Question

# As part of the process for improving the quality of their cars, Toyota engineers have identified...

As part of the process for improving the quality of their cars, Toyota engineers have identified a potential improvement to the process that makes a washer that is used in the accelerator assembly. The tolerances on the thickness of the washer are fairly large since the fit can be loose, but if it does happen to get too large, it can cause the accelerator to bind and create a potential problem for the driver. (Note: This part of the case has been fabricated for teaching purposes and none of these data were obtained from Toyota.)

Let’s assume that as a first step to improving the process, a sample of 40 washers coming from the machine that produces the washers was taken and the thickness measured in millimeters. The following table has the measurements from the sample:

 2.0 1.6 2.0 2.1 2.0 1.8 1.7 1.9 1.7 1.8 1.8 2.2 2.1 2.2 1.9 1.8 2.1 1.6 1.8 1.6 2.1 2.4 2.2 2.1 2.1 1.8 1.7 1.9 1.9 2.1 2.0 2.4 2.0 2.1 1.9 2.3 1.7 2.0 1.9 2.2

Use the appropriate Excel function to compute normal distribution probabilities in Parts a, b, e and f.

a. If the specification is such that no washer should be greater than 2.4 millimeters, assuming that the thicknesses are distributed normally, what percentage of output is expected to be greater than this thickness? (Round your answer to 2 decimal places.)

Percentage of output             %

b. What if there is an upper and lower specification, where the upper thickness limit is 2.4 and the lower thickness limit 1.5, what percentage of output is expected to be out of tolerance? (Round your answer to 2 decimal places.)

Percentage of the output             %

c. What is the Cpk for the process? (Round your answer to 4 decimal places.)

Cpk

d. What would be the Cpk for the process if it were centered (assume the process standard deviation is the same)? (Round your answer to 3 decimal places.)

Cpk

e. What percentage of output would be expected to be out of tolerance if the process were centered? (Round your answer to 2 decimal places.)

Percentage of output             %

f. If the process could be improved so that the standard deviation was only about 0.1 millimeters, what would be the best that could be expected with the processes relative to fraction defective?

The process to be centered at            (Click to select)1.952.41.5

g. Setup X-bar and Range control charts for the current process. Assume the operators will take samples of 10 washers at a time. (Round your answer to 3 decimal places.)

 X-bar chart Range chart Upper control limit Lower control limit

a) Mean of the sample = 1.9625

S.D of the sample = 0.210844

z =(2.4-1.9625)/.210844 = 2.08

p(z>2.08) = 1-p(z<=2.4) = 1-.98=.02

b) What if there is an upper and lower specification, where the upper thickness limit is 2.4 and the lower thickness limit 1.5, what percentage of output is expected to be out of tolerance?

z=(1.5-1.9625)/.210844 = -2.20

p(z<-2.20)=.0139

Out of tolerence percentage is = p(z>2.08)+p(z<-2.20) = .02+.0139 = .0339

c) cpk = min{ (USL-mean)/3σ,(mean-LSL)/3σ}

=min {.69,.73} = .69

d) If the process is centered Cp = Cpk

Cp= (USL-LSL)/6σ

Cp = .71

so Cpk when the process is centered is .711

g)

 X bar R UCL 2.071 0.623 LCL 1.854 0.077

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