SOLUTION: The length of a pencil is supposed to fit within the tolerance 7.25 Â± 0.05 in. However, it has been formally distributed with a tolerance of 7.2 Â± 0.02 in. However, due to ne

Question 1209556: The length of a pencil is supposed to fit within the tolerance 7.25 Â± 0.05 in. However, it has been formally distributed with a tolerance of 7.2 Â± 0.02 in. However, due to new machines, the tolerance has increased to 7.25 Â± 0.025 in. The mean has remained constant. What is the tolerance used to produce the same percentage of pencils before the increase?
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
Here's how to solve this problem:
**Understanding the Concept**
The tolerance represents the acceptable variation in pencil length. A smaller tolerance means more consistent lengths, but also potentially more rejects. We're looking for the tolerance that, when combined with the original mean, would result in the same *proportion* of acceptable pencils as the *new* tolerance.
**Steps**
1. **Determine the Z-score:** The tolerance is related to the standard deviation (a measure of how spread out the lengths are) and a Z-score. The Z-score tells us how many standard deviations away from the mean a particular length is. We need to find the Z-score that corresponds to the desired percentage of pencils within the tolerance. Since the problem implies we are looking at a 6-sigma process, we can assume that the percentage of pencils within the tolerance is 99.73%
2. **Calculate the Original Standard Deviation:** We can use the original tolerance to calculate the standard deviation of the process, using the Z-score we calculated above.
3. **Calculate the new standard deviation:** We can use the new tolerance to calculate the standard deviation of the process, using the Z-score we calculated above.
4. **Calculate the Equivalent Tolerance:** Now that we have the standard deviation of the new distribution, we can calculate the tolerance that would produce the same percentage of pencils under the original mean, using the Z-score we calculated above.
**Calculation**
1. **Z-score:** For a 99.73% confidence interval, the Z-score is approximately 3.
2. **Original Standard Deviation:** original_std = original_tolerance / 3
3. **New Standard Deviation:** new_std = new_tolerance / 3
4. **Equivalent Tolerance:** equivalent_tolerance = 3 * original_std
**Result**
The tolerance used to produce the same percentage of pencils before the increase is 0.05 in.

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