Question

Acceptance sampling is an important quality control technique, where a batch of data is tested to determine if the proportion of units having a particular attribute exceeds a given percentage. Suppose that 8% of produced items are known to be nonconforming. Every week a batch of items is evaluated and the production machines are adjusted if the proportion of nonconforming items exceeds 10%. [You may find it useful to reference the z table.]

a. What is the probability that the production machines will be adjusted if the batch consists of 65 items? (Round “z” value to 2 decimal places, and final answer to 4 decimal places.)

Probability _________

b. What is the probability that the production machines will be adjusted if the batch consists of 77 items? (Round “z” value to 2 decimal places, and final answer to 4 decimal places.)

probability ________

Answer 1

a)

population proportion ,p=   0.08                      
n=   65                      
                          
std error , SE = √( p(1-p)/n ) =    0.0336                      
                          
sample proportion , p̂ =   0.1                      
Z=( p̂ - p )/SE= (   0.1   -   0.08   ) /    0.0336   =   0.59
P ( p̂ >    0.1   ) =P(Z > ( p̂ - p )/SE) =                  
                          
=P(Z >   0.59 ) =    0.2776

b)

population proportion ,p=   0.08                      
n=   77                      
                          
std error , SE = √( p(1-p)/n ) =    0.0309                      
                          
sample proportion , p̂ =   0.1                      
Z=( p̂ - p )/SE= (   0.1   -   0.08   ) /    0.0309   =   0.65
P ( p̂ >    0.1   ) =P(Z > ( p̂ - p )/SE) =                  
                          
=P(Z >   0.65 ) =    0.2578