grade 12 data management

Question

1. Avery and Bradley work at a large electronics manufacturer that produces DVD players. The defective rate on the assembly line has gone up to 12% and the manager wants to know the probability that a skid of 50 DVD players will contain at least 3 defective units.

a)     Help Avery use the binomial distribution  to answer this question

b)     Help Bradley use the normal approximation to answer this question.                                      

Answer 1

(a) Using the binomial distribution:

The question asks specifically for the probability of at least 3 in 50 defects. So we must take the whole probability and remove the cases that do not fit that. Then, we have that our answer must be 1-p₀-p₁-p₂, or the total probability less the probability or 0, 1, or 2 defects.

Then,

p₀= (0 choose 50)*(0.12)^0*(0.88)^50=0.001675

p₁= (1 choose 50)*(0.12)^1*(0.88)^49=0.011424

p₂= (2 choose 50)*(0.12)^2*(0.88)^48=0.038165

After we subtract these values we are left with 94.87%.

(b) To apply the normal distribution. We must find the mean and variance of the sample set first.

mean=np=(50)(0.12)=6

variance=np(1-p)=50*0.12*0.88=5.28

Then, we have Pr(X<=3)=Pr(Z<= (3-6)/√5.28)=Pr(Z<=-1.31)= 0.0951.

Then, the probability that X>=3 is 1-0.0951=0.9049.