Question

West Battery Corp. has recently been receiving complaints from retailers that its​ 9-volt batteries are not lasting as long as other name brands. James​ West, head of the TQM program at​ West's Austin​ plant, believes there is no problem because his batteries have had an average life of

5050

​hours, about​ 10% longer than​ competitors' models. To raise the lifetime above this level would require a new level of technology not available to West. ​ Nevertheless, he is concerned enough to set up hourly assembly line checks.​ Previously, after ensuring that the process was running​ properly, West took samples of

55

​9-volt batteries for 25 test to establish the standards for control chart limits. Those 25 tests are shown in the following​ table:

                                                                                                                                                                                

	Sample Data								Sample Data
Hour Sample Taken	1	2	3	4	5	x overbarx	R	Hour Sample Taken	1	2	3	4	5	x overbarx	R
1	51	50	49	50	50	50.0	2	14	51	58	41	51	46	49.4	17
2	46	46	71	47	35	49.0	36	15	56	41	51	59	59	53.2	18
3	50	35	48	39	47	43.8	15	16	62	48	41	32	54	47.4	30
4	54	56	52	50	52	52.8	6	17	39	40	48	59	48	46.8	20
5	48	37	77	37	48	49.4	40	18	65	49	42	57	50	52.6	23
6	58	63	36	53	67	55.4	31	19	57	53	53	47	50	52.0	10
7	37	33	49	48	57	44.8	24	20	59	49	42	42	49	48.2	17
8	51	67	54	43	41	51.2	26	21	53	46	47	59	39	48.8	20
9	43	53	45	48	43	46.4	10	22	55	41	56	49	46	49.4	15
10	69	45	51	47	42	50.8	27	23	46	48	50	49	48	48.2	4
11	57	53	63	45	37	51.0	26	24	51	51	49	52	51	50.8	3
12	55	54	47	41	63	52.0	22	25	51	49	51	52	63	53.2	14
13	41	69	58	46	45	51.8	28

This exercise contains only parts a and b.

​a) For the given​ data, the

x double overbarx

​=

nothing

hours ​(round your response to three decimal​ places).

For the given​ data, mean range​ =

nothing

hours ​(round your response to two decimal​ places).

With

​z=3​,

the control limits for the mean chart​ are:

                                               

UCL Subscript x overbarUCLx

​=

nothing

hours ​(round your response to three decimal​ places).

                                               

LCL Subscript x overbarLCLx

​=

nothing

hours ​(round your response to three decimal​ places).

The control limits for the range chart​ are:

                                               

UCL Subscript Upper RUCLR

​=

nothing

hours ​(round your response to two decimal​ places).

                                               

LCL Subscript Upper RLCLR

​=

nothing

hours ​(round your response to two decimal​ places).

With these limits​ established, West now takes 5 more hours of​ data, which are shown in the following table. Calculate the mean and range for each hour that the sample data is taken. ​(Enter your responses for the mean to one decimal place and enter your responses for the range as whole​ numbers.)

                                                                                                                   

	Sample Data
Hour Sample Taken	1	2	3	4	5	x overbarx	R
26	47	52	40	57	62	nothing	nothing
27	45	53	47	45	65	nothing	nothing
28	63	50	50	45	53	nothing	nothing
29	57	71	46	52	62	nothing	nothing
30	46	39	47	55	53	nothing	nothing

​b) Do the samples for hours 26 through 30 indicate that the process is in​ control?

Based on the

x overbarx​-chart,

the process has been

▼

IN CONTROL

OUT OF CONTROL

.

Based on the​ R-chart, the process has been

▼

OUT OF CONTROL

IN CONTROL

.

Answer 1

x overbarx	R
50	2
49	36
43.8	15
52.8	6
49.4	40
55.4	31
44.8	24
51.2	26
46.4	10
50.8	27
51	26
52	22
51.8	28
49.4	17
53.2	18
47.4	30
46.8	20
52.6	23
52	10
48.2	17
48.8	20
49.4	15
48.2	4
50.8	3
53.2	14
49.936	19.360
X-bar-bar (average of Xbars)	R-bar (Average of Range)

Answer a: Xbarbar= 49.936

Rbar= 19.360

D4, D3 and A2 are taken from table of factors computing 3 sigma control limits, sample size=5

Fore Xbar control limits

A2 (n=5)			0.577
control limits for X bar,
CL or Xbarbar			49.936
UCL=Xbarbar+ (A2)*Rbar			61.107
LCL=Xbarbar- (A2)*Rbar			38.765

Fore Range control limits

D4 (n=5)=			2.115
D3 (n=5)=			0
control limits for Range,
CL or R-bar=			19.360
UCL=R-bar*D4			40.946
LCL=R-bar*D3			0.000

Answer b:

1	2	3	4	5	R= Max- Min	X-bar= average of sample
47	52	40	57	62	22.00	51.60
45	53	47	45	65	20.00	51.00
63	50	50	45	53	18.00	52.20
57	71	46	52	62	25.00	57.60
46	39	47	55	53	16.00	48.00

All Xbar values are within range of 61.107 and 38.765. the process has been in control

All Range values are within range of 40.946 and 0. the process has been in control

Please ask, if you have any doubts through the comment section. Do rate the answer