A robot that shapes metal needs overhauling if it is out of tolerance on 4.5% of

Question

A robot that shapes metal needs overhauling if it is out of tolerance on 4.5% of the items processed, and it is operating satisfactorily if it is off on only 0.8% of its output. A test is performed involving 50 sample items. If the sample proportion of out-of-tolerance items is greater than 0.02, the robot will be overhauled. Otherwise it will be allowed to continue operating. (a) What is the probability that a satisfactory robot will be overhauled unnecessarily? (b) What is the probability that a robot in need of overhauling will be left in operation?

Explanation / Answer

a) Given the robot is operating satisfactorily, probability of being off on its output is p = 0.8%= 0.008

Now, robot would be overhauled if sample proportion of off items is greater than 0.02, meaning out of 50 items, if number of off items is greater than 0.02*50 = 1.

So, we have to find probability of getting off items greater than 1 out of total of 50 items. This becomes a binomial case, with N=50, p=0.008, k=1, and we have to find P(x>1).

P(x>1)
= 1 - p(x=0) - p(x=1)
= 1 - 1*(0.008)^0* (1-0.008)^50 - 50*(0.008)^1*(1-0.008)^49
= 0.0609014

b) Given robot is in need of overhauling, probability of being off on its output is p = 4.5% = 0.045

Robot would be left in operation if number of off items is less than or equal to 1.

Again, this is a binomial case with p = 0.045, N=50, k=1, and we have to find P(x1)

P(x1)
= P(x=0) + P(x=1)
= 1*(0.045)^0*(1-0.045)^50 + 50*(0.045)^1*(1-0.045)^49
= 0.3357324

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.