Binomial distribution

Question

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

P(defective) = .12. We will just call this p.

P(not defective) = .88

X = number defective in a batch of 50

a. Avery

The probability is most easily found with the appropriate distribution function on a graphing calculator.

In a batch of 50, P(X ≥ 3) = 1 - P(X ≤ 2) = 1 - binomcdf(50,.12,2) = .949

b. Bradley

To use the normal approximation, we would first find the mean and the standard deviation for the number of defective units in a batch of 50. The mean is μ = np = 50(.12) = 6.

The standard deviation is σ = √np(1-p) = √50(.12)(.88) = 2.30.

In the normal approximation, the discrete 2's, are represented by the continuous range from 1.5 to 2.5

Putting that together, we want the area under the normal distribution curve from 0 to 2.5.

The z-score for 2.5 is (2.5 - 6)/ 2.3 = -1.52

Finding this z-score in a table of the normal distribution, or using the normdcf function on the graphing calculator, we find the P(X ≤ 2) = normcdf (-999,-1.52,0,1) = .064

Finally, P(X ≥ 3) = 1 - P(X ≤ 2) = 1 - .064 = .936