The Quality Control Manager Shuts Down An Automatic Lathe

The Quality Control Manager Shuts Down An Automatic Lathe For Correcti

The quality control manager shuts down an automatic lathe for corrective maintenance whenever a sample of the manufactured items has an average diameter outside the interval (2.01, 2.03) where these figures are in centimeters. The lathe is designed to produce items whose diameters are normally distributed with mean 2.02 cm and standard deviation 0.005 cm. a) Determine the chance of the control manager stopping the lathe when it is operating as designed. b) Suppose that the machine wears out so that it makes with a mean diameter of 2.04 cm. Determine the probability that the lathe will be allowed to continue. c) An adjustment error caused the lathe to produce items with a mean diameter of 2.01 cm. Determine the probability that the lathe will be stopped.

Paper For Above instruction

In the realm of manufacturing, ensuring the quality of produced items is critical not only for maintaining customer satisfaction but also for optimizing operational efficiency. Automatic lathes are widely used in industry to mass-produce cylindrical parts with precise dimensions. However, machine variability and wear can lead to deviations from nominal specifications. Therefore, implementing effective quality control procedures, such as sampling and statistical analysis, becomes essential for early detection of deviations and maintaining consistent product quality. This paper explores the probability of shutdown of an automatic lathe based on sampling, considering different operational scenarios, and employs statistical principles rooted in normal distribution to analyze the problem comprehensively.

Introduction

Quality control in manufacturing often involves monitoring the output of machines to detect any deviations from the desired specifications. The common practice is to use sampling methods where a sample of products is inspected, and decisions are made based on statistical criteria. In the case of the automatic lathe under discussion, the control manager inspects the diameter of items produced and shuts down the machine if the average diameter of a sample falls outside a predefined interval. This process relies on the theory of sampling distributions, specifically the properties of the sample mean and the normal distribution, given the underlying assumptions about the process.

Statistical Foundation

The process assumes that individual diameters are independently normally distributed with a known standard deviation (σ) and a specified mean (μ). When a sample of size n is drawn, the sampling distribution of the sample mean (x̄) also follows a normal distribution with mean μ and standard error (SE) given by:

SE = σ / √n

where σ is the population standard deviation, and n is the sample size. The control rule applies if the sample mean falls outside the interval (2.01, 2.03). The probabilities of false alarms (Type I error) or missed detections (Type II error) can be computed by standard normal probability calculations.

Scenario Analyses

a) Operation as Designed

In the baseline scenario, the lathe produces items with a mean diameter μ = 2.02 cm and standard deviation σ = 0.005 cm. Assuming the control manager inspects a sample of size n (not specified, but typically large enough to ensure precision), the probability of stopping the machine when it is operating correctly corresponds to the probability that the sample mean falls outside the interval (2.01, 2.03).

Calculating the probability entails standardizing the bounds using the normal distribution:

Z = (x̄ - μ) / (σ / √n)

Probability that the sample mean

P(x̄

Similarly, for > 2.03:

P(x̄ > 2.03) = P(Z > (2.03 - 2.02) / (0.005 / √n))

Adding these two probabilities gives the overall chance of unnecessary shutdowns or false alarms under normal operation.

b) Machine Wears Out (μ = 2.04 cm)

When machine wear causes the mean diameter to shift to 2.04 cm, the probability the lathe continues operating without being shut down is the probability that the sample mean falls inside the interval (2.01, 2.03). This can be computed similarly by standardizing the bounds with the new mean:

P(2.01

This probability indicates the likelihood of a failure to detect the deterioration, potentially leading to defective products reaching consumers.

c) Adjustment Error (μ = 2.01 cm)

In this scenario, the process is skewed with a mean of 2.01 cm, closer to the lower threshold. The probability that the process is stopped (i.e., the sample mean falls outside the interval) can be calculated as:

P(x̄ 2.03) = P(Z (2.03 - 2.01) / (σ / √n))

Given the shifted mean, the focus is on the increased likelihood of the machine being prematurely shut down due to this error.

Discussion and Implications

The probabilities calculated in each scenario offer critical insights into the effectiveness and limitations of the current quality control procedure. For the operating process, a low false alarm probability indicates a robust system with minimal unnecessary shutdowns. Conversely, under wear-out conditions, a high probability of continued operation signifies a risk of defective products escaping detection. The case of adjustment errors highlights the sensitivity of the sampling plan to process shifts, emphasizing the need for calibration and process control.

Adjustments to the sample size n can greatly influence these probabilities, with larger samples reducing variability and improving detection sensitivity. The choice of control limits (2.01 and 2.03) also affects the false alarm and detection rates, and often, statistical quality control employs techniques such as Shewhart control charts to optimize these parameters.

Conclusion

Statistical analysis rooted in the normal distribution provides a powerful methodology for monitoring manufacturing processes. By understanding the probabilities of machine shutdown under different operational states, managers can make informed decisions about process adjustments and maintenance schedules. Implementing appropriate sampling plans and control limits is vital for balancing the trade-offs between false alarms and missed detections, ultimately ensuring consistent product quality and operational efficiency.

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