A laptop manufacturing company has implemented a 2-step process to test the quality of each production batch. In the first step, a technician randomly selects 15 laptops from the batch and determines whether they meet specifications. The batch is considered acceptable provided no more than 1 laptop fails to meet specifications. Otherwise, the entire batch must be tested in the second step. Historical data shows that 95% of the laptops produced adhere to specifications. Discuss the following:

• What are the 4 characteristics of a binomial experiment?
• Can we use a binomial distribution to model this process?
• What is the probability that the entire batch unnecessarily has to be tested if in fact 95% of its laptops conform to specifications? (Hint: Use Excel's =BINOMDIST() function to find the probability.)
• What is the probability that the batch is incorrectly accepted if only 75% of its laptops conform to specifications?

Answer:

(a) The 4 characteristics of a binomial experiment are,

·       The number of trials is constant

·       Each trial is independent of the other

·       The outcome of every trial can be categorized into either, "success" or "failure".

·       The probability of success is equal in every trial.

(b) As 15 laptops are selected so number of trials(selecting a laptop) is constant, the laptops can be categorized as defective (success) or working (failure), each laptop's condition is independent of other laptops, the probability of getting a defective laptop is same for all laptops. Arguments are implying that all 4 conditions for Binomial distribution are satisfied and the considered case can be taken as a binomial distribution model.

(c) The entire batch unnecessarily has to be tested has to be tested if more than 1 laptop fails to meet specifications. The required probability would be, P(X > 1) which can be found using excel command "1-BINOMDIST(1,15,0.05,TRUE)". The resulted probability is 0.17095.

(d) The batch would be incorrectly accepted if number of defectives is less than or equal to 1 given he defective rate is 0.25. The required probability would be, P(X <= 1) which can be found using excel command "BINOMDIST(1,15,0.25,TRUE)". The resulted probability is 0.08018.