Quality Control A manufacturer of bolts has a qualitycontrol policy that requires it to destroy any bolts that are more than 2 standard deviations from the mean. The quality-control engineer knows that the bolts coming off the assembly line have a mean length of 8 cm with a standard deviation of 0.05 cm. For what lengths will a bolt be destroyed? 167

Verified step by step guidance

1. Identify the mean (\(\mu\)) and standard deviation (\(\sigma\)) of the bolt lengths. Here, \(\mu = 8\) cm and \(\sigma = 0.05\) cm.
2. Understand that bolts more than 2 standard deviations from the mean will be destroyed. This means bolts with lengths less than \(\mu - 2\sigma\) or greater than \(\mu + 2\sigma\) will be rejected.
3. Calculate the lower limit for acceptable bolt length using the formula: \(\mu - 2\sigma\).
4. Calculate the upper limit for acceptable bolt length using the formula: \(\mu + 2\sigma\).
5. Conclude that any bolt with length less than the lower limit or greater than the upper limit will be destroyed according to the quality control policy.

Key Concepts

Normal Distribution and Standard Deviation

The normal distribution is a symmetric, bell-shaped curve describing how data values are spread around the mean. Standard deviation measures the average distance of data points from the mean, indicating variability. In quality control, it helps determine acceptable ranges for product measurements.

Control Limits in Quality Control

Control limits define the boundaries within which a product's measurements are considered acceptable. Typically set at a certain number of standard deviations from the mean (e.g., ±2 SD), values outside these limits indicate defects or out-of-spec products that should be rejected or destroyed.

Application of Mean and Standard Deviation to Determine Rejection Criteria

Given the mean and standard deviation, the rejection criteria are calculated by adding and subtracting multiples of the standard deviation from the mean. For bolts, lengths beyond mean ± 2 standard deviations are destroyed, ensuring only bolts within the acceptable size range are kept.