A small electronic device is designed to emit a timing signal of 200 m

Question

A small electronic device is designed to emit a timing signal of 200 milliseconds (ms) duration. In the production of this device, 10 subgroups of four units are taken at periodic intervals and tested. The results are shown in the following table:

Subgroup Number	Duration of Automatic Signal (in ms)
	a	b	c	d
1	195	201	194	201
2	204	190	199	195
3	195	197	205	201
4	211	198	193	180
5	204	193	197	200
6	200	202	195	200
7	196	198	197	196
8	201	197	206	207
9	200	202	204	192
10	203	201	209	192

i) Estimate the process mean and standard deviation.
ii) Determine the centre line and control limits for the process mean and process variability.
iii) By plotting the charts on graph paper, determine that the process is stable or not with respect to the process mean and process variability. If necessary, compute revised control limits

14 Feb 2023

Answer :

Word Count : 1946

i) To estimate the process mean and standard deviation, we first need to calculate the sample means and ranges:

Subgroup Number Sample Mean Sample Range

Subgroup	Mean	Range
1	197.75	7
2	197.0	14
3	199.5	11
4	195.5	31
5	198.5	11
6	199.25	7
7	196.75	5
8	202.75	9
9	199.5	12
10	201.25	17

The sample mean of the subgroups is the estimate of the process mean, which is:

x̄ = (197.75 + 197.0 + 199.5 + 195.5 + 198.5 + 199.25 + 196.75 + 202.75 + 199.5 + 201.25) / 10 = 198.875

To estimate the process standard deviation, we use the average of the subgroup ranges. The estimate is:

R̄ = (7 + 14 + 11 + 31 + 11 + 7 + 2 + 9 + 12 + 17) / 10 = 11.1
s = R̄ / d2 = 11.1 / 2.059 = 5.38

ii) To determine the center line and control limits for the process mean and process variability, we use the following formulas:

Center line for X̄ chart: x̄
Upper control limit for X̄ chart: x̄ + A2s/sqrt(n)
Lower control limit for X̄ chart: x̄ - A2s/sqrt(n)

Center line for R chart: R̄
Upper control limit for R chart: D4R̄
Lower control limit for R chart: D3R̄

where n is the subgroup size, A2, D3, and D4 are constants from the control chart factors.

From the control chart factors table, with n=4, A2=0.729, D3=0, and D4=2.282.

Plugging in the values, we get:

Center line for X̄ chart: 198.875
Upper control limit for X̄ chart: 198.875 + 0.7295.38/sqrt(4) = 201.136
Lower control limit for X̄ chart: 198.875 - 0.7295.38/sqrt(4) = 196.614

Center line for R chart: 11.1
Upper control limit for R chart: 2.282*11.1 = 25.335
Lower control limit for R chart: 0

iii) To plot the charts and determine if the process is stable, we can plot the sample means on the X̄ chart and the sample ranges on the R chart. The charts are shown below:

X̄ chart:

The overall mean $ar{X}$ is calculated as:

$ar{X} = frac{1}{n} sumlimits_{i=1}^{n} ar{X}i = frac{1}{10} sumlimits{i=1}^{10} ar{X}_i = frac{1}{10} imes (197.75 + 197.00 + cdots + 201.25) = 198.30$

To calculate the standard deviation, we first need to calculate the average range $ar{R}$, which is given by:

$ar{R} = frac{1}{n} sumlimits_{i=1}^{n} R_i = frac{1}{10} sumlimits_{i=1}^{10} R_i = frac{1}{10} imes (7.00 + 14.00 + cdots + 17.00) = 12.70$

Next, we calculate the control limits for the $ar{X}$ chart using the formula:

$UCL = ar{X} + A_2 frac{ar{R}}{sqrt{n}}$

$LCL = ar{X} - A_2 frac{ar{R}}{sqrt{n}}$

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