1 Design specifications require that a key dimension on a product measure 102 ± 15 units. A process being considered for producing this product has a standard deviation of 6 units. (Format answers...

Attached - 10 questions.
Answered 2 days AfterJun 16, 2022

Answer To: 1 Design specifications require that a key dimension on a product measure 102 ± 15 units. A process...

Prince answered on Jun 19 2022
1
a.
Upper Specification Limit     117
Lower Specification Limit     87
Mean    102
Standard Dev    6
Cpk    0.8333333333
As Cpk < 1.33 the process is not expected to meet its desired specifications                                                    13
Now, in order to find the proportion or probability of defective units, we would find the z values for USL and LSL
Z_USL    2.5            0.9937903347
Z_LSL    2.5
% of units above USL = Probability value derived from z-statistic table for Z 2.5 = 0.0062
% of Units Below LSL = Probability value derived from z-statistic table for Z 2.5 = 0.0062
% of total units out of the desired specification limits = 0.0062 + 0.0062 = 0.0124
b.
Upper Specification Limit     117
Lower Specification Limit     87
Mean    96
Standard Dev    6

Cpk    0.5
As Cpk < 1.33 the process is not expected to meet its desired specifications
c.
Now, in order to find the proportion or probability of defective units, we would find the z values for USL and LSL
Z_USL    3.5
Z_LSL = (µ-L) / σ    1.5
% of units above USL = Probability value derived from z-statistic table for Z 3.5 = 0.0002
% of Units Below LSL = Probability value derived from z-statistic table for Z 1.5 = 0.0668
% of total units out of the desired specification limits = 0.0002 + 0.0668 = 0.0670
Thus, the total probability of defective units is increased by the shift of mean, by 0.0670 - 0.0124 = 0.0546
Design specifications require that a key dimension on a product measure 102 ± 15 units. A process being considered for producing this product has a standard deviation of 6 units.
(Format answers rounded to 4 decimal places when entering answers in the quiz, Leave the probabilty as decimal numbers, not %.)

a) Find the process capability index, Cpk, and the probability of defective output. Assume that the process is centered with respect to specifications, i.e., process mean is 102.
) Suppose the process mean shifts to 96. Calculate the new process capability index.
c) What is the probability of defective output after the process shift? Did it increase or decrease?
(For this question and the next, a relevant reference is Example 13.1 in the textbook, pp. 370-371. The Excel calculation of that example is shown in sheet Ex 13.1 of Excel SPC workbook. )
2
Thickness (mm)
1.9
1.8
2.1
2.1
2
2.2
2.4
2
1.9
2.1
2.2
2.4
1.8
2.2
2.1
1.7
2.2
1.9
2.1
2.2
1.7
1.8
2
2
2
2.1        a    Mean    1.9625
1.8            Stand Dev.    0.2096
1.6
1.9        b    Cpk    0.6957
1.6
1.7        c    z-score     2.0873091603
2            Probability of thickness of washer to be 2.4 mm, P(X = 2.4) = NORMSDIST(z-score) = NORMSDIST(2.0871) = 0.9816
1.7            Probability of thickness of washer greater outside tolerance limit, i.e. greater than 2.4 mm = 1-0.9816 = 0.0184 = 1.84%
1.8            Hence, 1.84% of output is expected to be greater than 2.4 mm.
1.9
2.1        d    Mean    1.9
1.8            Cpk    0.7951
1.6
1.9        e    z-score    2.3855
2.2            Probability of thickness of washer to be 2.4 mm, P(X = 2.4) = NORMSDIST(z-score) = NORMSDIST(2.3855) = 0.9915
Probability of thickness of washer greater outside tolerance limit, i.e. greater than 2.4 mm = 1-0.9915= 0.0085 = 0.85%
Hence, 0.85% of output is expected to be greater than 2.4 mm.
f    Mean    1.9
Stand Dev.    0.1000
Cpk    1.6667
g    The process with Cpk >= 1.33 is highly capable, hence we should select the situation mentioned in part (f).
To the right is a picture of a washer that is supposed to be 1.9 mm thick.
The tolerances on the thickness are 0.5 mm, so the thickness should be between 1.4 mm and 2.4 mm.
You are given here the thickness in mm for a sample of 40 washers. Assume the thickness is distributed normally.
a) Verify that the mean = 1.9625 and the standard deviation from this data is 0.2096. (Recall STDEV.S is the function for computing the sample standard deviation.)
) What is the Cpk for the process?
c) What % of the output is expected to be out of tolerance (outside the specification limits = defective)? (Express the probability as % with 2 decimal digits, e.g., 12.34%.)
d) If the process were centered, i.e., sample mean were equal to 1.9 mm (with the standard deviation unchanged), what would be the Cpk?
e) If the process were centered as in (d), what percentage of output would be expected to be out of tolerance?

