3) A company that manufactures oil seals found the population mean to be 49.15 mm (1.935 in.), the population standard deviation to be 0.51 mm (0.020 in.), and the data to be normally distributed. If the internal diameter of the seal is below the lower specification limit of 47.80 mm, the part is reworked. However, if it is above the upper specification limit of 49.80 mm, the seal is scrapped. (a) What percentage of the seals is reworked? What percentage is scrapped? (b) For various reasons, the process average is changed to 48.50 mm. With this new mean or process center, what percentage of the seals is reworked? What percentage is scrapped? If rework is economically feasible, is the change in the process center a wise decision?
(a) Well, if the average seal diameter is 49.15 mm and the standard deviation is 0.51 mm, we can calculate the z-score for the lower specification limit of 47.80 mm and the upper specification limit of 49.80 mm.
For the lower specification limit:
z = (x - μ) / σ
z = (47.80 - 49.15) / 0.51
For the upper specification limit:
z = (x - μ) / σ
z = (49.80 - 49.15) / 0.51
Using a z-table or calculator, we can find the corresponding percentage (area under the curve) for each z-value. The area to the left of the z-value represents the percentage of seals that will be reworked (since they are below the lower limit) and the area to the right represents the percentage that will be scrapped (since they are above the upper limit).
(b) Now, let's consider the new mean or process center of 48.50 mm. To calculate the new percentages, we repeat the same process but using the new mean instead:
For the lower specification limit:
z = (47.80 - 48.50) / 0.51
For the upper specification limit:
z = (49.80 - 48.50) / 0.51
Again, we can use a z-table or calculator to find the corresponding percentages.
As for whether the change in the process center is a wise decision, that depends on various factors such as the cost of rework and scrapping, customer requirements, and the company's overall objectives. So, I can't provide a definitive answer to that. But I can provide lots of clown jokes if you're looking to brighten your day!
To solve this problem, we can use the z-score formula to calculate the proportion of seals that fall below the lower specification limit (rework) and above the upper specification limit (scrapped).
(a) Calculate the z-score for the lower specification limit of 47.80 mm:
z = (X - μ) / σ
= (47.80 - 49.15) / 0.51
= -2.65
Using a standard normal distribution table or calculator, we can find the percentage of seals that have a z-score less than -2.65. This represents the percentage of seals that need rework.
Percentage of seals reworked = Percentage of z < -2.65
Similarly, calculate the z-score for the upper specification limit of 49.80 mm:
z = (X - μ) / σ
= (49.80 - 49.15) / 0.51
= 1.27
Using a standard normal distribution table or calculator, find the percentage of seals that have a z-score greater than 1.27. This represents the percentage of seals that are scrapped.
Percentage of seals scrapped = Percentage of z > 1.27
(b) Now, let's calculate the new percentages after the process average is changed to 48.50 mm.
For the lower specification limit:
z = (X - μ) / σ
= (47.80 - 48.50) / 0.51
= -1.37
Find the percentage of seals with a z-score less than -1.37.
For the upper specification limit:
z = (X - μ) / σ
= (49.80 - 48.50) / 0.51
= 2.55
Find the percentage of seals with a z-score greater than 2.55.
To determine if the change in process center is a wise decision, compare the percentage of scrapped seals before and after the change. If the percentage of scrapped seals decreases, then changing the process center can be considered a wise decision, as it reduces waste and saves costs.
To answer these questions, we need to use the concept of Z-scores and the normal distribution.
(a) What percentage of the seals is reworked? What percentage is scrapped?
To find the percentage of seals that are reworked, we need to calculate the Z-score for the lower specification limit of 47.80 mm and then find the corresponding percentage in the standard normal distribution table.
The Z-score can be calculated using the formula:
Z = (X - μ) / σ
where X is the value (lower specification limit), μ is the population mean, and σ is the population standard deviation.
Calculating Z for the lower specification limit:
Z = (47.80 - 49.15) / 0.51
To find the corresponding percentage in the standard normal distribution table, we can look up the Z-score we calculated. Let's assume the Z-score is -2.10, which means a value that is 2.10 standard deviations below the mean.
From the standard normal distribution table, we can find that the area to the left of -2.10 is approximately 0.0179 or 1.79%.
Therefore, the percentage of seals that are reworked is approximately 1.79%.
To find the percentage of seals that are scrapped, we need to calculate the Z-score for the upper specification limit of 49.80 mm and find the area to the right of that Z-score using the standard normal distribution table.
Calculating Z for the upper specification limit:
Z = (49.80 - 49.15) / 0.51
Assuming we get a Z-score of 1.27, which means a value that is 1.27 standard deviations above the mean.
From the standard normal distribution table, we can find that the area to the right of 1.27 is approximately 0.1020 or 10.20%.
Therefore, the percentage of seals that are scrapped is approximately 10.20%.
(b) With the new mean of 48.50 mm, we follow the same process as in part (a) to determine the percentage of seals that are reworked and scrapped.
Calculating Z for the lower specification limit:
Z = (47.80 - 48.50) / 0.51
Assuming we get a Z-score of -1.37.
From the standard normal distribution table, we find that the area to the left of -1.37 is approximately 0.0853 or 8.53%.
Therefore, the percentage of seals that are reworked with the new mean is approximately 8.53%.
Calculating Z for the upper specification limit:
Z = (49.80 - 48.50) / 0.51
Assuming we get a Z-score of 2.55.
From the standard normal distribution table, we find that the area to the right of 2.55 is approximately 0.0060 or 0.60%.
Therefore, the percentage of seals that are scrapped with the new mean is approximately 0.60%.
If rework is economically feasible, we need to compare the percentages of reworked seals before and after the change in the process center. In this case, the percentage of reworked seals decreased from 1.79% to 8.53%, which indicates a higher number of seals requiring rework with the new mean. Therefore, the change in the process center may not be a wise decision if rework is economically feasible.