Final answer:
The probability of finding exactly two defective bulbs in a sample of 15 can be calculated using the binomial probability formula, and the mean and standard deviation for 1,000 bulbs can be found using the normal approximation to the binomial. For the insurance claim, the Z-score is used to determine the probability of a claim being greater than $10,000.
Step-by-step explanation:
For part (a), we need to use the binomial probability formula to determine the probability that exactly two bulbs are defective out of 15:
The formula is P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where:
- 'n' is the number of trials (15 bulbs)
- 'k' is the number of successes (2 defective bulbs)
- 'p' is the probability of a single success (0.02 or 2% chance of a bulb being defective)
For part (b), since we have a large sample size, we can use the normal approximation to the binomial distribution:
The mean (μ) of the number of defective bulbs is np, and the standard deviation (σ) is the square root of np(1-p), where 'n' is 1000 and 'p' is 0.02.
The insurance claim question involves using the properties of the normal distribution:
To find the probability that a claim is greater than $10,000, we need to find the Z-score and use a normal distribution table or calculator. The Z-score is calculated by (X - μ) / σ where X is $10,000, μ is $5,950 and σ is $1,750.