In: Statistics and Probability
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 |
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