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HW 2 Problems (10 points each)

(1) A manufacturer of magnetic tapes is interested in reducing the variability of the thickness of the coating on the tape.  It is estimated that the loss to the consumer is $10 per reel if the thickness exceeds 0.005 ± 0.0004 mm.  Each reel has 200 m of tape.  A random sample of 10 yields the following thickness (in millimeters):  0.0048, 0.0053, 0.0051, 0.0051, 0.0052, 0.0049, 0.0051, 0.0047, 0.0054, and 0.0052.  Find the average loss per reel. (Use = 0.00507, s = 0.00023)

(2) The manufacturer is considering adopting a new process to reduce the variability in the thickness of coating.  It is estimated that the additional cost for this improvement is $0.03 per linear meter.  The annual production is 10,000 reels.  Each reel has 200 m of tape.  A random sample of size 8 from the new process yielded the following thickness (in millimeters): 0.0051, 0.0048, 0.0049, 0.0052, 0.0052, 0.0051, 0.0050, and 0.0049.  Is it cost effective to use the new process?  What is the annual savings or loss?  (Use = 0.00503, s = 0.00015)

(3) Studies on a bottle-filling machine indicates it fill bottles to a mean of 16 ounces with a standard deviation of 0.10 ounces. What is the process specification, assuming the C_p index of 1?

• Studies on a machine that molds plastic water pipe indicate that when it is injecting 1-inch diameter pipe, the process standard deviation is 0.05 inches. The one-inch pipe has a specification of 1-inch plus or minus 0.10 inch. What is the process capability index (C_p) if the long-run process mean is 1 inch?
• The specification limit for a product is 8 cm and 10 cm. A process that produces the product has a mean of 9.5 cm and a standard deviation of 0.2 cm. What is the process capability, C_pk?
• The specifications for a product are 6 mm ±1 mm. The process is known to operate at a mean of 6.05 with a standard deviation of 0.01 mm. What is the C_pk for this process?
• Design specifications require that a key dimension on a product measure 100 plus minus 10 units. A process being considered for producing this product has a standard deviation of 5 units.

(a) What can you say (quantitatively) regarding the process capability?

(b) Suppose the process average shifts to 102.  Calculate the new process capability.

(c) What can you say about the process after the shift?  Approximately what percentage of the items produced will be defective?