Question

In: Statistics and Probability

Solve step-by-step the following: 1) X-bar and R control Chart Subgroup X1 X2 X3 X4 X-bar...

Solve step-by-step the following:


1) X-bar and R control Chart

Subgroup

X1

X2

X3

X4

X-bar

R

1

1.2

1.1

1.1

2.3

2

3.2

1.3

1.3

3.3

3

1.3

1.4

1.3

1.1

4

3.2

1.1

3.3

2

5

2.1

3.2

2.4

2.3

6

3.1

3.1

3.2

3.3

7

3

1

2.2

2.1

8

2.3

1.3

3.1

2.4

9

4

3.3

4

4.3

10

1.2

3.1

1.3

1.3

11

2.2

1.4

1.1

3

12

3.2

2.3

3.1

1.2

13

1.1

3

1

3.1

14

3.1

1.3

2.2

1.2

15

2.3

2

1

2.3

16

2.2

3.1

3.4

3.2

17

1.2

1.2

2.1

2

18

1.3

3.3

1.2

2.1

19

2.2

2

1.3

3

20

0.7

1.1

1.1

1.2

21

3

3.1

2.2

2.2

22

2.3

2.3

3.1

2.2

23

2.3

1.3

1.2

3.1

24

3.1

3

1.4

1.4

25

3.1

3.2

2.1

2.1

a. Determine the Trial UCL, CL, and LCL for both charts of X-bar and R

b. Determine the Revised UCL, CL, and LCL for both charts of X-bar and R

Solutions

Expert Solution

a) The control limits for X-bar chart is,

UCL= (X-bar) + A2 R = 2.400151

CL = X-bar = 2.1701

LCL = (X-bar) - A2 R = 1.940049

Where A2 is table value corresponding to n = 25

The control limits for R chart is

UCL = D3 R = 2.317048

CL = R = 1.5036

LCL = D4 R = 0.690192

Where D3 & D4 is table value corresponding to n = 25.

A2 = 0.153, D3 = 0.459, D4 = 1.541

Sub group x1 x2 x3 x4 X bar R
1 1.2 1.1 1.1 2.3 1.425 1.2
2 3.2 1.3 1.3 3.3 2.275 2
3 1.3 1.4 1.3 1.1 1.275 0.3
4 3.2 1.1 3.3 2 2.4 2.2
5 2.1 3.2 2.4 2.3 2.5 1.1
6 3.1 3.1 3.2 3.3 3.175 0.2
7 3 1 2.2 2.1 2.075 2
8 2.3 1.3 3.1 2.4 2.275 1.8
9 4 3.3 4 4.3 3.9 1
10 1.2 3.1 1.3 1.3 1.725 1.9
11 2.2 1.4 1.1 3 1.925 1.9
12 3.2 2.3 3.1 1.2 2.45 2
13 1.1 3 1 3.1 2.05 2.1
14 3.1 1.3 2.2 1.2 1.95 1.9
15 2.3 2 1 2.3 1.9 1.3
16 2.2 3.1 3.4 3.2 2.975 1.2
17 1.2 1.2 2.1 2 1.625 0.9
18 1.3 3.3 1.2 2.1 1.975 2.1
19 2.2 2 1.3 3 2.125 1.7
20 0.7 1.1 1.1 1.2 1.025 0.5
21 3 0.31 2.2 2.2 1.9275 2.69
22 2.3 2.3 3.1 2.2 2.475 0.9
23 2.3 1.3 1.2 3.1 1.975 1.9
24 3.1 3 1.4 1.4 2.225 1.7
25 3.1 3.2 2.1 2.1 2.625 1.1
Average 2.1701 1.5036

From above graph we see that the process of X-bar chart and R chart is out of control because in both of graph we see that some points goes above UCL line and some points are below LCL line.

b) We see that the process of X-bar and R chart is out of control. Hence we revised control limits of X-bar and R chart.

For revising control limits we delete the points which are above UCL line and below LCL line in both X-bar and R chart and again compute control limits for both of these chart hence we get revised control limits.

Sub group x1 x2 x3 x4 X bar R
1 3.2 1.3 1.3 3.3 2.275 2
2 3.2 1.1 3.3 2 2.4 2.2
3 3 1 2.2 2.1 2.075 2
4 2.3 1.3 3.1 2.4 2.275 1.8
5 1.2 3.1 1.3 1.3 1.725 1.9
6 2.2 1.4 1.1 3 1.925 1.9
7 3.2 2.3 3.1 1.2 2.45 2
8 1.1 3 1 3.1 2.05 2.1
9 3.1 1.3 2.2 1.2 1.95 1.9
10 2.3 2 1 2.3 1.9 1.3
11 1.3 3.3 1.2 2.1 1.975 2.1
12 2.2 2 1.3 3 2.125 1.7
13 3 0.31 2.2 2.2 1.9275 2.69
14 2.3 2.3 3.1 2.2 2.475 0.9
15 2.3 1.3 1.2 3.1 1.975 1.9
16 3.1 3 1.4 1.4 2.225 1.7
Average 2.107969 1.880625

