Question

In: Statistics and Probability

Problem#: Consider the data shown. Assume that the specifications on this component are 74.05 and 73.95...

Problem#: Consider the data shown. Assume that the specifications on this component are 74.05 and 73.95 mm. (a)Estimate process capability for the piston-ring process, for both Cp &Cpk and Estimate the percentage of piston rings produced that will be outside of the specifications?

Sample Number Observation 1 Observation 2 Observation 3 Observation 4 Observation 5
1 74.03 74.002 74.019 73.992 74.008
2 73.995 73.992 74.001 74.011 74.004
3 73.988 74.024 74.021 74.005 74.002
4 74.002 73.996 73.993 74.015 74.009
5 73.992 74.007 74.015 73.989 74.014
6 74.009 73.994 73.997 73.985 73.993
7 73.995 74.006 73.994 74 74.005
8 73.985 74.003 73.993 74.015 73.988
9 74.008 73.995 74.009 74.005 74.004
10 73.998 74 73.99 74.007 73.995
11 73.994 73.998 73.994 73.995 73.99
12 74.004 74 74.007 74 73.996
13 73.983 74.002 73.998 73.997 74.012
14 74.006 73.967 73.994 74 73.984
15 74.012 74.014 73.998 73.999 74.007
16 74 73.984 74.005 73.998 73.996
17 73.994 74.012 73.986 74.005 74.007
18 74.006 74.01 74.018 74.003 74
19 73.984 74.002 74.003 74.005 73.997
20 74 74.01 74.013 74.02 74.003
21 73.982 74.001 74.015 74.005 73.996
22 74.004 73.999 73.99 74.006 74.009
23 74.01 73.989 73.99 74.009 74.014
24 74.015 74.008 73.993 74 74.01
25 73.982 73.984 73.995 74.017 74.013

Solutions

Expert Solution

a)

The process capability measure by process capability ratio,

Where,

The standard deviation for the sample is obtained in excel in following steps,

Step 1: First calculate the average for each sample. The screenshot is shown below,

Step 2: Now, the standard deviation for sample averages is obtained using the function =STDEV(). The screenshot is shown below,

Now,

and the process capability index,

Where, the grand average (average of sample averages) is obtained in excel using the function =AVERAGE().

Since,

Since the Cpk>>1, the process is within specification limit.

The percentage of piston rings produced that will be outside of the specifications is obtained by calculating the z score,

Using the standard normal distribution table for z=-10.5075 and z=10.0246, the probabilities are


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