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Natural Tolerance Limits (NTLs) are a range that describes the natural spread of a process. This range is critical to understanding the variability in quality characteristics. For a process that follows a normal distribution, NTLs are determined by the formula: \[ \mu \pm 3\sigma \]Where \( \mu \) is the process mean, and \( \sigma \) is the standard deviation. The limits hence cover 99.73% of the process outputs. It's like drawing a boundary within which most of the data points will fall under normal operations. In our exercise, when the process mean is 100 and \( \sigma = 6 \), the NTLs are calculated as:* Lower Limit: \( 100 - 18 = 82 \)* Upper Limit: \( 100 + 18 = 118 \)These calculations are crucial because they help in predicting whether a process is capable of meeting set specifications, such as \( 100 \pm 20 \). By comparing NTLs with specification limits, we get a clear idea of the process's natural variability and how likely it is to produce defects.

Fraction Defective refers to the proportion of products or outputs that fall outside the specified limits. Essentially, it shows how many units are not meeting the quality standards and may require rework or rejection. To find the fraction defective, we consider the specification limits and assess where the natural tolerance limits fall in relation to these. If NTLs lie within the specification limits, the fraction defective can be negligible. For example, in part (a) of our exercise, the specification limits are 80 to 120, and the NTLs (82 to 118) fall within these bounds, meaning the fraction defective is very low, almost zero. For part (b), with the mean adjusted to 106, the new NTLs are 88 to 124. In this case, the upper NTL exceeds the upper specification limit (120), indicating a potential for outputs to exceed quality expectations. Here, the fraction defective is computed using a Z-score: \[ Z = \frac{120 - 106}{6} = 2.33 \]The probability of Z exceeding 2.33 can be found using standard normal distribution tables or software, leading to an approximate defect rate of 0.99%.

The Process Capability Ratio (PCR) is a simple yet powerful way to assess a process's ability to produce outputs within specified limits. It is defined as:\[ PCR = \frac{USL - LSL}{6 \sigma} \]Where \( USL \) is the Upper Specification Limit, \( LSL \) is the Lower Specification Limit, and \( \sigma \) is the standard deviation. This ratio effectively measures how well the 6-sigma range fits within the specification intervals. In the exercise, using \( USL = 120 \), \( LSL = 80 \), and \( \sigma = 6 \), we compute:\[ PCR = \frac{40}{36} \approx 1.11 \]A PCR greater than 1 suggests that the process is generally capable of staying within limits, as there is a buffer between process variation and specification extremes. However, PCR doesn't account for whether the process mean is centered or shifted towards one limit, which could result in a different interpretation if not checked alongside other metrics.

Process Capability Index (PCR_k), sometimes denoted as Cpk, offers a more comprehensive insight into process capability by measuring not only the spread but also any potential shift in the process mean. It is calculated as:\[ PCR_k = \frac{1}{3} \min\left(\frac{USL - \mu}{\sigma}, \frac{\mu - LSL}{\sigma}\right) \]PCR_k considers both the upper and lower bound of the process performance concerning the mean. This makes it suitable for understanding how off-centered the process might be. For exercise (a), when \( \mu = 100 \), the PCR_k calculates to:\[ PCR_k = 1.11 \]This indicates a centered and capable process. However, in part (b) with mean adjusted to 106, PCR_k evaluates as:\[ PCR_k = 0.78 \]This lower value signifies that though the process capability, as highlighted by PCR, remains unaltered (1.11), the mean shift decreases its effectiveness due to higher defect probabilities, particularly at the upper specification limit.