Quality: Normal Distribution and Corresponding Probability Measures
IE423 PROBLEM SESSION III SOLUTIONS
Oct 19, 2012 1. According to Western Electric sensitizing rules for Shewhart Control Charts, the action signals are also defined according to the following rules:
i) Two out of three consecutive points fall beyond the 2σ limit on the same side of the centerline. ii) Four out of five consecutive points fall beyond the 1σ limit, on the same side of the centerline. iii) Eight consecutive points falling on one side of the centerline
When the process is in statistical control, find the false alarm probability (Type-I error) for each case. The corresponding probability measures are obtained from the Normal table as
P(3 " Z) = 0.00135 P(2 " Z) = 0.02275 P(1 " Z) = 0.1587
Solution:
!
i) Use the Binomial distribution to ! calculate the probability measures.
!
3!
3!
P(Y ! 2 n = 3, p = 0.02275) =
(0.02275)2 (1" 0.02275) +
(0.02275)3 = 0.00153
2!1!
3!0!
Type-1 Risk considering both sides: ! = 0.00306 ii) 5!
5!
(0.1587)4 (1" 0.1587) +
(0.1587)5 = 0.00277
4!1!
5!0!
Type-1 Risk considering both sides: ! = 0.00554
P(Y ! 4 n = 5, p = 0.1587) =
iii)
8!
(0.5)8 = 0.00391
8!0!
Type-1 Risk considering both sides: ! = 0.00782
P(Y = 8 n = 8, p = 0.5) =
2. There are four parallel machines that manufacture the same product with a certain quality
2
characteristic, X ~ Normal( µ j ," j ) , j=1,2,3,4. Each machine produces 100 products per hour, and these products are transported to a storage area and labeled with a lot number
(Date/Hour). Note that each lot is composed of 400 units.
The managers are thinking of implementing an SPC methodology on this shop floor. They are
!
considering several potential methods of subgrouping the data to be used in control charts.
i) Every hour, select one sample from each machine to form the subgroup of four. ii) Every hour, select four samples from the blended stream (each lot) in the storage
Oct 19, 2012 1. According to Western Electric sensitizing rules for Shewhart Control Charts, the action signals are also defined according to the following rules:
i) Two out of three consecutive points fall beyond the 2σ limit on the same side of the centerline. ii) Four out of five consecutive points fall beyond the 1σ limit, on the same side of the centerline. iii) Eight consecutive points falling on one side of the centerline
When the process is in statistical control, find the false alarm probability (Type-I error) for each case. The corresponding probability measures are obtained from the Normal table as
P(3 " Z) = 0.00135 P(2 " Z) = 0.02275 P(1 " Z) = 0.1587
Solution:
!
i) Use the Binomial distribution to ! calculate the probability measures.
!
3!
3!
P(Y ! 2 n = 3, p = 0.02275) =
(0.02275)2 (1" 0.02275) +
(0.02275)3 = 0.00153
2!1!
3!0!
Type-1 Risk considering both sides: ! = 0.00306 ii) 5!
5!
(0.1587)4 (1" 0.1587) +
(0.1587)5 = 0.00277
4!1!
5!0!
Type-1 Risk considering both sides: ! = 0.00554
P(Y ! 4 n = 5, p = 0.1587) =
iii)
8!
(0.5)8 = 0.00391
8!0!
Type-1 Risk considering both sides: ! = 0.00782
P(Y = 8 n = 8, p = 0.5) =
2. There are four parallel machines that manufacture the same product with a certain quality
2
characteristic, X ~ Normal( µ j ," j ) , j=1,2,3,4. Each machine produces 100 products per hour, and these products are transported to a storage area and labeled with a lot number
(Date/Hour). Note that each lot is composed of 400 units.
The managers are thinking of implementing an SPC methodology on this shop floor. They are
!
considering several potential methods of subgrouping the data to be used in control charts.
i) Every hour, select one sample from each machine to form the subgroup of four. ii) Every hour, select four samples from the blended stream (each lot) in the storage