The probability of at least 11 defects on a 24-square-foot metal sheet, modeled by a Poisson distribution with λ = 15.43, is approximately 0.1434, calculated using the cumulative probability formula.
To find the probability that a 24 square foot metal sheet has at least 11 defects, we can use the Poisson distribution. The average number of defects per 14 square feet is 9 defects, so the average number of defects per 1 square foot is 9/14.
We can use this information to find the parameter λ for the Poisson distribution, which represents the average number of defects per unit area:
λ = (9/14) * 24 = 15.43
Now we can calculate the probability of at least 11 defects using the Poisson probability formula:
P(X ≥ 11) = 1 - P(X ≤ 10)
We can use a calculator or software to find the Poisson cumulative probability for X ≤ 10 with parameter λ = 15.43. Let's round our answer to four decimals:
P(X ≥ 11) ≈ 1 - 0.8566 = 0.1434