Question

A process manufacturing resistors whose specifications are 1 kΩ ± 0.2 kΩ. The resistance of these parts is known to follow a normal distribution with a mean of 1.03 kΩ and a standard deviation of 0.08 kΩ. to.

What is the probability that a randomly selected resistance does not meet the specifications?

b. If 10 resistors are selected at random, what is the probability that there will be at least 1 out of specification?

c. If 16 resistors are selected at random, what is the probability that the average of the 16 resistors is less than 1.05 kΩ?

d. If 99.73% is desired to be within specifications, by how much will the standard deviation need to be reduced?

Answer 1

a)

µ =    1.03              
σ =    0.08              
P (   0.8   < X <   1.2   )  
=P( (0.8-1.03)/0.08 < (X-µ)/σ < (1.2-1.03)/0.08 )                  
                  
P (    -2.88   < Z <    2.13   )   
= P(Z<2.13) - P(Z<-2.88) =        0.9832-0.002=      0.9812  

probability that a randomly selected resistance does not meet the specifications = 1-0.9812 = 0.0188 (answer)

b)

P ( X =    0   ) = C(16,0) * 0.0188^0 * (1-0.0188)^16 =        0.7379

P(at least 1) = 1 - P(X=0) = 0.2621

c)

µ =    1.03      
σ =    0.08      
n=   16      
          
X =   1.05      
Z =(X - µ )/(σ/√n) =    (1.05-1.03)/(0.08/√16)=   1.00  
P(X<1.05) = P(Z<1) =        0.8413   (answer)

d) std dev should be reduced to 0.061