Question

1) A process for manufacturing an electronic component yields items of which 1% are defective. A quality control plan is to select 100 items from the process, and if none are defective, the process continues. Use the normal approximation to the binomial to find (a) the probability that the process continues given the sampling plan described; (b) the probability that the process continues even if the process has gone bad (i.e., if the frequency of defective components has shifted to 5.0% defective)

2) Researchers at George Washington University and the National Institutes of Health claim that approximately 75% of people believe “tranquilizers work very well to make a person more calm and relaxed.” Of the next 80 people interviewed, what is the probability that (a) at least 50 are of this opinion? (b) at most 56 are of this opinion?

Answer 1

Answer:

1.

Given,

p = 1%

= 0.01

n = 100

mean = np

= 100*0.01

= 1

standard deviation = sqrt(npq)

= sqrt(100*0.01*0.99)

= 0.995

a)

To give the required probability that process continues given the sampling plan described

P(process continues given the sampling plan described) = P(((x+0.5) - mu) / s <= ((0+0.5) - 1) / 0.995)

= P(z <= (-0.5/0.995))

= P(z <= - 0.50)

= 0.3085 [since from z table]

P(process continues given the sampling plan described) = 0.3085

b)

Here it is given ,

p = 5%

= 0.05

q = 1 - 0.05

= 0.95

mean = np

= 100*0.05

= 5

standard deviation = sqrt(npq)

= sqrt(100*0.05*0.95)

= 2.18

Now consider,

P(process continues even if the process has gone bad) = P(((x+0.5)-mu)/s <= ((0+0.5) - 5)/2.18)

= P(z <= -2.06)

= 0.0197 [since from z table]

P(process continues even if the process has gone bad) = 0.0197