of a potential shift during the first 7 points but returned to stable behavior.

This is correct because standard run rules for control charts include the rule that 7 or more consecutive points on one side of the centerline indicate a process shift or special cause variation. In this scenario, the first 7 points being above the centerline triggers this run rule, suggesting a potential shift occurred during that period. However, the subsequent 6 points being randomly distributed around the centerline indicates the process returned to normal, stable behavior.

The second option is incorrect because simply having all points within control limits does not guarantee process stability. Run rules exist specifically to detect patterns that indicate special causes even when points remain within limits. Seven consecutive points on one side of the centerline is a classic run rule violation that signals potential instability.

The third option is incorrect because the pattern of 7 consecutive points above the centerline actually indicates special cause variation, not common cause variation. There is no basis for recalculating control limits based on this observation.

The fourth option is incorrect because while the run of 7 points does indicate special cause variation for that portion of the data, the investigation should focus on that specific time period when the shift occurred. The subsequent 6 points showing random distribution around the centerline suggest the process has stabilized, so investigating all 20 subgroups equally would not be the most efficient approach. The focus should be on understanding what caused the shift during the first 7 subgroups.