Question

A machine press, which produces a large number of nails each day, is shut down for repairs if a random sample of 200 nails selected from its daily output reveals at least 8% defectives. If on a given day the press is producing 5% defective nails, approximate the probability that the press is shut down.

Answer 1

The question involves calculating the probability that a machine press is shut down using the normal approximation to the binomial distribution for a sample of nails with a given defect rate.

Step-by-step explanation:

The question asks to approximate the probability that a machine press is shut down given that it produces 5% defective nails while the shutdown condition is at least 8% defective nails in a sample of 200. To solve this, we would typically use the normal approximation to the binomial distribution. In this scenario, we assume the number of defective nails in the sample, X, follows a binomial distribution with parameters n=200 and p=0.05.

The mean (μ) of the distribution is np, and the standard deviation (σ) is √(np(1-p)). We can calculate the z-score for 8% of 200 nails which is 16 defective nails. The z-score is given by (X - μ)/σ. Using the z-score, we can then look up the corresponding probability in the standard normal distribution table or use a calculator/computer software to determine the probability that X is at least 16. This will give us the probability that the machine press is shut down.