Question

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 C_{pk} for the process?
**(Round your answer to 4 decimal places.)**

C_{pk
}

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

C_{pk
}

**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 | ||

Answer #1

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 |

The home run percentage is the number of home runs per 100 times
at bat. A random sample of 43 professional baseball players gave
the following data for home run percentages.
1.6
2.4
1.2
6.6
2.3
0.0
1.8
2.5
6.5
1.8
2.7
2.0
1.9
1.3
2.7
1.7
1.3
2.1
2.8
1.4
3.8
2.1
3.4
1.3
1.5
2.9
2.6
0.0
4.1
2.9
1.9
2.4
0.0
1.8
3.1
3.8
3.2
1.6
4.2
0.0
1.2
1.8
2.4
(a) Use a calculator with mean...

The home run percentage is the number of home runs per 100 times
at bat. A random sample of 43 professional baseball players gave
the following data for home run percentages. 1.6 2.4 1.2 6.6 2.3
0.0 1.8 2.5 6.5 1.8 2.7 2.0 1.9 1.3 2.7 1.7 1.3 2.1 2.8 1.4 3.8 2.1
3.4 1.3 1.5 2.9 2.6 0.0 4.1 2.9 1.9 2.4 0.0 1.8 3.1 3.8 3.2 1.6 4.2
0.0 1.2 1.8 2.4 (a) Use a calculator with mean...

The home run percentage is the number of home runs per 100 times
at bat. A random sample of 43 professional baseball players gave
the following data for home run percentages.
1.6
2.4
1.2
6.6
2.3
0.0
1.8
2.5
6.5
1.8
2.7
2.0
1.9
1.3
2.7
1.7
1.3
2.1
2.8
1.4
3.8
2.1
3.4
1.3
1.5
2.9
2.6
0.0
4.1
2.9
1.9
2.4
0.0
1.8
3.1
3.8
3.2
1.6
4.2
0.0
1.2
1.8
2.4
(a) Use a calculator with mean...

The home run percentage is the number of home runs per 100 times
at bat. A random sample of 43 professional baseball players gave
the following data for home run percentages.
1.6
2.4
1.2
6.6
2.3
0.0
1.8
2.5
6.5
1.8
2.7
2.0
1.9
1.3
2.7
1.7
1.3
2.1
2.8
1.4
3.8
2.1
3.4
1.3
1.5
2.9
2.6
0.0
4.1
2.9
1.9
2.4
0.0
1.8
3.1
3.8
3.2
1.6
4.2
0.0
1.2
1.8
2.4
(a) Compute a 90% confidence interval...

Using the data below, calculate the correlation between
temperature and growth rate for these samples. Round your answer to
3 decimal place.
Temperature (C)
Growth Rate (divisions per hour)
2.6
854
1.4
200
1.6
354
1.9
395
2.2
401
2.8
590
2.2
587
2
465
2.3
450
2.5
456
2
560
2.5
932
2.8
835
2.6
789
2.4
660
2.1
615
1.8
589
2.2
532
2.3
541
2.1
550
(a) The correlation coefficient r = [a] (report
your answer to...

American Express Company has long believed that its cardholders
tend to travel more extensively than others-both on business and
for pleasure. As part of a comprehensive research effort undertaken
by a New York marketing research firm on behalf of American
Express, a study was conducted to determine the relationship
between miles traveled (x) and charges made on the American Express
card (y). Eight cardholders were randomly selected and their total
charges recorded for a specified period. A questionnaire was then...

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