table values corresponding to n = 16 is,

A2 = 0.212, D3 = 0.363, D4 = 1.637

The revised control limits for X-bar chart is,

UCL= (X-bar) + A2 R = 2.50662

CL = X-bar = 2.107969

LCL = (X-bar) - A2 R = 1.709277

The revised control limits for R chart is

UCL = D3 R = 3.078583

CL = R = 1.880625

LCL = D4 R = 0.870729


Related Solutions

Solve step-by-step: X-bar and S control Chart Subgroup X1 X2 X3 X4 X-bar S 1 1.2...
Solve step-by-step: X-bar and S control Chart Subgroup X1 X2 X3 X4 X-bar S 1 1.2 1.3 1.2 2.2 2 3.3 1.2 1.2 3.2 3 1.2 1.3 1 1.3 4 3.4 1.2 3.1 2.1 5 2.3 3.3 2.2 2.3 6 3.1 3.3 3 3.1 7 3.1 1.2 2.1 2.4 8 2.4 1.4 3 2.4 9 4.1 4.2 4.1 4.2 10 1.3 3 1.3 1.3 11 2.3 1 1.4 3.3 12 3.3 2.4 3.3 1.1 13 1.4 3.4 1.3 3.2 14...
Let U = {(x1,x2,x3,x4) ∈F4 | 2x1 = x3, x1 + x4 = 0}. (a) Prove...
Let U = {(x1,x2,x3,x4) ∈F4 | 2x1 = x3, x1 + x4 = 0}. (a) Prove that U is a subspace of F4. (b) Find a basis for U and prove that dimU = 2. (c) Complete the basis for U in (b) to a basis of F4. (d) Find an explicit isomorphism T : U →F2. (e) Let T as in part (d). Find a linear map S: F4 →F2 such that S(u) = T(u) for all u ∈...
The prices of inputs (x1,x2,x3,x4) are (4,1,3,2): (a) If the production function is given by f(x3,x4)...
The prices of inputs (x1,x2,x3,x4) are (4,1,3,2): (a) If the production function is given by f(x3,x4) =min⁡{x1+x2,x3+x4} what is the minimum cost of producing one unit of output? (b) If the production function is given by f(x3,x4)=x1+x2 +min⁡{x3+x4} what is the minimum cost of producing one unit of output?
We have a random sample of observations on X: x1, x2, x3, x4,…,xn. Consider the following...
We have a random sample of observations on X: x1, x2, x3, x4,…,xn. Consider the following estimator of the population mean: x* = x1/2 + x2/4 + x3/4. This estimator uses only the first three observations. a) Prove that x* is an unbiased estimator. b) Derive the variance of x* c) Is x* an efficient estimator? A consistent estimator? Explain.
4. Determine the number of integer solutions of x1 + x2 + x3 + x4 =...
4. Determine the number of integer solutions of x1 + x2 + x3 + x4 = 17, where a. xi ≥ 0, 1 ≤ i ≤ 4 b. x1, x2 ≥ 3 and x3, x4 ≥ 1 c. xi ≥ -2, 1 ≤ i ≤ 4 d. x1 , x2 , x3 > 0 and 0 < x4 ≤ 10
The values of X1, X2, X3, X4 and X5 are .1, .2, .08, .2 and .3....
The values of X1, X2, X3, X4 and X5 are .1, .2, .08, .2 and .3. Explicate the meaning of those numbers for the Z score. What are the prospects of this firm? Discuss thoroughly.
Let X1, X2, X3, and X4 be a random sample of observations from a population with...
Let X1, X2, X3, and X4 be a random sample of observations from a population with mean ? and variance ?2. Consider the following two point estimators of ?: b1= 0.30 X1 + 0.30 X2 + 0.30 X3 + 0.30 X4 and b2= 0.20 X1 + 0.40 X2 + 0.40 X3 + 0.20 X4 . Which of the following constraints is true? A. Var(b1)/Var(b2)=0.76 B. Var(b1)Var(b2) C. Var(b1)=Var(b2) D. Var(b1)>Var(b2)
Find a basis and the dimension of the subspace: V = {(x1, x2, x3, x4)| 2x1...
Find a basis and the dimension of the subspace: V = {(x1, x2, x3, x4)| 2x1 = x2 + x3, x2 − 2x4 = 0}
Using Y as the dependent variable and X1, X2, X3, X4 and X5 as the explanatory...
Using Y as the dependent variable and X1, X2, X3, X4 and X5 as the explanatory variables, formulate an econometric model for data that is (i) time series data (ii) cross-sectional data and (iii) panel data – (Hint: please specify the specific model here not its general form).
How many ordered quadruples (x1, x2, x3, x4) of positive integers satisfy the conditions x1 <...
How many ordered quadruples (x1, x2, x3, x4) of positive integers satisfy the conditions x1 < x2 < x3 < x4 < 15 and xi+1 − xi ≥ i for each i = 1, 2, 3.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT