Question

A machine produces bolts of which 10% are defective Using approximation; find the probability that in a random sample of 400 bolts produced by the machine between 30 and 50

Answer 1

To find the probability of getting between 30 and 50 defective bolts in a sample of 400 with a defect rate of 10%, we use the normal approximation to the binomial distribution, calculating the mean and standard deviation, converting to z-scores, and using the standard normal distribution to find the probability range.

Step-by-step explanation:

The student has asked to find the probability that in a random sample of 400 bolts produced by a machine, between 30 and 50 bolts are defective, given that on average 10% are defective. This situation can be approached using the normal approximation to the binomial distribution because the sample size is large (n=400) and the probability of success (p=0.10) is fixed. To use normal approximation, we also need to check if np(1-p) is greater than 10, which it is in this case (400*0.10*0.90 = 36).

First, we calculate the mean (μ) and the standard deviation (σ) for the binomial distribution:

1. Mean (μ) = np = 400 * 0.10 = 40
2. Standard Deviation (σ) = √(np(1-p)) = √(400 * 0.10 * 0.90) ≈ 6

To find the probability of getting between 30 and 50 defective bolts, we convert the numbers to z-scores and use the standard normal distribution:

1. For 30 defective bolts: z = (30-40)/6 ≈ -1.67
2. For 50 defective bolts: z = (50-40)/6 ≈ 1.67

We then look up these z-scores in the standard normal distribution table or use a calculator to find the probabilities associated with these z-scores and calculate the probability for the range between them.