Question 344456: Please could you kindly help me solve this question.Kindly show me all the working stages. Thank you very very much for your help.
The question: A company suuplies pins to a customer. It uses an automatic lathe to produce the pins. Due to factors such as vibration, temperature and wear and tear, the lengths of the pins are normally distributed with a mean of 25.30 mm and a standard deviation of 0.45 mm. The customer will only buy those pins with lengths in teh intervals of 25.00 +/- 0.50mm.
a. What percentage of the pins will be acceptable to the customer?
b. In order to improve the percentage accepted, management considers adjusting the population mean and standard deviation of the length of the pins. If the lathe can be adjusted to have any desired mean of the lenghts, what should it be adjusted to? Why?
c. Suppose that the mean cannot bea djusted but the standard deviation can be reduced. What maximum value of the standard would make 90% of thi pins acceptable? (Assume the mean to be 25.30mm).
d. The production manager the considers the costs involved. The cost of resetting the machine to adjust the population mean involves engineering costs and the cost of production time lost. The cost of reducing the population standard deviation involves, in addition to these costs, the cost of overhauling the machine and reengineering the process. Assume it costs $150 Xsquared to decrease the standard deviation by (x/40)mm. Find the cost of reducing the standard deviation to the values found in C.
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! the lengths of the pins are normally distributed with a mean of 25.30 mm and a standard deviation of 0.45 mm. The customer will only buy those pins with lengths in the intervals of 25.00 +/- 0.50mm.
a. What percentage of the pins will be acceptable to the customer?
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z(25.5) = (25.5-25.30)/0.45 = 0.2/0.45 = 0.4444
z(24.5) = (24.5-25.30)/0.45 = -0.8/0.45 = -1.7778
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P(24.4 < x < 25.5) = P(-1.7778< z < 0.4444) = 0.5522
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b. In order to improve the percentage accepted, management considers adjusting the population mean and standard deviation of the length of the pins. If the lathe can be adjusted to have any desired mean of the lenghts, what should it be adjusted to? Why?
Make the mean 25 mm.
Then 73.35 % of the pins will acceptable to the customer
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c. Suppose that the mean cannot bea djusted but the standard deviation can be reduced. What maximum value of the standard deviation would make 90% of the pins acceptable? (Assume the mean to be 25.30mm).
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Comment: I'll have to think about this one.
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d. The production manager the considers the costs involved. The cost of resetting the machine to adjust the population mean involves engineering costs and the cost of production time lost. The cost of reducing the population standard deviation involves, in addition to these costs, the cost of overhauling the machine and reengineering the process. Assume it costs
$150x^2 to decrease the standard deviation by (x/40)mm. Find the cost of reducing the standard deviation to the values found in C.
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If I had the answer to part c, part d would be a snap.
But I don't. Sorry about that.
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Cheers,
Stan H.
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