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 X bar = 0.00507, s = 0.00023)

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 x bar = 0.00503, s = 0.00015)

Solution and Explanation:

The Upper specification Limit of magnetic tape = USL = 0.005 + 0.0004 = 0.0054 MM

Lower Specification Limit of magnetic tape = LSL = 0.005 – 0.0004 = 0.0046 MM

The process average = m = 0.00507

The process standard deviation = Sd = 0.00023

Next step would be to find out corresponding Z values such that:

    M + Z.Sd = USL

Or, 0.00507 + 0.00023.Z = 0.0054

Or, 0.00023.Z = 0.00033

Or, Z = 1.434

Corresponding probability for value of Z = 1.434 as derived from Normal distribution table = 0.92364.

This means that probability that process parameter will not exceed USL = 0.92364

Therefore, probability that process parameter will exceed USL = 1 – 0.92364 = 0.0763

    M + Z.Sd = LSL Or, 0.00507 + 0.00023.Z = 0.0046

Or, 0.00023.Z = 0.0046 – 0.00507 = - 0.00047

Or, Z = - 2.043

Corresponding probability for value of Z = - 2.043 as derived from Normal distribution table = 0.02063

This means that probability that process parameter will be LSL or below = 0.02063

Therefore , probability that the thickness exceeds USL as well as thickness goes below USL = 0.0763 + 0.02063 = 0.09693

Therefore average loss per reel

= $10 per reel x probability that thickness exceeds USL as well as goes below LSL

= $10 multiply 0.09693

= $0.9693

AVERAGE LOSS PER REEL = $0.9693