Question

An xbar chart on a normally distributed quality characteristic is to be established with the standard values μ = 100. σ = 8, and n = 4.

Find the following: (a) The two-sigma control limits (b) The 0.005 probability limits.

Answer 1

SOLUTION:

From given data,

An x bar chart on a normally distributed quality characteristic is to be established with the standard values μ = 100. σ = 8, and n = 4.

Where,

Mean = μ = 100.

Standard deviation = σ = 8,

Sample size = n = 4.

(a) The two-sigma control limits

2 control limits are,

μ - 2< X < μ + 2

(μ - 2 , μ + 2​​​​​​​ )

By substituting the values,

(100- 2*8 , 100+ 2*8​​​​​​​ )

(100- 16 , 100+16​​​​​​​ )

(84, 116​​​​​​​ )

The  two-sigma control limits are (84, 116​​​​​​​ )

(b) The 0.005 probability limits.

Critical value = Z_0.005 = 2.5758

P(μ - Z_0.005 *(σ / sqrt(n)) < X < μ + Z_0.005 *(σ / sqrt(n)))

(μ - Z_0.005 *(σ / sqrt(n)) , μ + Z_0.005 *(σ / sqrt(n)))

By substituting all values

(100 - 2.5758 *(8/ sqrt(4)) , 100 + 2.5758 *(8 / sqrt(4)))

(100 - 2.5758​​​​​​​ *(8/ 2) , 100 + 2.5758 *(8 / 2))

(100 - 2.5758​​​​​​​ *4, 100 + 2.5758 *4)

(100 - 10.3032, 100 + 10.3032)

(89.6968, 110.3032)

The 0.005 probability limits are  (89.6968, 110.3032)