Question

1. The net weight (in oz.) of a dry bleach product is to be monitored by X and R and X and S control charts using a sample size of n = 5. Data for 20 preliminary samples are shown below.

Subgroup Number	X1	X2	X3	X4	X5
1	12	10	13	15	12
2	10	7	16	14	8
3	11	12	10	16	14
4	8	11	15	13	7
5	9	10	14	13	16
6	12	11	10	14	13
7	16	10	14	9	12
8	14	9	10	14	13
9	9	10	13	12	10
10	15	8	10	13	12
11	10	12	14	16	11
12	16	13	9	11	8
13	16	11	13	15	10
14	15	13	10	12	9
15	12	14	8	16	11
16	15	4	10	14	12
17	9	11	12	10	14
18	15	12	10	11	13
19	14	16	9	12	10
20	13	5	12	15	8

Determine the trial central line and control limits for the X and R.

UCLX=

CLX=

LCLX=

UCLR=

CLR=

LCLR=

Answer 1

Ans : From the data first we calculate and

Let X : Quality Variable under Study

k= random sample each of size n are available on variable X

Ri : Sample range of ith random sample

= max(xij) - min(Xij)  

  

Now we calculate using following table :

No.	X1	X2	X3	X4	X5	Ri	Xi-bar
1	12	10	13	15	12	5	12.4
2	10	7	16	14	8	9	11
3	11	12	10	16	14	6	12.6
4	8	11	15	13	7	8	10.8
5	9	10	14	13	16	7	12.4
6	12	11	10	14	13	4	12
7	16	10	14	9	12	7	12.2
8	14	9	10	14	13	5	12
9	9	10	13	12	10	4	10.8
10	15	8	10	13	12	7	11.6
11	10	12	14	16	11	6	12.6
12	16	13	9	11	8	8	11.4
13	16	11	13	15	10	6	13
14	15	13	10	12	9	6	11.8
15	12	14	8	16	11	8	12.2
16	15	4	10	14	12	11	11
17	9	11	12	10	14	5	11.2
18	15	12	10	11	13	5	12.2
19	14	16	9	12	10	7	12.2
20	13	5	12	15	8	10	10.6

= 134/20

= 6.7

= 236/20

= 11.8

Control limits for R - chart

Upper Control Limit

UCL_R = D4 *

where D4 is calculate from statistical constants for control chart for n=5

UCL_R = 2.155 * 6.7

UCL_R =14.1705

Central Line

CL_R =

CL_R = 6.7

Lower Control Limit

LCL_R = D3 *

= 0 * 67

LCL_R = 0

Control limits for - chart

Upper Control Limit

UCL = + A2 *

where A2 is calculate from statistical constants for control chart for n=5

UCL = 11.8 + 0.577*6.7

UCL =15.665

Central Line

CL =

CL= 11.8

Lower Control Limit

LCL = - A2 *

= 11.8 -0.577*6.7

LCL = 7.9341

Final Answer :

central line and control limits for the and R

UCL =15.665

CL= 11.8

LCL = 7.9341

R chart :

UCL_R =14.1705

CL_R = 6.7

LCL_R = 0