Question

1. An automatic CNC machine produces bolts which are stamped out of a length of metal. Various faults may generally occur during the production process, such as, the heads or the threads may be incorrectly formed, the length might be incorrect, and so on.

Assuming that 4 bolts out of every 100 produced are defective in some way. If a sample of 50 bolts is drawn at random, then the manufacturer might be satisfied that his defect rate is still 4% provided that there are no more than 2 defective bolts in the sample. The manufacturer’s quality manager has formulated the hypotheses: H₀ : p=0.04 The manufacturer has also made a decision that should the defect rate rise to 6% or more, he will take some action to protect the companies good name.

Using the binomial distribution determine the probabilities of getting 0,1,2 or 3 defective bolts in the sample and interpret your statistical hypothesis test results (refer to the error types) in the provision of justified advice for the manufacturer

Answer 1

We have:

n = 50

p=0.04

r = 0, 1, 2 or 3

For r = 0,

P(r = 0) = ⁵⁰C₀(0.04)⁰(1 - 0.04)^{50 - 0} = 0.1299

For r = 1,

P(r = 1) = ⁵⁰C₁(0.04)¹(1 - 0.04)^{50 - 1} = 0.2706

For r = 2,

P(r = 2) = ⁵⁰C₂(0.04)²(1 - 0.04)^{50 - 2} = 0.2762

For r = 3,

P(r = 3) = ⁵⁰C₃(0.04)³(1 - 0.04)^{50 - 3} = 0.1842

Since all the probabilities of defect rates are greater than 6%, the manufacturer will take some action to protect the companies good name.