Question

A process manufactures 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.

a. 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)

probability that a randomly selected resistance does not meet the specifications

=1-P(0.8<X<1.2)=1-P((0.8-1.03)/0.08)<Z<(1.2-1.03)/0.08)=1-P(-2.88<Z<2.13)=1-(0.9834-0.002)=0.0186

b)

probability that there will be at least 1 out of specification =1-P(none out of spec)

=1-(1-0.0186)^10

=0.1712

c)

std error=σx̅=σ/√n=0.08/√16 =

0.0200

probability that the average of the 16 resistors is less than 1.05 kΩ :

probability =P(X<1.05)=(Z<(1.05-1.03)/0.02)=P(Z<1)=0.8413

d)

since 99.73 % values are within 3 standard deviaiton ; therefore standard deviation =0.2/3= 0.061

(required decrease =0.019)