Statistical Approaches

❶ A quality control engineer is in charge of testing whether or not 95% of the Blu-ray disc players produced by his company conform to specifications. To do this, the engineer randomly selects a batch of 13 Blu-ray players from each day’s production. The day’s production is acceptable provided no more than 1 Blu-ray player fails to meet specifications. Otherwise, the entire day’s production has to be tested. (a) What is the probability that the engineer incorrectly passes a day’s production as acceptable if only 85% of the day’s Blu-ray players actually conform to specification? From the given information, we can know both problems are n Bernoulli Trials, where n  13 . Suppose p is probability that the engineer passes a day’s production as acceptable; and x is the quantity of defectives, then:

p = P( x  1) = P ( x  0)  P ( x  1) =

13! 13! (1  0.85)0 (0.85)130  (1  0.85)1 (0.85)131 0! (13  0)! 1! (13  1)!

= 0 . 1 2 +0 . 2 7 7 4 . 3 9 = 3 9 . 8 3 % 09 =0 8

Thus, the probability that the engineer incorrectly passes a day’s production as acceptable if only 85% of the day’s Blu-ray players actually conform to specification is 39.83%. (b) What is the probability that the engineer unnecessarily requires the entire day’s production to be tested if in fact 95% of the Blu-ray players conform to specifications? Suppose p is the probability that the engineer requires the entire day’s production to be tested, and

x is the quantity of defectives, then:

p  P( x  2)  1  P( x  1)  1  P( x  0)  P( x  1)  1 

13! 13! (1  0.95) 0 (0.95)130  (1  0.95)1 (0.95)131 0!(13  0)! 1!(13  1)!

 1 - 0.5133 - 0.3512  0.1355  13.55%

Thus, the probability that the engineer unnecessarily requires the entire day’s production to be tested if in fact 95% of the Blu-ray players conform to specifications is 13.55%. ❷ The reliability of an electrical fuse is the probability that a fuse, chosen at random from production, will function under its designed...