A company produces pens which it packages into cartons of 500. The average number of defective pens per carton is 5.5. Defective pens are produced randomly and independently of each other.

a. If the random variable X, the number defective pens per carton, has a Poisson distribution, write down the value/s of its parameter/s.

b. Suppose a purchaser will stop buying from this company if a carton contains more than six defective pens. What is the probability a carton contains more than six defective pens?

c. In these challenging times the company cannot afford to lose any of its existing customers. You are responsible for quality control and are aware of the strict conditions imposed by the purchaser mentioned in part b. Would you be concerned? Explain.

d. The company implements some improvements in its processes and has managed to reduce the average number of defective pens per carton to 2.5 What is the probability of now randomly selecting 100 pens from a carton and finding at most two defective pens?

e. Verify your answers to parts b., and d. using the appropriate Excel statistical function and demonstrate you have done this by including the Excel formula used.