A manufacturing company produces parts of which 97% are withinspecifications and

Question

A manufacturing company produces parts of which 97% are withinspecifications and 3% are defective (outside specifications). Thereis apparently no pattern to the production of defective parts;thus, we assume that whether or not a part is defective isindependent of other parts.

a) What is the probability that there will be no defective parts ina box of five?

b) What is the probability that there will be exactly two defectiveparts in a box of five?

c) What is the probability that there will be two or more defectiveparts in a box of five?

d) Use the Poisson distribution to approximate the probability thatthere will be 4 or more defective
     parts in a box of 40.

e) Use the normal distribution to approximate the probability thatthere will be 20 or more defective parts in a box of 400.

Explanation / Answer

A manufacturingcompany produces parts of which 97% are within specifications and3% are defective (outside specifications). There is apparently nopattern to the production of defective parts; thus, we assume thatwhether or not a part is defective is independent of otherparts. a) What is theprobability that there will be no defective parts in a box offive? = Binomial{N = 5, k = 0, p = 0.03} = (1 -0.03)^5 = 0.8587 b) What is theprobability that there will be exactly two defective parts in a boxof five? = Binomial{N= 5, k = 2, p = 0.03} = C(5,2)*((0.03)^2)*(1 -0.03)^3 = 0.008214 c) What is theprobability that there will be two or more defective parts in a boxof five? = Binomial{N = 5, k > 1, p = 0.03} = 0.008472 d) Use thePoisson distribution to approximate the probability that there willbe 4 or more defective      parts in a box of40. = Poisson{k > 3, =(0.03)*(40) = 1.2} = 0.0314 e) Use the normaldistribution to approximate the probability that there will be 20or more defective parts in a box of 400.= Normal{X> 19, = (0.03)*(400) = 12, =((400)*(0.03)*(0.97)) = 3.4} = 0.0196 .

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