Weights of parts are normally distributed with varianceσ2. measurement error is normally distributedwith mean zero and variance 0.5σ2, independent ofthe part weights, and adds to the part weight. Upper andlower specifications are centered at 3σ about the processmean.
a) Without measurement error, what is the probability that apart does not meet the specifications
b) Without measurement error, what is the probability that apart is measured as beyond specifications? Does this imply itis truly beyond specifications?
To answer these questions, we need to understand the concept of the normal distribution, as well as how to calculate probabilities using this distribution.
a) Without measurement error, the weight of the parts follows a normal distribution with variance σ^2. The upper and lower specifications are centered at 3σ about the process mean. Let's denote the mean of the part weights as μ.
To calculate the probability that a part does not meet the specifications, we need to find the area under the normal distribution curve that lies outside the region defined by the upper and lower specifications. Since the specifications are centered at 3σ about the mean, any part weight that falls outside this range will not meet the specifications.
To calculate this probability, we need to find the areas under the normal distribution curve beyond the upper and lower specification limits. We can use a standard normal distribution table or a statistical calculator to find these probabilities.
For example, if we assume a standard normal distribution (mean = 0 and variance = 1), and the upper specification limit is 3σ and the lower specification limit is -3σ, then we can find the probability as follows:
P(Part does not meet specifications) = P(Z < -3) + P(Z > 3)
Here, Z represents the standardized value obtained by subtracting the mean (μ) and dividing by the standard deviation (σ). In this case, since we assumed a standard normal distribution, Z represents the number of standard deviations away from the mean.
Using a standard normal distribution table or a statistical calculator, we can find the probabilities associated with the values -3 and 3 (or their absolute values if using a one-sided table). These probabilities give us an estimate of the probability that a part does not meet the specifications, assuming no measurement error.
b) Without measurement error, the probability that a part is measured as beyond specifications depends on how close the measured weight is to the specification limits. If the measured weight is beyond the specification limits, it implies that the measurement is outside the acceptable range set by the specifications.
However, it's important to note that a measurement beyond the specifications does not necessarily mean the part is truly beyond the specifications. This is because measurement error can introduce variability and affect the accuracy of the measurement. In this case, the measurement error is normally distributed with mean zero and variance 0.5σ^2, independent of the part weights.
To calculate the probability that a part is measured as beyond specifications, we need to consider the probability of both the part weight being beyond the specifications and the measurement error adding to or subtracting from the weight to yield a value beyond the specifications.
The probability calculation becomes more complex as we need to consider two distributions: one representing the part weights and the other representing the measurement error. We would need to convolve these distributions to obtain the joint distribution, which can then be used to calculate the probability.
Ultimately, the probability that a part is measured as beyond specifications does not directly imply that it is truly beyond specifications. It is important to carefully consider the measurement error and its impact on the measurement. Statistical techniques such as tolerance intervals and process capability analysis can be used to provide a more robust analysis of whether a part is truly beyond specifications.