Problem 73 - Applied Statistics and Probability for Engineers

Question

Weights of parts are normally distributed with variance \(\sigma^{2}\). Measurement error is normally distributed with mean 0 and variance \(0.5 \sigma^{2}\), independent of the part weights, and adds to the part weight. Upper and lower specifications are centered at \(3 \sigma\) about the process mean. (a) Without measurement error, what is the probability that a part exceeds the specifications? (b) With measurement error, what is the probability that a part is measured as being beyond specifications? Does this imply it is truly beyond specifications? (c) What is the probability that a part is measured as being beyond specifications if the true weight of the part is \(1 \sigma\) below the upper specification limit?

Step-by-Step Solution

Answer

a) 0.27% part exceeds limits without error. b) Increased probability with error, may not truly exceed. c) Measure probability using shifted distribution.

Step 1: Understanding the Specifications

The upper and lower specifications are set at \(3\sigma\) from the process mean, meaning the interval from \(-3\sigma\) to \(+3\sigma\) covers 99.73% of the data without measurement error.

Step 2: Calculating Probability Without Error

The probability that a part exceeds \(-3\sigma\) to \(+3\sigma\) limits without error is simply the probability outside this range in a standard normal distribution. This is \(1 - 0.9973 = 0.0027\), or 0.27%.

Step 3: Introducing Measurement Error

Measurement error has a variance of \(0.5\sigma^2\) and adds to the part weight, which changes the overall variance to \(1.5\sigma^2\) when measuring parts. Thus, the total standard deviation while measuring is \(\sqrt{1.5}\sigma\).

Step 4: Calculating Probability With Error

With the new variance, calculate the Z-score using the formula \(Z = \frac{x - \mu}{\sqrt{1.5}\sigma}\) for the interval limits \([-3\sigma, +3\sigma]\). The probability outside this range now increases because the interval size is effectively reduced.

Step 5: Calculating True Beyond Specifications

A part measured beyond \(\pm3\sqrt{1.5}\sigma\) may not truly be beyond \(\pm3\sigma\) due to measurement error. Hence, when a part is measured beyond specifications, it does not automatically mean it is truly beyond specifications, as errors could lead to false positives.

Step 6: Considering Specific True Weight

For a part truly at \(1\sigma\) below the upper specification (which is at \(2\sigma\) from the mean), use the normal distribution with measurement error variance. The probability that a part is measured as beyond \(3\sigma\) can be found by integrating the likelihood past \(3\sqrt{1.5}\sigma - 2\sigma\), considering the error.

Step 7: Short Conclusion

With measurement error, measurement extends the limits and increases the probability of measuring beyond specifications, though this may not imply the part truly exceeds without error.

Key Concepts

Normal distribution

Normal distribution is a crucial concept in probability theory often represented by a bell-shaped curve. Its symmetrical pattern around the mean indicates that data near the mean are more frequent in occurrence than data far from the mean. In many natural phenomena, such as weights of manufactured parts, measurements tend to follow this pattern.

Key characteristics of a normal distribution include:

• The mean, median, and mode are equal and located at the center of the distribution.
• It is described by two parameters: the mean (\( \mu \)) which determines the location, and the standard deviation (\( \sigma \)) which impacts the spread.
• The total area under the normal distribution curve is equal to 1, which represents the total probability.

In our exercise, the weights of parts are normally distributed with a variance \( \sigma^2 \). The specification limits are calculated using this distribution to ensure quality control, thus covering 99.73% of all parts within \( \pm 3\sigma \) from the mean.

Measurement error

Measurement error refers to the variation between the true value and the measured value of a part. In many scenarios, it is assumed to be normally distributed and is an inherent issue in measurement processes.

Key aspects of measurement error:

• It has its own variance, which describes the spread or distribution of the error around the true value.
• Our exercise suggests that the variance of the measurement error is \( 0.5\sigma^2 \), indicating its independency and randomness.
• Measurement error can significantly affect the reliability of measurements if not taken into account, as seen in instruments or processes with limited precision or accuracy.

This error adds to the part weight, modifying the overall variance and impacting the probability calculations significantly for determining whether a part meets specifications.

Specification limits

Specification limits define the range within which a product or part must fall to be considered acceptable. These boundaries are crucial in quality control to ensure products meet consumer expectations and regulatory requirements.

Points about specification limits:

• These limits are typically set based on the process mean and specific multiples of the standard deviation.
• In our case, the limits are set at \( \pm 3\sigma \) from the mean, encapsulating 99.73% of all acceptable data without error.
• They act as thresholds for parts to either pass or fail quality checks based on measurements.

When measurement errors are introduced, parts can appear to fall outside these specification limits because the effective range is reduced, leading to potential incorrect rejections of acceptable parts.

Z-score

The Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean.

Highlights of Z-score:

• It helps understand how far or close a specific data point is from the average.
• The formula used is \( Z = \frac{x - \mu}{\sigma} \), where \( x \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
• In cases where measurement error is present, the standard deviation would adjust, hence the original formula adapts as \( Z = \frac{x - \mu}{\sqrt{1.5}\sigma} \).

The Z-score becomes an essential tool in the exercise to determine the probability of a part's weight exceeding specification limits and assessing the true consistency of part measurements.