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 C_p &C_pk 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

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

orchestra answered 2 years ago

Consider the piston ring data in the following table. Assume that specifications are 74.00 ± 0.035...

Consider the piston ring data in the following table. Assume that specifications are 74.00 ± 0.035 mm. Estimate the process capability (Cp and Cpk) using: Sample Range Measurements Sample Std Dev. Measurements Convert the Cp found above into approximate dpm. Inside Diameter Measurements (mm) for Automobile Piston Rings Sample ID 1 74.03 1 74.002 1 74.019 1 73.992 1 74.008 2 73.995 2 73.992 2 74.001 2 74.011 2 74.004 3 73.988 3 74.024 3 74.021 3 74.005 3 74.002...

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