AaBbCcD AaBbCcDdEe No 1) Specification for the length of a bolt is 16.191 centim

Question

AaBbCcD AaBbCcDdEe No 1) Specification for the length of a bolt is 16.191 centimeters +/-0.188 centimeters. The part is fabricated using a normally distributed process with a mean of 16.264 centimeters and a population standard deviation of 0.078. What is the sigma capability for this process? 2) Specification for the length of a bolt is 16.297 centimeters+0.207 centimeters. The part is fabricated using a normally distributed process with a mean of 16.230 centimeters and a population standard deviation of 0.077. What is the Cok for this process? 3) Tolerances for a new part require weights between 240.4 and 269.4 pounds. The part is made using a process that has a mean of 256.4 pounds with a standard deviation of 2.4 pounds. The process population is normally distributed. What is the Cok for this process? (use four decimal places) 4) Tolerances for a new part require weights between 243.9 and 260.1 pounds. The part is made using a process that has a mean of 256.0 pounds with a standard deviation of 3.8 pounds. The process population is normally distributed. What is the sigma capability for this process? (use four decimal places) 5) Tolerances for a new part require weights between 249.6 and 268.4 pounds. The part is made using a process that has a mean of 256.8 pounds with a standard deviation ot 28 pounds. The process population is normaly distributed Ir the process can be adjusted so that it is centered, what will be the sigma capability for the adjusted process? use four decimal piaces)

Explanation / Answer

1. Sigma Capability of a process is represented by Cp

Cp = (Cpu+Cpl)/2

Cpu = allowable upper spread / actual upper spread = (USL-M)/3S

Cpl = allowable lower spread /actual lower spread = (M-LSL)/3S

Here USL = Upper Specification Limit = Product Specified Dimension + Upper tolerence = 16.191 + 0.186 = 16.377 Centimeter

LSL = Lower Specification Limit = Product Specified Dimension - Lower tolerence 16.191-0.186 = 16.005 Centimeter

M = Mean = 16.264 Centimeter

S = Standard Deviation = 0.078

Cpu = ( 16.377-16.264)/(3*0.078) = 0.113/0.234 = 0.4829

Cpl = (16.264-16.005)/(3*0.078) = 0.259/0.234= 1.1068

Cp = (0.4829+1.1068)/2 = 0.7949

If we solve Cp = (Cpu+Cpl)/2 further we will have

Cp = ((USL-M)/3*S+(M-LSL)/3*S)/2 = (USL-LSL)/(6*S)=(16.377-16.005)/(6*0.078) = 0.372/0.468= 0.79487949=0.7949

So value of Cp = 0.7949

Now Cp indicates whether the process is able to produce parts within the specifications . It does not indicates its ability to adhere to its target value . For this another index is used which is Cpk

Cpk is lower of Cpu and Cpl and in this case it is 0.4829

2. Here we have to caluculate Cpk

Cpk is lower of Cpu & Cpl

We have defined clearly how Cpu and Cpl are calculated above.

Cpu = (USL-M)/(3*S) = ((16.297+0.207)-16.230)/(3*0.077) = 0.274/0.231 = 1.1861

Cpl = (M-LSL)/(3*S)= (16.230-(16.297-0.207))/(3*0.077) = 0.14/0.231 =0.6060

So Cpk = 0.6060

3. Here USL = 269.4 , LSL = 240.4 , M = 256.4 , S = 2.4

All above parameters in pounds

Cpu = (269.4-256.4)/(3*2.4) = 13/7.2 = 1.8056

Cpl = (256.4-240.4)/(3*2.4) = 16/7.2 = 2.2222

Here Cpk is lower of Cpu and Cpl = 1.8056

I was supposed to do 1 here . I have done 3 . Due to time constraints I am stopping here. Will reattempt if the questions comes back again.

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