The Diameters Of Ball Bearings Manufactured By A Process Are

The diameters of ball bearings manufactured by a process are normally

The diameters of ball bearings manufactured by a process are normally distributed with a mean of 4.52 cm and a standard deviation of 0.04 cm. The specification limits for the diameter of a bolt are 4.50 ± 0.05 cm, meaning acceptable diameters are between 4.45 cm and 4.55 cm. Based on this information, the tasks are to determine the proportion of ball bearings that meet specifications, find the 99th percentile of the diameters, assess the impact of shifting the process mean to 4.50 cm on the proportion meeting specifications, and to determine the required standard deviation to ensure 99% of the bearings meet the specifications when the mean is set to 4.50 cm.

Paper For Above instruction

Understanding the variability and quality of manufactured products such as ball bearings is crucial for ensuring conformance to specifications that meet industry standards. When the diameters of ball bearings are normally distributed, statistical tools such as the normal distribution, Z-scores, and percentiles facilitate the analysis of how well the process performs in relation to customer requirements. This paper explores the application of these tools to evaluate the proportion of acceptable bearings, identify key percentiles, and analyze the effects of process adjustments on quality outputs.

Proportion of Ball Bearings Meeting Specifications

The initial step is to determine the proportion of ball bearings produced by the current process that meet the specified diameter limits of 4.45 cm to 4.55 cm. Given the mean (μ) of 4.52 cm and standard deviation (σ) of 0.04 cm, the probabilities associated with the limits can be calculated using the standard normal distribution (Z-distribution).

Calculating the Z-scores for the lower and upper specification limits:

  • Z for 4.45 cm: Z = (4.45 - 4.52) / 0.04 = -1.75
  • Z for 4.55 cm: Z = (4.55 - 4.52) / 0.04 = 0.75

Using standard normal distribution tables or software, find the probabilities corresponding to these Z-scores:

  • P(Z
  • P(Z

The proportion of bearings within the specification limits is the difference between these two probabilities:

Proportion = 0.7734 - 0.0401 = 0.7333 or approximately 73.33%. This indicates that under the current process settings, about 73.33% of the ball bearings meet the specified diameter requirements.

Calculating the 99th Percentile of Diameter

The 99th percentile diameter is the value below which 99% of the observations fall. To find this, identify the Z-score corresponding to the 99th percentile, which is approximately 2.33 (from Z-tables). Using the Z-score formula:

Diameter = μ + Z σ = 4.52 + 2.33 0.04 = 4.52 + 0.0932 = 4.6132 cm

Therefore, the 99th percentile diameter is approximately 4.613 cm. This means that 99% of the bearings will have diameters less than or equal to 4.613 cm under the current process conditions.

Impact of Setting the Mean to 4.50 cm on Proportion Meeting Specifications

Adjusting the process mean to 4.50 cm, while keeping the original standard deviation of 0.04 cm, shifts the distribution of diameters. The new Z-scores for the specification limits are:

  • Z for 4.45 cm: (4.45 - 4.50) / 0.04 = -1.25
  • Z for 4.55 cm: (4.55 - 4.50) / 0.04 = 1.25

Corresponding probabilities are:

  • P(Z
  • P(Z

The proportion now meeting specifications is:

Proportion = 0.8944 - 0.1056 = 0.7888 or approximately 78.88%. This improvement from 73.33% indicates that shifting the mean closer to the target increases the proportion of bearings that meet specifications.

Required Standard Deviation to Ensure 99% Conformance at Mean of 4.50 cm

To guarantee that at least 99% of the bearings meet the specification limits when the mean is 4.50 cm, determine the maximum allowable standard deviation (σ). The limits are 4.45 to 4.55 cm, with the mean at 4.50 cm, so the process must be tightly controlled.

Because the distribution should cover at least 99% within the limits, the total range (0.10 cm) should encompass approximately ±2.58 standard deviations (since 99% of data lies within ±2.58 Z-scores):

2.58 * σ = (4.55 - 4.50) = 0.05 cm (upper limit distance from mean)

Solving for σ: σ = 0.05 / 2.58 ≈ 0.0194 cm

Similarly, checking the lower limit confirms the symmetry, and this standard deviation ensures 99% of the bearings will be within the specification limits when the mean is accurately calibrated to 4.50 cm.

Therefore, to achieve this high level of quality, the process standard deviation must be reduced to approximately 0.0194 cm, highlighting the importance of tight process control in manufacturing.

Conclusion

This analysis underscores the critical role of statistical methods in manufacturing quality control. By calculating the proportion of components meeting specifications, identifying key percentiles, and understanding how process adjustments influence outcomes, manufacturers can optimize their processes for consistency and customer satisfaction. Adjustments in the process mean and variance significantly affect the percentage of acceptable products, emphasizing the need for precise calibration and control in production settings. These statistical insights support continuous improvement strategies, ultimately leading to higher quality products and reduced waste.

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