f) If the process were centered AND the standard deviation was only about .10 millimeter, what would be the Cpk and percent defective?
(This percent defective is very close to 0. Increase decimal digits until you can see 2 significant figures (e.g., 0.0000012%))
g) Out of situations in (b), (d), and (f) which had the process considered capable?
3
Sample    n    Number of Defective Items    Fraction Defects    UCL    LCL
1    15    1    0.0666666667    0.3717141236    -0.1317141236
2    15    2    0.1333333333    0.3717141236    -0.1317141236
3    15    3    0.2    0.3717141236    -0.1317141236
4    15    2    0.1333333333    0.3717141236    -0.1317141236
5    15    1    0.0666666667    0.3717141236    -0.1317141236
6    15    2    0.1333333333    0.3717141236    -0.1317141236
7    15    1    0.0666666667    0.3717141236    -0.1317141236
8    15    3    0.2    0.3717141236    -0.1317141236
9    15    2    0.1333333333    0.3717141236    -0.1317141236
10    15    1    0.0666666667    0.3717141236    -0.1317141236
18    1.2
a    p-Bar    0.1200
stand.dev    0.0839
UCL    0.3717141236
LCL    -0.1317141236

c    Since, all points are within UCL and LCL, process is under Control
Ten samples of 15 parts each were taken from an ongoing process to establish a p-chart. The samples and the number of defectives in each are shown in the following table.
a) Determine the p-bar, Sp, UCL and LCL for a p-chart with z = 3 (Format your answers with 4 decimal places.)
) Plot the p-chart.
c) Based on the plotted data points, what comments can you make? (i.e., is the process in control or out of control?) Decide based on whether at least one point is outside the control limits. Recall a point is outside the control limits if it is < LCL or > UCL.
p Chart
Fraction Defects    6.6666666666666666E-2    0.13333333333333333    0.2    0.13333333333333333    6.6666666666666666E-2    0.13333333333333333    6.6666666666666666E-2    0.2    0.13333333333333333    6.6666666666666666E-2    UCL    0.37171412356083633    0.37171412356083633    0.37171412356083633    0.37171412356083633    0.37171412356083633    0.37171412356083633    0.37171412356083633    0.37171412356083633    0.37171412356083633    0.37171412356083633    LCL    -0.13171412356083634    -0.13171412356083634    -0.13171412356083634    -0.13171412356083634    -0.13171412356083634    -0.13171412356083634    -0.13171412356083634    -0.13171412356083634    -0.13171412356083634    -0.13171412356083634
4
Sample    # Defects    Cbar    UCL    LCL
1    6    4    10    -2
2    5    4    10    -2
3    0    4    10    -2
4    1    4    10    -2
5    4    4    10    -2
6    2    4    10    -2
7    5    4    10    -2
8    3    4    10    -2
9    3    4    10    -2
10    2    4    10    -2
11    6    4    10    -2
12    1    4    10    -2
13    8    4    10    -2
14    7    4    10    -2
15    5    4    10    -2
16    4    4    10    -2
17    11    4    10    -2
18    3    4    10    -2
19    0    4    10    -2
20    4    4    10    -2
80
C-Bar    4                    Here the problem is out of control because 17th sample is out of control having more than UCL value ie 10
UCL    10                    So the process is out of control
LCL    -2
At Data Systems Services company owned by Donna, she wants to see if the data entry process is in control. She collected 100 records entered by each of the 20 clerks. She counted the number of inco
ectly entered records...
SOLUTION.PDF