Question

We have m = 5 preliminary samples of size n = 3 (some numbers have unfortunately been erased by accident by a clumsy co-op student): 24,2 1 27.1 29.4 2 30.6 32.5 3 25.7 35.5 4 31.1 23.2 5 24.1 34.2 11,3 27.9 32.4 31.83 1.9 30 30.4 25 26.43 7.9 27.4 28.57 10.1 total: 145.13 32 Si 1.3 1.07 4.91 5.15 16.57 What is the control chart (give the interval) for X from 5?

Answer 1

ANSWER:

Here we are given that m=5 and n= sample size= 3. Now we complete the following incomplete table:

i	xi,1	xi,2	xi,3	Ci	ri	si
1	27.1	29.4	27.2	27.9	2.2	1.3
2	30.6	32.5	32.4	31.83	1.9	1.07
3	25.7	35.5	30	30.4	5.5	4.91
4	31.1	23.2	25	26.43	7.9	4.14
5	24.1	34.2	27.4	28.57	10.1	5.15
			Total	145.13	32	16.57

Now we have to construct the control chart for
X
from $\overline{R}$ :

x_i,3:

3

27.9 * 3 = 27.1 + 29.4 + x_i,3

83.7 = 56.5 + x_i,3

x_i,3 = 83.7 - 56.5

x_i,3 = 27.2

r₁ = max - min = 29.4 - 27.2 = 2.2

r₃ = 35.5 - 30 = 5.5

s₄ = 16.57 - s₁ + s₂ + s₃ + s₅

₌ 16.57 - 12.43

s₄ = 4.14

x bars s.d chart formula:

CL = S = ΣS, k

= 16.57 / 5 = 3.314

$\small UCL =B_{4} \overline{S}$

B₄ = 2.089 (constant)

UCL = 2.089 * 3.314 = 6.922946

$\small LCL =B_{3} \overline{S}$

B₃ = 0

LCL =0

For $\small \overline{X}$ chart

$\small UCL=\overline{\overline{X}}+A_{3}\overline{S}$ (where A₃ constant)

$\small \overline{\overline{X}}=\frac{\sum \overline{X_{i}}}{5}$

$\small \overline{\overline{X}}=29.026$

A₃ = 1.427

UCL = 29.026 + 1.427 * 3.314

UCL = 33.755078

$\small LCL=\overline{\overline{X}}-A_{3}\overline{S}$

LCL = 29.026 - 1.427 * 3.314

LCL = 24.296922

Control chart for$\small \overline{X}$ from $\small \overline{S}$ is

(24.296922 , 33.755078)

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