Chapter 6
Statistical Quality Control
Montgomery’s 6th edition
Solutions for Chapter 06
Jan Rohlén jan.rohlen@hb.se Question 6.04
Sample No.
Ri
Xi
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
9
7
5
7
6
2
8
6
5
6
8
7
7
6
9
5
4
8
6
4
10
7.75
7.5
9
9.75
10.75
10.75
6.5
9
13.5
12.5
9.75
13.25
10.5
11
12.5
9.75
10.75
8.75
13.25
Table 1: Table 6E.4
1
LaTeX Typesetting by : Amirkiarash Kiani
Jan Rohlén
Statistical Quality Control
Chapter 06
(a)
R=
Ri
= 6.25 m Sample Size
4
X=
A2
0.729
D3
0
Xi
= 10.325 m D4
2.282
U CL = X + A2 R = 10.325 + 0.729 × 6.25 = 14.88
X−R Chart CL = X = 10.325
LCL = X − A2 R = 10.325 − 0.729 × 6.25 = 5.77
U CL = D4 × R = 2.282 × 6.25 = 14.26
R Chart CL = R = 6.25
LCL = D4 × R = 0 × 6.25 = 0
The process is in-control.
2
LaTeX Typesetting by : Amirkiarash Kiani
Jan Rohlén
Statistical Quality Control
Chapter 06
(b)
Specs. (350V ± 5V ) σ= The real σ can be calculated:
Cp =
R d2 =
6.25
2.059
= 3.0355
U SL − LSL
3550 − 3450
=
= 5.491
6×σ
6 × 3.0355
The minimum capability index for existing processes is 1.33 (i.e., 43 ). Obviously, this process is higher (5.49
1.33).
(c)
3
LaTeX Typesetting by : Amirkiarash Kiani
Jan Rohlén
Statistical Quality Control
Chapter 06
Question 6.15
(a)
X=
1000
72
= 20 s =
= 1.44
50
50
Sample Size
4
X −S
A3
1.628
B3
0
B4
2.266
U CL = X + A3 R = 20 + 1.628 × 1.44 = 22.34
Chart CL = X = 10.325
LCL = X − A3 R = 205 − 1.628 × 1.44 = 17.66
S
U CL = B4 s = 2.266 × 1.44 = 3.26
Chart CL = s = 1.44
LCL = B s = 0 × 1.44 = 0
3
(b)
Natural Tolerance Limits
First of all, we need to calculate the real σ: σ =
s
C4
=
1.44
0.9213
= 1.563
U N T L = X + 3σ = 20 + 3 × 1.563 = 24.69
LN T L = X − 3σ = 20 − 3 × 1.563 = 15.31
(c)
Specs Limits: 19 ± 4
U SL − LSL
23 − 15
=
= 0.85
6σ
6 × 1.563
=⇒ the process is not capable!
Cp =
Cp < 1.33
(d)
Prework = P (S > U SL) = 1 − P (X ≤ U SL) = 1 − P (
4
LaTeX Typesetting by : Amirkiarash Kiani
U SL − µ
)=
Montgomery’s 6th edition
Solutions for Chapter 06
Jan Rohlén jan.rohlen@hb.se Question 6.04
Sample No.
Ri
Xi
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
9
7
5
7
6
2
8
6
5
6
8
7
7
6
9
5
4
8
6
4
10
7.75
7.5
9
9.75
10.75
10.75
6.5
9
13.5
12.5
9.75
13.25
10.5
11
12.5
9.75
10.75
8.75
13.25
Table 1: Table 6E.4
1
LaTeX Typesetting by : Amirkiarash Kiani
Jan Rohlén
Statistical Quality Control
Chapter 06
(a)
R=
Ri
= 6.25 m Sample Size
4
X=
A2
0.729
D3
0
Xi
= 10.325 m D4
2.282
U CL = X + A2 R = 10.325 + 0.729 × 6.25 = 14.88
X−R Chart CL = X = 10.325
LCL = X − A2 R = 10.325 − 0.729 × 6.25 = 5.77
U CL = D4 × R = 2.282 × 6.25 = 14.26
R Chart CL = R = 6.25
LCL = D4 × R = 0 × 6.25 = 0
The process is in-control.
2
LaTeX Typesetting by : Amirkiarash Kiani
Jan Rohlén
Statistical Quality Control
Chapter 06
(b)
Specs. (350V ± 5V ) σ= The real σ can be calculated:
Cp =
R d2 =
6.25
2.059
= 3.0355
U SL − LSL
3550 − 3450
=
= 5.491
6×σ
6 × 3.0355
The minimum capability index for existing processes is 1.33 (i.e., 43 ). Obviously, this process is higher (5.49
1.33).
(c)
3
LaTeX Typesetting by : Amirkiarash Kiani
Jan Rohlén
Statistical Quality Control
Chapter 06
Question 6.15
(a)
X=
1000
72
= 20 s =
= 1.44
50
50
Sample Size
4
X −S
A3
1.628
B3
0
B4
2.266
U CL = X + A3 R = 20 + 1.628 × 1.44 = 22.34
Chart CL = X = 10.325
LCL = X − A3 R = 205 − 1.628 × 1.44 = 17.66
S
U CL = B4 s = 2.266 × 1.44 = 3.26
Chart CL = s = 1.44
LCL = B s = 0 × 1.44 = 0
3
(b)
Natural Tolerance Limits
First of all, we need to calculate the real σ: σ =
s
C4
=
1.44
0.9213
= 1.563
U N T L = X + 3σ = 20 + 3 × 1.563 = 24.69
LN T L = X − 3σ = 20 − 3 × 1.563 = 15.31
(c)
Specs Limits: 19 ± 4
U SL − LSL
23 − 15
=
= 0.85
6σ
6 × 1.563
=⇒ the process is not capable!
Cp =
Cp < 1.33
(d)
Prework = P (S > U SL) = 1 − P (X ≤ U SL) = 1 − P (
4
LaTeX Typesetting by : Amirkiarash Kiani
U SL − µ
)=