A statistical process control chart example. Samples of 20 parts from a metal punching process are selected every hour. Typically, 1% of the parts require rework. Let X denote the number of parts in the sample of 20 that require rework. A process problem is expected if X exceeds its mean by more than three standard deviations. a)If the percentage of parts that require rework remains at 1%, what is the probability that X exceeds its mean by more than three standard deviations?
b)If the rework percentage increases to 4%, what is the probability that X exceeds 1?
c)If the rework percentage increases to 4%, what is the probability that X exceeds 1 in at least one of the next five hours of samples?

a) 0.0169

b) 0.19

c) 0.651                  

Step-by-step explanation:

Given:

Number of samples = 20

X denote the number of parts in the sample of 20 that require rework

A process problem is expected if X exceeds its mean by more than three standard deviations

Solution:

Let X be the random variable that denotes the number of parts that require rework

Then X follows a binomial distribution with parameters:

n = 20

p = 0.01

Probability mass function of X:

Pr(X=k) =

where k = 0, 1, 2, ..., 20

and

a)

Compute mean and standard deviation:

percentage of parts that require rework remains at 1% so p = 0.01

n = 20

Compute Mean:

E(X) = np = 20 * 0.01 = 0.2

Compute Standard Deviation:

σ = √np(1-p) = √20 * 0.01 ( 1 - 0.01 ) = √0.2 (0.99) = √0.198 = 0.444972

Compute upper bound:

U = E(X) + 3 * σ = 0.2 + 3 ( 0.444972 ) = 0.2 + 1.334916 = 1.534916 = 1.53

Compute probability of X exceeds its mean by more than three standard deviations.:

P(X> 1.53 ) = 1 - P(X = 0) - P(X = 1)  

P ( X = 0 ) =  

                = 20! / 0! ( 20 - 0 ) ! 1 ( 0.99 )²⁰

                = 1 * 1 * 0.99²⁰

                = 1 * 1 * 0.817907

                = 0.817907

P ( X = 1 ) =  

                =  20! / 1! ( 20 - 1 ) ! 0.01 (0.99 )¹⁹

                = 20 * 0.01 * 0.826169

                = 0.165234

P(X> 1.53 ) = 1 - P(X=0) - P(X=1)= 1 - 0.817907 - 0.165234 = 0.016859 = 0.0169

b)

Compute mean and standard deviation:

rework percentage increases to 4% so p = 0.04

Compute the probability that X exceeds 1?

P(X>1) = 1 - P(X=0) - P(X=1)

P (X=0) =  

            = 20! / 0! ( 20 - 0 ) ! 1 ( 0.96 )²⁰

            = 1 * 1 * 0.96²⁰

            = 0.442002

P (X=1) =  

                =  20! / 1! ( 20 - 1 ) ! 0.04 (0.96 )¹⁹

                = 20 * 0.04 * 0.460419

                = 0.368335

P(X>1) = 1 - P(X=0) - P(X=1) = 1 - 0.442002 - 0.368335 = 0.189663 = 0.19

c)

Let Y be a random variable which denotes the number of hours in which X exceeds 1.  

Since the probability that X exceeds 1 is constant in every hour, so compute p and n for Y

p =  P(X>1)

From part (b) P(X>1) = 0.19 So,

p =  P(X>1) = 0.19

Then Y follows a binomial distribution with parameters:

n = 5

p = 0.19

Probability mass function of Y:

Pr(Y=k) =

Pr (Y=k) = (1-p

where k = 0, 1, 2, ... ,5

and

Compute the probability that X exceeds 1 in at least one of the next five hours of samples:

P (Y≥1) = 1 - P (Y=0)

P(Y=0) =

           = 1 * 1 * 0.81⁵

           = 0.348678

P (Y≥1) = 1 - P (Y=0) = 1 - 0.348678 = 0.651322 =  0.651

answer: on kudzu, being female and being green are independent events.

step-by-step explanation: